International Journal of Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document defines integration as the inverse operation of differentiation or the antiderivative. Integration finds the function given its derivative, while differentiation finds the derivative of a function. The key points are:
1) Integration is denoted by the integral sign ∫ and finds the antiderivative F(x) of a function f(x) plus a constant c.
2) Some basic integration rules and theorems are presented, including formulas for integrating polynomials and trigonometric functions.
3) The substitution rule is described for performing integral substitutions to solve integrals that can't be solved with basic formulas. Examples of integrating trigonometric functions and expressions involving square roots are provided.
To find the complete solution to the second order PDE
(i.e) To find the Complementary Function & Particular Integral for a second order (Higher Order) PDE
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document defines integration as the inverse operation of differentiation or the antiderivative. Integration finds the function given its derivative, while differentiation finds the derivative of a function. The key points are:
1) Integration is denoted by the integral sign ∫ and finds the antiderivative F(x) of a function f(x) plus a constant c.
2) Some basic integration rules and theorems are presented, including formulas for integrating polynomials and trigonometric functions.
3) The substitution rule is described for performing integral substitutions to solve integrals that can't be solved with basic formulas. Examples of integrating trigonometric functions and expressions involving square roots are provided.
To find the complete solution to the second order PDE
(i.e) To find the Complementary Function & Particular Integral for a second order (Higher Order) PDE
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
A semi analytic method for solving two-dimensional fractional dispersion equa...Alexander Decker
This document presents a semi-analytic method called the modified decomposition method for solving two-dimensional fractional dispersion equations. The method is applied to solve a two-dimensional fractional dispersion equation subject to initial and boundary conditions. The numerical results obtained from the modified decomposition method are shown to closely match the exact solution, demonstrating the accuracy of this approach. The method provides an efficient means of obtaining analytical solutions to fractional differential equations.
The study is concerned with a different perspective which the numerical solution of the singularly
perturbed nonlinear boundary value problem with integral boundary condition using finite difference method on
Bakhvalov mesh. So, we show some properties of the exact solution. We establish uniformly convergent finite
difference scheme on Bakhvalov mesh. The error analysis for the difference scheme is performed. The numerical
experiment implies that the method is the first order convergent in the discrete maximum norm, independently of
휀- singular perturbation parameter with effective and efficient iterative algorithm. The numerical results are
shown in table and graphs.
This document proposes a modified Newton's method for solving nonlinear equations that uses harmonic mean. It begins by reviewing Newton's method and some existing variants that use arithmetic mean or other integration rules to modify Newton's method and achieve cubic convergence without using second derivatives. It then presents the new Harmonic-Simpson-Newton method, which replaces the arithmetic mean in an existing Simpson Newton's method with harmonic mean. The method is proven to have cubic convergence. Numerical examples are provided to compare the efficiency of the new method to other cubic convergent methods.
This document discusses partial differential equations (PDEs). It begins by defining PDEs as equations that involve partial derivatives with respect to two or more independent variables. Next, it provides examples of how PDEs can be formed and classified based on characteristics like order, degree, whether they are linear or nonlinear. Then, it discusses methods for solving common types of PDEs like linear PDEs. Finally, it derives the one-dimensional wave equation and shows its solution as a product of functions involving the independent variables x and t.
A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equat...IOSR Journals
This document summarizes a two grid discretization method for decoupling time-harmonic Maxwell's equations. The method discretizes the coupled partial differential equations using continuous 1H-conforming finite elements on two grids - a fine grid and a coarser grid. This allows solving the coupled equations on the coarse grid along with decoupled equations on the fine grid, reducing computational costs compared to solving on just the fine grid. The document reviews relevant fundamentals, formulates the variational problem, and proves a regularity result stating the solution has higher regularity if the right hand side is more regular.
Calculus is the study of change and is divided into differential and integral calculus. Differential calculus studies rates of change using derivatives, while integral calculus uses integration to find accumulated change. These concepts build on limits and algebra/geometry. Leibniz developed the notation and principles of calculus in the 1670s. Differential calculus uses derivatives to determine how quantities change, and integral calculus uses integrals and antiderivatives to determine quantities from rates of change. Differential equations relate functions to their derivatives and have general solutions representing families of curves.
The document summarizes key concepts in number theory, including:
- Divisibility and the greatest common divisor (GCD)
- Prime numbers and the fundamental theorem of arithmetic
- Congruences and theorems related to congruences such as Euler's theorem
- Functions including the totient function and Mobius function
All presented by Dr. N. Chandramowliswaran in an invited talk on secure schemes for secret sharing and key distribution using graph theory.
1. The document discusses differential equations of higher order. It defines a general differential equation of order n with variable coefficients and explains when it can be considered a constant coefficient differential equation.
2. Methods for solving constant coefficient higher order differential equations are presented, including finding the characteristic equation and general solution involving exponential functions of roots of the characteristic equation.
3. Examples of solving second and third order differential equations with the given methods are provided. The complete solutions are written in terms of exponential functions and arbitrary constants.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
The document discusses differential equations, which are equations involving derivatives of functions with respect to independent variables. It defines ordinary and partial differential equations, provides examples, and discusses methods for solving differential equations, including finding general and particular solutions. It also describes the relationship between solutions of differential equations and their integrals.
Divulgamos um simples e direto método de resolução de equações diferenciais parciais lineares de ordem única. A vantagem do método é ser aplicável a ordens quaisquer e, a grande desvantagem, é ser restrito a uma única ordem, de cada vez. Por ser muito fácil em comparação com os métodos clássicos, possui grande valor didático.
1. The work done by a force F acting along a material path L can be calculated using the integral A = ∫_0^S F(s) ds, where s is the parameter of the path.
2. The mass of an object with non-uniform density ρ(x) along the x-axis between a and b can be found using the integral m = ∫_a^b ρ(x) dx.
3. The coordinates (xc, yc) of the centroid of a curve given in parametric form x=φ(t), y=ψ(t) between α and β can be determined using the integrals xc = ∫_α^
its my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time hav
Affordable housing in Cavite rush rush for sale/brand new houses rush for sal...Ma Erica Victoria Sacdalan
The document provides details of a property for sale including specifications, payment plans, and contact information for the real estate agent Cora Sacdalan. The property has 4 bedrooms, 4 bathrooms, and 1 carport on a 164 square meter lot. It has a down payment of 680,000 pesos and reservation fee of 30,000 pesos, which can be paid through in-house or bank financing plans. Interested parties are instructed to call the listed phone numbers or email for additional information.
Kiver Surf Camp es una empresa situada en El Palmar dedicada al ocio y al surf que ofrece alojamiento en bungalows, alquiler de equipamiento de surf y skate, clases de surf y un campamento para todas las edades. También cuenta con una tienda de moda y accesorios relacionados con las actividades acuáticas.
1. El documento describe la estructura y función del sistema nervioso central y periférico en humanos. 2. El sistema nervioso central está formado por el encéfalo y la médula espinal, mientras que el periférico conecta el central a los órganos y músculos. 3. Juntos, regulan funciones corporales internas y externas para mantener el equilibrio fisiológico.
The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
A semi analytic method for solving two-dimensional fractional dispersion equa...Alexander Decker
This document presents a semi-analytic method called the modified decomposition method for solving two-dimensional fractional dispersion equations. The method is applied to solve a two-dimensional fractional dispersion equation subject to initial and boundary conditions. The numerical results obtained from the modified decomposition method are shown to closely match the exact solution, demonstrating the accuracy of this approach. The method provides an efficient means of obtaining analytical solutions to fractional differential equations.
The study is concerned with a different perspective which the numerical solution of the singularly
perturbed nonlinear boundary value problem with integral boundary condition using finite difference method on
Bakhvalov mesh. So, we show some properties of the exact solution. We establish uniformly convergent finite
difference scheme on Bakhvalov mesh. The error analysis for the difference scheme is performed. The numerical
experiment implies that the method is the first order convergent in the discrete maximum norm, independently of
휀- singular perturbation parameter with effective and efficient iterative algorithm. The numerical results are
shown in table and graphs.
This document proposes a modified Newton's method for solving nonlinear equations that uses harmonic mean. It begins by reviewing Newton's method and some existing variants that use arithmetic mean or other integration rules to modify Newton's method and achieve cubic convergence without using second derivatives. It then presents the new Harmonic-Simpson-Newton method, which replaces the arithmetic mean in an existing Simpson Newton's method with harmonic mean. The method is proven to have cubic convergence. Numerical examples are provided to compare the efficiency of the new method to other cubic convergent methods.
This document discusses partial differential equations (PDEs). It begins by defining PDEs as equations that involve partial derivatives with respect to two or more independent variables. Next, it provides examples of how PDEs can be formed and classified based on characteristics like order, degree, whether they are linear or nonlinear. Then, it discusses methods for solving common types of PDEs like linear PDEs. Finally, it derives the one-dimensional wave equation and shows its solution as a product of functions involving the independent variables x and t.
A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equat...IOSR Journals
This document summarizes a two grid discretization method for decoupling time-harmonic Maxwell's equations. The method discretizes the coupled partial differential equations using continuous 1H-conforming finite elements on two grids - a fine grid and a coarser grid. This allows solving the coupled equations on the coarse grid along with decoupled equations on the fine grid, reducing computational costs compared to solving on just the fine grid. The document reviews relevant fundamentals, formulates the variational problem, and proves a regularity result stating the solution has higher regularity if the right hand side is more regular.
Calculus is the study of change and is divided into differential and integral calculus. Differential calculus studies rates of change using derivatives, while integral calculus uses integration to find accumulated change. These concepts build on limits and algebra/geometry. Leibniz developed the notation and principles of calculus in the 1670s. Differential calculus uses derivatives to determine how quantities change, and integral calculus uses integrals and antiderivatives to determine quantities from rates of change. Differential equations relate functions to their derivatives and have general solutions representing families of curves.
The document summarizes key concepts in number theory, including:
- Divisibility and the greatest common divisor (GCD)
- Prime numbers and the fundamental theorem of arithmetic
- Congruences and theorems related to congruences such as Euler's theorem
- Functions including the totient function and Mobius function
All presented by Dr. N. Chandramowliswaran in an invited talk on secure schemes for secret sharing and key distribution using graph theory.
1. The document discusses differential equations of higher order. It defines a general differential equation of order n with variable coefficients and explains when it can be considered a constant coefficient differential equation.
2. Methods for solving constant coefficient higher order differential equations are presented, including finding the characteristic equation and general solution involving exponential functions of roots of the characteristic equation.
3. Examples of solving second and third order differential equations with the given methods are provided. The complete solutions are written in terms of exponential functions and arbitrary constants.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
The document discusses differential equations, which are equations involving derivatives of functions with respect to independent variables. It defines ordinary and partial differential equations, provides examples, and discusses methods for solving differential equations, including finding general and particular solutions. It also describes the relationship between solutions of differential equations and their integrals.
Divulgamos um simples e direto método de resolução de equações diferenciais parciais lineares de ordem única. A vantagem do método é ser aplicável a ordens quaisquer e, a grande desvantagem, é ser restrito a uma única ordem, de cada vez. Por ser muito fácil em comparação com os métodos clássicos, possui grande valor didático.
1. The work done by a force F acting along a material path L can be calculated using the integral A = ∫_0^S F(s) ds, where s is the parameter of the path.
2. The mass of an object with non-uniform density ρ(x) along the x-axis between a and b can be found using the integral m = ∫_a^b ρ(x) dx.
3. The coordinates (xc, yc) of the centroid of a curve given in parametric form x=φ(t), y=ψ(t) between α and β can be determined using the integrals xc = ∫_α^
its my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time havits my personal file. please you guys dont have to read these or download these. sorry for your time waste. next time hav
Affordable housing in Cavite rush rush for sale/brand new houses rush for sal...Ma Erica Victoria Sacdalan
The document provides details of a property for sale including specifications, payment plans, and contact information for the real estate agent Cora Sacdalan. The property has 4 bedrooms, 4 bathrooms, and 1 carport on a 164 square meter lot. It has a down payment of 680,000 pesos and reservation fee of 30,000 pesos, which can be paid through in-house or bank financing plans. Interested parties are instructed to call the listed phone numbers or email for additional information.
Kiver Surf Camp es una empresa situada en El Palmar dedicada al ocio y al surf que ofrece alojamiento en bungalows, alquiler de equipamiento de surf y skate, clases de surf y un campamento para todas las edades. También cuenta con una tienda de moda y accesorios relacionados con las actividades acuáticas.
1. El documento describe la estructura y función del sistema nervioso central y periférico en humanos. 2. El sistema nervioso central está formado por el encéfalo y la médula espinal, mientras que el periférico conecta el central a los órganos y músculos. 3. Juntos, regulan funciones corporales internas y externas para mantener el equilibrio fisiológico.
El documento resume brevemente la historia temprana de la informática, desde los intentos iniciales de Babbage de crear una máquina analítica mecánica hasta el desarrollo de los primeros ordenadores electromecánicos y electrónicos en el siglo XX. Se describe cómo Hollerith desarrolló equipos basados en tarjetas perforadas para procesar datos censales, y cómo esto condujo al desarrollo de IBM. También se explica cómo máquinas como la de Zuse y la Mark I de Harvard utilizaron relés electromecánicos y otros
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In just one sentence, it pitches the idea of using Haiku Deck to easily design slideshows.
This document summarizes a study of CEO succession events among the largest 100 U.S. corporations between 2005-2015. The study analyzed executives who were passed over for the CEO role ("succession losers") and their subsequent careers. It found that 74% of passed over executives left their companies, with 30% eventually becoming CEOs elsewhere. However, companies led by succession losers saw average stock price declines of 13% over 3 years, compared to gains for companies whose CEO selections remained unchanged. The findings suggest that boards generally identify the most qualified CEO candidates, though differences between internal and external hires complicate comparisons.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve porous medium equation. This method is a combination of the new integral transform “ELzaki transform” and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. Some cases of the porous medium equation are solved as examples to illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. The results reveal that the combination of ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems. Key words: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and porous medium equation
Elzaki transform homotopy perturbation method for solving gas dynamics equationeSAT Journals
Abstract E-learning is a modern way of learning now days. It includes electronic media in the field of education. E-learning makes use of (ICT) information and communication technology. Now a day, In every field like learning in medical or learning in agriculture, e-learning is going popular to use. E-learning involves various types of media that provide video, audio, images and text. E-learning uses the intranet/ extranet / internet, and widens the horizon of traditional learning. This article explores the time when the concept of e-learning was introduced, mentions its basic principles, discuss the ways in which it is superior than the traditional educational system. E-Agriculture is a rising field that specialize in the improvement of agricultural and rural development through improved and updated information and communication processes. Specifically, e-Agriculture involves the conceptualization, design, development, analysis and application of innovative ways in which to use information and communication technologies (IT) within the rural domain, with a primary specialize in agriculture. E-Agriculture may be a comparatively new term and that we totally expect its scope to alter and evolve as our understanding of the realm grows. Keywords: E-Education in agriculture, E-learning, E-Education system, Internet-Education, Impact of E-Learning in Education.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Iterative procedure for uniform continuous mapping.Alexander Decker
This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
A Fast Numerical Method For Solving Calculus Of Variation ProblemsSara Alvarez
This document presents a numerical method called differential transform method (DTM) for solving calculus of variation problems. DTM finds the solution of variational problems in the form of a polynomial series without discretization. The method is applied to obtain the solution of the Euler-Lagrange equation arising from variational problems by considering it as an initial value problem. Some examples are presented to demonstrate the efficiency and accuracy of DTM for solving calculus of variation problems.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
The document discusses the calculus of variations and derives Euler's equation. It explains that the fundamental problem is to find an unknown function that makes a functional (function of a function) an extremum. Euler's equation provides the necessary condition for a functional to be extremized. As an example, it derives the Euler-Lagrange equation for the shortest distance between two points in Cartesian coordinates, showing the path is a straight line.
The document applies the variational iteration method (VIM) to solve linear and nonlinear ordinary differential equations (ODEs) with variable coefficients. It emphasizes the power of the method by using it to solve a variety of ODE models of different orders and coefficients. The document also uses VIM to solve four scientific models - the hybrid selection model, Thomas-Fermi equation, Kidder equation for unsteady gas flow through porous media, and the Riccati equation. The VIM provides efficient iterative approximations for both analytic solutions and numeric simulations of real-world applications in science and engineering.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
1) The document presents a wavelet collocation method for numerically solving nth order Volterra integro-differential equations. It expands the unknown function as a series of Chebyshev wavelets of the second kind with unknown coefficients.
2) It states and proves a uniform convergence theorem that establishes the convergence of approximating the solution using truncated Chebyshev wavelet series expansions.
3) The paper demonstrates the validity and applicability of the proposed method through some illustrative examples of solving integro-differential equations using the Chebyshev wavelet collocation approach.
This document discusses Frullani integrals, which are integrals of the form ∫01 f(ax)−f(bx)x dx = [f(0)−f(∞)]ln(b/a). It provides 11 examples of integrals from Gradshteyn and Ryzhik that can be reduced to this Frullani form by appropriate choice of the function f(x). It also lists 9 examples found in Ramanujan's notebooks. One example, involving logarithms of trigonometric functions, requires a more complex approach. The document concludes by deriving the solution to this more delicate example.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
This document presents derivations of several integrals involving the error function that are contained in the table of Gradshteyn and Ryzhik. It begins by introducing the error function and some of its basic properties. It then derives recurrences and explicit formulas for integrals of the form Fn(v) = ∫v0 tn e−t2 dt. Using these results and elementary changes of variables, it evaluates several entries in the Gradshteyn and Ryzhik table. It also presents a series representation for the error function and evaluates Laplace's classical integral involving the error function.
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Mt3621702173
1. P. G. Bhadane et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.2170-2173
RESEARCH ARTICLE
www.ijera.com
OPEN ACCESS
Elzaki Transform Homotopy Perturbation Method for Solving
Fourth Order Parabolic PDE with Variable Coefficients
Prem Kiran G. Bhadane*, V. H. Pradhan**
* (Department of Applied Sciences, RCPIT, Shirpur-425405, India)
** (Department of Applied Mathematics and Humanities, SVNIT, Surat-395007, India)
ABSTRACT
In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve
one dimensional fourth order parabolic linear partial differential equations with variable coefficients. This
method is a combination of the new integral transform “ELzaki transform” and the homotopy perturbation
method. Some cases of one dimensional fourth order parabolic linear partial differential equations are solved to
illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. We have
compared the obtained analytical solution with the available Laplace decomposition solution and homotopy
perturbation method solution which is found to be exactly same. The results reveal that the combination of
ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in
such problems.
Keywords – ELzaki transform, homotopy perturbation method, linear partial differential equation.
I.
INTRODUCTION
Many problems of physical interest are
described by linear partial differential equations with
initial and boundary conditions. One of them is fourth
order parabolic partial differential equations with
variable coefficients; these equations arise in the
transverse vibration problems[1]. In recent years,
many research workers have paid attention to find the
solution of these equations by using various methods.
Among these are the variational iteration method
[Biazar
and
Ghazvini
(2007)],
Adomian
decomposition method [Wazwaz (2001) and Biazar et
al (2007)], homotopy perturbation method [Mehdi
Dehghan and Jalil Manafian (2008)], homotopy
analysis method [Najeeb Alam Khan, Asmat Ara,
Muhammad Afzal and Azam Khan (2010)] and
Laplace decomposition algorithm [Majid Khan,
Muhammad Asif Gondal and Yasir Khan (2011)]. In
this paper we use coupling of new integral transform
“ELzaki transform” and homotopy perturbation
method. This method is a useful technique for solving
linear and nonlinear differential equations. The main
aim of this paper is to consider the effectiveness of the
ELzaki transform homotopy perturbation method in
solving higher order linear partial differential
equations with variable coefficients. This method
provides the solution in a rapid convergent series
which leads the solution in a closed form.
II.
ELZAKI TRANSFORM HOMOTOPY
PERTURBATION METHOD [1, 2, 3, 4]
Consider a one dimensional linear
nonhomogeneous fourth order parabolic partial
differential equation with variable coefficients of the
form
www.ijera.com
∂2 u
∂4 u
+ψ x
= ϕ x, t ,
(1)
∂t 2
∂x 4
where ψ x is a variable coefficient, with the
following
initial
conditions
∂u
u x, 0 = f x and
x, 0 = h x .
(2)
∂t
and boundary conditions are
u a, t = β1 t , u b, t = β2 t ,
∂2 u
∂2 u
a, t = β3 t , 4 b, t = β4 t ,
(3)
∂x 4
∂x
Apply ELzaki transform on both sides of Eq. (1)
∂2 u
∂4 u
E
+ψ x
= E ϕ x, t ,
(4)
∂t 2
∂x 4
But, ELzaki transform of partial derivative is given by
[3]
∂2 f
1
∂f
E 2 (x, t) = 2 E f x, t − f x, 0 − v
x, 0 ,
∂t
v
∂t
Using this property, Eq. (3) can be written as
1
∂u
E u x, t − u x, 0 − v
x, 0
2
v
∂t
∂4 u
= E ϕ x, t − E ψ x
,
(5)
∂x 4
Put the values of initial conditions in Eq. (5), we get
1
E u x, t − f x − vh x
v2
∂4 u
= E ϕ x, t − E ψ x
,
∂x 4
By simple calculations, we have
E u x, t = g x, v
∂4 u
− v2 E ψ x
,
(6)
∂x 4
where g x, v = v 2 f x + v 3 h x + E ϕ x, t .
2170 | P a g e
2. P. G. Bhadane et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.2170-2173
Applying ELzaki inverse on both sides of Eq. (6), we
get
∂4 u
u(x, t) = K x, t − E −1 v 2 E ψ x
,
(7)
∂x 4
where K x, t = E −1 g x, v , represents the term
arising from the source term and the prescribed initial
conditions.
Now, we apply the homotopy perturbation method.
∞
pn un (x, t) ,
u x, t =
(8)
n=0
By substituting Eq. (8) into Eq. (7), we get
∞
pn un x, t = K x, t
n=0
∞
−pE −1 v 2 E ψ x
pn un (x, t)
n=0
,
(9)
xxxx
This is the coupling of the ELzaki transform and the
homotopy perturbation method. Comparing the
coefficient of like powers of p , the following
approximations are obtained.
p0 : u0 x, t = K x, t ,
p1 : u1 x, t = −E −1 v 2 E ψ x u0xxxx x, t ,
p2 : u2 x, t = −E −1 v 2 E ψ x u1xxxx x, t ,
p3 : u3 x, t = −E −1 v 2 E ψ x u2xxxx x, t ,
… … … … … … … … … … … … … … … … … … … ….
In general recursive relation is given by,
pm : um x, t = −E −1 v 2 E ψ x u(m−1)xxxx x, t ,
Then the solution is
u x, t = u0 x, t + u1 x, t + u2 x, t
+ …………
(10)
III.
APPLICATION
To demonstrate the effectiveness of the
method we have solved homogeneous and
nonhomogeneous one dimensional fourth order linear
partial differential equations with initial and boundary
conditions.
Example 1. Consider fourth order homogeneous
partial differential equation as [1, 5]
∂2 u
1
x 4 ∂4 u
1
+
+
= 0,
< 𝑥 < 1, 𝑡 > 0 (11)
2
4
∂t
x 120 ∂x
2
with the following initial conditions
∂u
x5
u x, 0 = 0 ,
x, 0 = 1 +
,
(12)
∂t
120
and boundary conditions
1 + (0.5)5
121
u 0.5, t =
sint, u 1, t =
sint,
120
120
∂2 u
∂2 u
1
0.5, t = 0.02084sint , 2 1, t = sint. (13)
2
∂x
∂x
6
Applying ELzaki transform to Eq. (11), we get
∂2 u
1
x 4 ∂4 u
E
+E
+
= 0,
∂t 2
x 120 ∂x 4
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www.ijera.com
1
∂u
E u x, t − u x, 0 − v
x, 0
v2
∂t
1
x 4 ∂4 u
+E
+
= 0,
(14)
x 120 ∂x 4
Using initial conditions from Eq. (12), we get
x5
E u x, t = v 3 1 +
120
1
x 4 ∂4 u
− v2 E
+
,
(15)
x 120 ∂x 4
Now taking ELzaki inverse on both sides of above Eq.
(15), we have
x5
1
x 4 ∂4 u
u x, t = 1 +
t − E −1 v 2 E
+
,
120
x 120 ∂x 4
(16)
Now, we apply the homotopy perturbation method.
∞
pn un (x, t),
u x, t =
n=0
Putting this value of u(x, t) into Eq. (16), we get
∞
x5
pn un x, t = 1 +
t
120
n=0
−pE −1 v 2 E
1
x4
+
x 120
∞
pn un (x, t)
n=0
xxxx
(17)
Comparing the coefficient of like powers of p , in Eq.
(17) the following approximations are obtained.
x5
p0 : u0 x, t = K x, t = 1 +
t,
120
1
x4
p1 : u1 x, t = −E −1 v 2 E
+
u
x, t
x 120 0xxxx
x5 t3
=− 1+
,
120 3!
1
x4
p2 : u2 x, t = −E −1 v 2 E
+
u
x, t
x 120 1xxxx
x5 t5
= 1+
,
120 5!
1
x4
p3 : u3 x, t = −E −1 v 2 E
+
u
x, t
x 120 2xxxx
x5 t7
=− 1+
,
120 7!
⋮
1
x4
pm : um x, t = −E −1 v 2 E
+
u
x 120 (m−1)xxxx
x5
t 2m+1
= −1 m 1 +
,
120 2m + 1 !
m = 1,2,3, … … … .
And so on in the same manner the rest of the
components of iteration formula can be obtained and
thus solution can be written in closed form as
x5
x5 t3
u x, t = 1 +
t− 1+
120
120 3!
5
5
x
t
x5 t7
+ 1+
− 1+
+ ⋯ … … … . ..
120 5!
120 7!
2171 | P a g e
3. P. G. Bhadane et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.2170-2173
x5
t3 t5 t7
t− + − + ⋯..
120
3! 5! 7!
5
x
u x, t = 1 +
sint,
(18)
120
which is an exact solution of Eq. (11) and can be
verified through substitution.
u x, t = 1 +
Example 2. Consider fourth order homogeneous
partial differential equation as [5]
∂2 u
x
∂4 u
+
−1
= 0, 0 < 𝑥 < 1, 𝑡 > 0 (19)
∂t 2
sinx
∂x 4
with the following initial conditions
∂u
u x, 0 = x − sinx ,
x, 0 = −x + sinx,
(20)
∂t
and boundary conditions
u 0, t = 0, u 1, t = e−t 1 − sin1 ,
∂2 u
∂2 u
0, t = 0 , 2 1, t = e−t sin1.
(21)
∂x 2
∂x
Applying ELzaki transform to Eq. (19), we get
∂2 u
x
∂4 u
E
+E
−1
= 0,
∂t 2
sinx
∂x 4
1
∂u
E u x, t − u x, 0 − v
x, 0
2
v
∂t
x
∂4 u
+E
−1
= 0,
(22)
sinx
∂x 4
Using initial conditions from Eq. (20), we get
E u x, t = v 2 x − sinx + v 3 −x + sinx
x
∂4 u
− v2 E
−1
,
(23)
sinx
∂x 4
Now taking ELzaki inverse on both sides of above Eq.
(23), we have
u x, t = x − sinx + (−x + sinx)t
x
∂4 u
−E −1 v 2 E
−1
,
(24)
sinx
∂x 4
Now, we apply the homotopy perturbation method.
∞
pn un (x, t),
u x, t =
n=0
Putting this value of u(x, t) into Eq. (24), we get
∞
pn un x, t = x − sinx + (−x + sinx)t
n=0
−pE −1 v 2 E
x
−1
sinx
∞
pn un (x, t)
n=0
,
xxxx
(25)
Comparing the coefficient of like powers of p , in Eq.
(25) the following approximations are obtained.
p0 : u0 x, t = K x, t = x − sinx + (−x + sinx)t
x
p1 : u1 x, t = −E −1 v 2 E
− 1 u0xxxx x, t
sinx
2
t
t3
= x − sinx
−
,
2! 3!
x
p2 : u2 x, t = −E −1 v 2 E
− 1 u1xxxx x, t
sinx
4
t
t5
= x − sinx
−
,
4! 5!
www.ijera.com
p3 : u3 x, t = −E −1 v 2 E
= x − sinx
www.ijera.com
x
− 1 u2xxxx x, t
sinx
6
t
t7
−
,
6! 7!
⋮
x
− 1 u(m−1)xxxx
sinx
2m
t
t 2m +1
= x − sinx
−
,
2m!
2m + 1 !
m = 1,2,3,4 … … ..
And so on in the same manner the rest of the
components of iteration formula can be obtained and
thus solution can be written in closed form as
u x, t = x − sinx + −x + sinx t
t2 t3
t4 t5
+ x − sinx
−
+ x − sinx
−
2! 3!
4! 5!
t6 t7
+ x − sinx
−
… … … … . ..
6! 7!
t2 t3 t4 t5
u x, t = x − sinx 1 − t + − + − +. . ,
2! 3! 4! 5!
u x, t = x − sinx e−t ,
(26)
which is an exact solution of Eq. (19) and can be
verified through substitution.
pm : um x, t = −E −1 v 2 E
Example 3. Consider fourth order nonhomogeneous
partial differential equation as [1, 5]
∂2 u
∂4 u
6
+ 1+x
= x 4 + x 3 − x 7 cost,
∂t 2
∂x 4
7!
0 < 𝑥 < 1, 𝑡 > 0
(27)
with the following initial conditions
6
∂u
u x, 0 = x 7 ,
x, 0 = 0.
(28)
7!
∂t
and boundary conditions
6
u 0, t = 0, u 1, t = cost ,
7!
∂2 u
∂2 u
1
0, t = 0 , 2 1, t =
cost.
(29)
∂x 2
∂x
20
Applying ELzaki transform to Eq. (27), we get
∂2 u
∂4 u
E
+E 1+x
2
∂t
∂x 4
6
= E x 4 + x 3 − x 7 cost ,
7!
1
∂u
E u x, t − u x, 0 − v
x, 0
v2
∂t
2
6
v
∂4 u
= x4 + x3 − x7
−E 1+x
,
7!
1 + v2
∂x 4
Using initial conditions from Eq. (28), we get
6
6
v4
E u x, t = v 2 x 7 + x 4 + x 3 − x 7
7!
7!
1 + v2
∂4 u
− v2 E 1 + x
,
(30)
∂x 4
Now taking ELzaki inverse on both sides of above Eq.
(30), we have
6
6
u x, t = x 7 + x 4 + x 3 − x 7 (1 − cost)
7!
7!
∂4 u
−1
2
−E
v E 1+x
,
(31)
∂x 4
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4. P. G. Bhadane et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.2170-2173
Now, we apply the homotopy perturbation method.
∞
n=0
Putting this value of u(x, t) into Eq. (31), we get
∞
6
pn un x, t = x 7 +
7!
n=0
6 7
x4 + x3 − x
1 − cost
7!
∞
−pE
−1
2
pn un (x, t)
v E 1+x
n=0
(32)
REFERENCES
[1] Majid Khan, Muhammad Asif Gondal and
xxxx
Here, we assume that
6
6
K x, t = x 7 + x 4 + x 3 − x 7 1 − cost
7!
7!
can be divided into the sum of two parts namely
K 0 (x, t) and K1 (x, t), therefore we get [6]
K x, t = K 0 x, t + K1 x, t
Under this assumption, we propose a slight variation
only in the components u0 , u1 . The variation we
propose is that only the part K 0 (x, t) be assigned to
the u0 , whereas the remaining part K1 (x, t), be
combined with the other terms to define u1 .
6
K 0 x, t = x 7 cost ; K1 x, t = x 4 + x 3 (1 − cost)
7!
In view of these, we formulate the modified
recursive algorithm as follows:
6
p0 : u0 x, t = x 7 cost,
7!
p1 : u1 x, t = x 4 + x 3 (1 − cost)
−E −1 v 2 E 1 + x u0xxxx x, t = 0,
2
p : u2 x, t = −E −1 v 2 E 1 + x u1xxxx x, t = 0,
And so on in the same manner the rest of the
components of iteration formula can be obtained
um x, t = 0, for m ≥ 1.
Thus solution can be written in closed form as
6
u x, t = x 7 cost + 0 + 0 + 0 + 0 + ⋯ ….
7!
6
u x, t = x 7 cost .
(33)
7!
which is an exact solution of Eq. (27) and can be
verified through substitution.
IV.
CONCLUSION
The main goal of this paper is to show the
applicability of the mixture of new integral transform
“ELzaki transform” with the homotopy perturbation
method to solve one dimensional fourth order
homogeneous and nonhomogeneous linear partial
differential equations with variable coefficients. This
combination of two methods successfully worked to
give very reliable and exact solutions to the equation.
This method provides an analytical approximation in a
rapidly convergent sequence with in exclusive manner
computed terms. Its rapid convergence shows that the
method is trustworthy and introduces a significant
improvement in solving linear partial differential
equations over existing methods.
www.ijera.com
ACKNOWLEDGEMENT
I am deeply grateful to the management of
Shirpur Education Society, Shirpur (Maharashtra)
without whose support my research work would not
have been possible. I would also like to extend my
gratitude to the Prin. Dr. J. B. Patil and Mr. S. P.
Shukla, Head of Department of Applied Sciences,
RCPIT for helping and inspiring me for the research
work.
pn un (x, t),
u x, t =
V.
www.ijera.com
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