This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for handling partial derivatives.
[2] It then introduces the projected differential transform method and its fundamental theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression in terms of the nonlinear terms, which are then decomposed using the projected differential transform method.
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
11.[104 111]analytical solution for telegraph equation by modified of sumudu ...Alexander Decker
1) The document presents a new integral transform method called the Elzaki transform to solve the general linear telegraph equation.
2) The Elzaki transform is used to obtain analytical solutions for the telegraph equation. Definitions and properties of the transform are provided.
3) Several examples are presented to demonstrate the method. Exact solutions to examples of the telegraph equation are obtained using the Elzaki transform.
11.[95 103]solution of telegraph equation by modified of double sumudu transf...Alexander Decker
1. The document presents a new mathematical transform called the double Elzaki transform.
2. This transform is used to solve the general linear telegraph equation, which is an important partial differential equation in physics.
3. The key steps are: taking the double Elzaki transform of the telegraph equation, taking the single Elzaki transform of the boundary and initial conditions, substituting these into the transformed equation, and taking the inverse transforms to obtain the solution.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
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.
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Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
11.[104 111]analytical solution for telegraph equation by modified of sumudu ...Alexander Decker
1) The document presents a new integral transform method called the Elzaki transform to solve the general linear telegraph equation.
2) The Elzaki transform is used to obtain analytical solutions for the telegraph equation. Definitions and properties of the transform are provided.
3) Several examples are presented to demonstrate the method. Exact solutions to examples of the telegraph equation are obtained using the Elzaki transform.
11.[95 103]solution of telegraph equation by modified of double sumudu transf...Alexander Decker
1. The document presents a new mathematical transform called the double Elzaki transform.
2. This transform is used to solve the general linear telegraph equation, which is an important partial differential equation in physics.
3. The key steps are: taking the double Elzaki transform of the telegraph equation, taking the single Elzaki transform of the boundary and initial conditions, substituting these into the transformed equation, and taking the inverse transforms to obtain the solution.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
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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.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
Numerical solution of boundary value problems by piecewise analysis methodAlexander Decker
This document presents a numerical method called Piecewise-Homotopy Analysis Method (P-HAM) for solving fourth-order boundary value problems. P-HAM is based on the Homotopy Analysis Method (HAM) but uses multiple auxiliary parameters, with each parameter applied over a sub-range of the domain for improved accuracy. The document outlines the basic steps of P-HAM, including constructing the zero-order deformation equation and deriving the governing equations. It then applies P-HAM to solve two example problems and compares the results to other numerical methods.
A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
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1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
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b
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a
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This document provides an introduction to stochastic calculus. It begins with a review of key probability concepts such as the Lebesgue integral, change of measure, and the Radon-Nikodym derivative. It then discusses information and σ-algebras, including filtrations and adapted processes. Conditional expectation is explained. The document concludes by introducing random walks and their connection to Brownian motion through the scaled random walk process. Key concepts such as martingales and quadratic variation are defined.
This document summarizes a presentation on rigorously verifying the accuracy of numerical solutions to semi-linear parabolic partial differential equations using analytic semigroups. It introduces the considered problem of finding the solution to a semi-linear parabolic PDE. It then discusses using a piecewise linear finite element discretization in space and time to obtain an initial numerical solution. The goal is to rigorously enclose the true solution within a radius ρ of this numerical solution in the function space L∞(J;H10(Ω)). Key steps involve using properties of the analytic semigroup generated by the operator A and estimating discretization errors to compute the enclosure radius ρ.
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This document proposes a linear programming (LP) based approach for solving maximum a posteriori (MAP) estimation problems on factor graphs that contain multiple-degree non-indicator functions. It presents an existing LP method for problems with single-degree functions, then introduces a transformation to handle multiple-degree functions by introducing auxiliary variables. This allows applying the existing LP method. As an example, it applies this to maximum likelihood decoding for the Gaussian multiple access channel. Simulation results demonstrate the LP approach decodes correctly with polynomial complexity.
Regularity and complexity in dynamical systemsSpringer
This chapter discusses how variational methods have been used to analyze three classes of snakelike robots: 1) hyper-redundant manipulators guided by backbone curves, 2) flexible steerable needles, and 3) concentric tube continuum robots. Variational methods provide a means to determine optimal backbone curves for manipulators, generate optimal plans for needle steering, and model equilibrium conformations for concentric tube robots based on elastic mechanics principles. The chapter reviews how variational formulations using Euler-Lagrange and Euler-Poincare equations are applied in each case.
Este documento presenta una introducción a los blogs, incluyendo definiciones, programas gratuitos para crear blogs, las características y elementos clave de los blogs, las diferencias entre blogs, páginas web y foros, y curiosidades sobre blogs. También incluye una conclusión y bibliografía.
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- Enzyme treatment, particularly with protease, improved oil extraction yields from the seeds compared to untreated controls.
- Modification of the oil using lipase increased the oleic acid content and decreased the melting point, indicating potential for food/culinary applications.
- Due to its similarities to olive oil and stability properties, M. oleifera seed oil shows promise
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This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
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1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
Estimation of the score vector and observed information matrix in intractable...Pierre Jacob
This document discusses methods for estimating the score vector and observed information matrix for intractable models. It begins with an overview of using derivatives in sampling algorithms. It then discusses iterated filtering, a method for estimating derivatives in hidden Markov models when the likelihood is not available in closed form. Iterated filtering introduces a perturbed model and relates the posterior mean to the score and posterior variance to the observed information matrix. The document outlines proofs that show this relationship as the prior concentration increases.
A new approach to constants of the motion and the helmholtz conditionsAlexander Decker
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2) It shows that constants of motion determined by this new approach vanish when the symmetry group is related to an actual invariance of the Lagrangian, as in this case the classical Noether invariants can be used.
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b
I(x) = F (x(t), x (t), t) dt,
a
with various types of boundary conditions. It proves the Euler-Lagrange equation for the simplest case where the curves are allowed to vary between two fixed points x(a) and x(b). The Euler-Lagrange equation is a differential equation that any extremal curve must satisfy. An example is worked out to demonstrate finding the curve that minimizes a given functional by solving the Euler-Lagrange equation subject to the boundary conditions.
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1) Markov's inequality and Chebyshev's inequality, which relate the probability that a random variable exceeds a value to its expected value and variance.
2) The weak law of large numbers and central limit theorem, which describe how the means of independent random variables converge to the expected value and follow a normal distribution as the number of variables increases.
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Ian.petrow【transcendental number theory】.Tong Leung
This document provides an introduction and overview of the course "Math 249A Fall 2010: Transcendental Number Theory" taught by Kannan Soundararajan. It discusses topics that will be covered, including proving that specific numbers like e, π, and combinations of them are transcendental. Theorems are presented on approximating algebraic numbers and showing linear independence of exponential functions of algebraic numbers. Examples are given of using an integral technique to derive contradictions and prove transcendence.
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This document defines and provides examples of expectation, or the average value, of random variables. It discusses properties of expectations including how the expectation of a function of a random variable is calculated. It also defines and gives properties of variance, covariance, conditional expectation, and conditional variance. Examples are provided throughout to illustrate key concepts.
This document proposes a linear programming (LP) based approach for solving maximum a posteriori (MAP) estimation problems on factor graphs that contain multiple-degree non-indicator functions. It presents an existing LP method for problems with single-degree functions, then introduces a transformation to handle multiple-degree functions by introducing auxiliary variables. This allows applying the existing LP method. As an example, it applies this to maximum likelihood decoding for the Gaussian multiple access channel. Simulation results demonstrate the LP approach decodes correctly with polynomial complexity.
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Este documento presenta una introducción a los blogs, incluyendo definiciones, programas gratuitos para crear blogs, las características y elementos clave de los blogs, las diferencias entre blogs, páginas web y foros, y curiosidades sobre blogs. También incluye una conclusión y bibliografía.
This document summarizes a study investigating the potential for Moringa oleifera seed oil to be a new source of oleic acid-type oil for Malaysia. Key findings include:
- M. oleifera seed oil contains high levels of oleic acid (67.9%), similar to olive oil, and remains liquid at room temperature.
- Enzyme treatment, particularly with protease, improved oil extraction yields from the seeds compared to untreated controls.
- Modification of the oil using lipase increased the oleic acid content and decreased the melting point, indicating potential for food/culinary applications.
- Due to its similarities to olive oil and stability properties, M. oleifera seed oil shows promise
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Cynthia Xiu Ehien is a British citizen studying for an MEng in Civil Engineering at the University of London. She has relevant work experience in project management and a client-focused environment from a sandwich placement. Her interests include music, travel, and sports. She is seeking a job in the service sector where she can apply her degree, communication skills, and client-focused experience.
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The document provides guidance on creating an effective mission statement for a company. It explains that a mission statement should convey the company's purpose, business, and values to clients. It emphasizes that the mission statement should focus on assisting clients, providing innovative high-quality digital media products and services, and supporting clients with responsive advice. A draft mission statement is presented as an example.
The document discusses dimensional analysis, similitude, and model analysis. It provides background on how dimensional analysis and model testing are used to study fluid mechanics problems. Dimensional analysis uses the dimensions of physical quantities to determine which parameters influence a phenomenon. Model testing in a laboratory allows measurements to be applied to larger scale systems using similitude. Buckingham's π-theorem is introduced as a way to non-dimensionalize variables when there are more variables than fundamental dimensions. Rayleigh's and Buckingham's methods are demonstrated on an example of determining the resisting force on an aircraft.
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10 Tips for Making Beautiful Slideshow Presentations by www.visuali.seEdahn Small
1. Know your goal | make each slide count
2. Plan it out | in some detail
3. Avoid templates | they have the uglies
4. Choose a color scheme | 4 colors, 1 accent
5. Choose a font scheme | match tone
6. Choose a layout scheme | comprehension
7. Use images (wisely) | they’re more memorable
8. 15 words per slide | this slide had 16 words
9. Play with typography | impact, interest, hierarchy
10. Don’t overdo it | white space
Hope you enjoy!
SEE MORE OF MY WORK: http://www.visuali.se
10 Tips for Making Beautiful Slideshow Presentations by www.visuali.se
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11.solution of linear and nonlinear partial differential equations using mixture of elzaki transform and the projected differential transform method
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
Solution of Linear and Nonlinear Partial Differential
Equations Using Mixture of Elzaki Transform and the
Projected Differential Transform Method
Tarig. M. Elzaki1* & Eman M. A. Hilal2
1. Mathematics Department, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University,
Jeddah-Saudi Arabia.
Mathematics Department, Faculty of Sciences, Sudan University of Sciences and Technology-Sudan.
2. Mathematics Department, Faculty of Sciences for Girles King Abdulaziz University
Jeddah-Saudi Arabia
* E-mail of the corresponding author: Tarig.alzaki@gmail.com
Abstract
The aim of this study is to solve some linear and nonlinear partial differential equations using the new
integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can
be handled by using of projected differential transform method; this method is more efficient and easy to
handle such partial differential equations in comparison to other methods. The results show the efficiency
and validation of this method.
Keywords: Elzaki transform, projected differential transform method, nonlinear partial differential
equations.
1. Introduction
Many problems of physical interest are described by linear and nonlinear partial differential equations with
initial or boundary conditions, these problems are fundamental importance in science and technology
especially in engineering. Some valuable contributions have already been made to showing differential
equations such as Laplace transform method [lslam, Yasir Khan, Naeem Faraz and Francis Austin (2010),
Lokenath Debnath and Bhatta (2006)], Sumudu transform [A.Kilicman and H.E.Gadain (2009), (2010)],
differential transform method etc. Elzaki transform method is very effective tool for solve linear partial
differential equations [Tarig Elzaki & Salih Elzaki (2011)]. In this study we use the projected differential
transform method [Nuran Guzel and Muhammet Nurulay (2008), Shin- Hsiang Chang , Ling Chang (2008)
] to decompose the nonlinear terms, this means that we can use both Elzaki transform and projected
differential transform methods to solve many nonlinear partial differential equations.
1.1. Elzaki Transform:
The basic definitions of modified of Sumudu transform or Elzaki transform is defined as follows, Elzaki
transform of the function f (t ) is
50
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
∞ t
−
E [ f (t ) ] = v ∫ f (t ) e v dt , t >0
0
Tarig M. Elzaki and Sailh M. Elzaki in [2011], showed the modified of Sumudu transform [Kilicman &
ELtayeb (2010)] or Elzaki transform was applied to partial differential equations, ordinary differential
equations, system of ordinary and partial differential equations and integral equations.
In this paper, we combined Elzaki transform and projected differential transform methods to solve linear
and nonlinear partial differential equations.
To obtain Elzaki transform of partial derivative we use integration by parts, and then we have:
∂f (x , t ) 1 ∂ 2 f (x , t ) 1 ∂f (x , 0)
E = T (x ,v ) −vf (x , 0) E = v 2 T (x ,v ) − f (x , 0) −v
∂t v ∂t ∂t
2
∂f (x , t ) d ∂ 2f (x , t ) d 2
E = [T (x ,v )] E = 2 [T (x ,v )]
∂x dx ∂x
2
dx
Proof:
To obtain ELzaki transform of partial derivatives we use integration by parts as follows:
∂f ∞ ∂f −t
p −t
∂f −t
p p −t
Ε ( x ,t ) = ∫ v e dt = lim ∫ve
v v
dt = lim ve f (x , t ) − ∫ e v f (x , t )dt
v
∂t 0 ∂t p →∞
0
∂t p →∞
0 0
T ( x ,v )
= −vf ( x , 0 )
v
We assume that f is piecewise continuous and it is of exponential order.
∂f
∞ −t
∂f ( x , t ) ∂
∞ −t
= ∫ ve v ve v f ( x , t ) dt , using the Leibnitz rule to find:
∂x ∫
Now Ε dt =
∂x 0 ∂x 0
∂f d
Ε = T ( x ,v )
∂x dx
∂ 2f d 2
By the same method we find: Ε 2 = 2 T ( x ,v )
∂x dx
∂2 f
To find: Ε 2 ( x, t )
∂t
∂f
Let = g , then we have:
∂t
∂ 2f ∂g ( x , t )
Ε 2 ( x , t ) = Ε g ( x , t ) −vg x , 0
=Ε ( )
∂t ∂t v
51
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
∂ 2f 1 ∂f
Ε 2 ( x , t ) = 2 T ( x , v ) − f ( x , 0 ) −v ( x , 0)
∂t v ∂t
We can easily extend this result to the nth partial derivative by using mathematical induction.
1.2. Projected Differential Transform Methods:
In this section we introduce the projected differential transform method which is modified of the
differential transform method.
Definition:
The basic definition of projected differential transform method of function f (x 1 , x 2 ,LL , x n ) is
defined as
1 ∂ k f (x 1 , x 2 ,LL , x n )
f (x 1 , x 2 ,LL , x n −1 , k ) = (1)
k ! ∂x n k
x n =0
Such that f (x 1 , x 2 ,LL , x n ) is the original function and f (x 1 , x 2 ,LL , x n −1 , k ) is projected
transform function.
And the differential inverse transform of f (x 1 , x 2 ,LL , x n −1 , k ) is defined as
∞
f (x 1 , x 2 ,LL , x n ) = ∑ f (x 1 , x 2 ,LL , x n )(x − x 0 ) k (2)
k =o
The fundamental theorems of the projected differential transform are
52
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
Theorems:
(1) If z ( x1 , x2 ,......, xn ) = u ( x1 , x2 ,......, xn ) ± v ( x1 , x2 ,......, xn )
Then z ( x1 , x2 ,......, xn −1 , k ) = u ( x1 , x2 ,......, xn −1 , k ) ± v ( x1 , x2 ,......, xn −1 , k )
( 2) Ifz ( x1 , x2 ,......, xn ) = c u ( x1 , x2 ,......, xn )
Then z ( x1 , x2 ,......, xn −1 , k ) = cu ( x1 , x2 ,......, xn −1 , k )
du ( x1 , x2 ,......, xn )
( 3) Ifz ( x1 , x2 ,......, xn ) =
dxn
Then z ( x1 , x2 ,......, xn −1 , k ) = ( k + 1) u ( x1 , x2 ,....., xn −1 , k + 1)
d n u ( x1 , x2 ,......, xn )
( 4 ) Ifz ( x1 , x2 ,......, xn ) = n
dxn
Then z ( x1 , x2 ,......, xn−1 , k ) =
( k + n )! u
( x1 , x2 ,....., xn−1 , k + n )
k!
( 5) If z ( x1 , x2 ,......, xn ) = u ( x1 , x2 ,......, xn ) v ( x1 , x2 ,......, xn )
k
Then z ( x1 , x2 ,......, xn −1 , k ) = ∑ u ( x1 , x2 ,......, xn −1 , m ) v ( x1 , x2 ,......, xn −1 , k − m )
m =0
( 6 ) If z ( x1 , x2 ,......, xn ) = u1 ( x1 , x2 ,......, xn ) u2 ( x1 , x2 ,......, xn ) ..... un ( x1 , x2 ,......, xn ) Then
k k n−1 k3 k2
z ( x1 , x2 ,......, xn −1 , k ) = ∑ ∑ ...... ∑ ∑ u ( x , x ,......, x
1 1 2 n −1 , k1 ) u2 ( x1 , x2 ,......, xn −1 , k2 − k1 )
kn −1 = 0 k n− 2 = 0 k 2 = 0 k1 = 0
×.....un −1 ( x1 , x2 ,......, xn −1 , kn −1 − kn −2 ) un ( x1 , x2 ,......, xn −1 , k − kn −1 )
(7) If z ( x1 , x2 ,......, xn ) = x1q1 x2 2 ........xn n
q q
1 k = qn
Then z ( x1 , x2 ,......, xn −1 , k ) = δ ( x1 , x2 ,......, xn−1 , qn − k ) =
0 k ≠ qn
Note that c is a constant and n is a nonnegative integer.
2. Applications:
Consider a general nonlinear non-homogenous partial differential equation with initial conditions of the
form:
Du (x , t ) + Ru (x , t ) + Nu (x , t ) = g (x , t ) (3)
u (x , 0) = h (x ) , u t (x , 0) = f (x )
53
5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
Where D is linear differential operator of order two, R is linear differential operator of less
order than D , N is the general nonlinear differential operator and g ( x , t ) is the source term.
Taking Elzaki transform on both sides of equation (3), to get:
E [Du (x , t )] + E [Ru (x , t )] + E [Nu (x , t )] = E [ g (x , t )] (4)
Using the differentiation property of Elzaki transforms and above initial conditions, we have:
E [u (x , t )] = v 2 E [ g (x , t )] + v 2 h (x ) + v 3f (x ) −v 2 E [Ru (x , t ) + Nu (x , t )] (5)
Appling the inverse Elzaki transform on both sides of equation (5), to find:
u (x , t ) = G (x , t ) − E −1 {v 2 E [ Ru (x , t ) + Nu (x , t )]} (6)
Where G ( x , t ) represents the term arising from the source term and the prescribed initial conditions.
Now, we apply the projected differential transform method.
u (x , m + 1) = −E −1 {v 2 E [ A m + B m ]} , u (x , 0) = G (x , t ) (7)
Where A m , B m are the projected differential transform of Ru (x , t ) , Nu (x , t ) .
From equation (7), we have:
u (x , 0) = G (x , t ) , u (x ,1) = − E −1 {v 2 E [ A 0 + B 0 ]}
u (x , 2) = −E −1 {v 2 E [ A1 + B 1 ]} , u (x ,3) = − E −1 {v 2 E [ A 2 + B 2 ]}
.....
Then the solution of equation (3) is
u (x , t ) = u (x , 0) + u (x ,1) + u (x , 2) + ........
To illustrate the capability and simplicity of the method, some examples for different linear and nonlinear
partial differential equations will be discussed.
Example 2.1:
Consider the simple first order partial differential equation
∂y ∂y
=2 +y , y (x , 0) = 6e −3x (8)
∂x ∂t
Taking Elzaki transform of (8), leads to
v
E [ y (x , t )] = 6v 2e −3 x + E [Am − B m ]
2
Take the inverse Elzaki transform to find,
v
y (x , m + 1) = E −1 { E [ Am − B m ] } , y (x , 0) = 6e −3x (9)
2
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∂y (x , m )
Where Am = , B m = y (x , m ) are the projected differential transform of
∂x
∂y (x , t )
, y (x , t ) .
∂x
The standard Elzaki transform defines the solution y (x , t ) by the series
∞
y (x , t ) = ∑ y (x , m )
m =0
From equation (9) we find that: y (x , 0) = 6e −3x
A 0 = −18e −3x , B 0 = 6e −3x , y (x ,1) = E =1 −12v 3e −3x = −12te −3x
A1 = 36te −3x , B 1 = −12e −3x , y (x , 2) = E =1 24v 4e −3x = 12t 2e −3x
.................................................................. y (x ,3) = −8t 3e −3x
The solution in a series form is given by
y (x , t ) = 6e −3x − 12te −3x + 12t 2e −3x + ........ = 6e −3x e −2t = 6e − (3x + 2t )
Example 2.2:
Consider the following linear second order partial differential equation
u xx + u tt = 0 , u (x , 0) = 0, u t (x , 0) = cos x (10)
To find the solution we take Elzaki transform of equation (10) and making use of the conditions to find,
∂ 2u (x , m )
E [u (x , t ) ] = v cos x − v E [ A m ] ,
3 2
Am =
∂x 2
Take the inverse Elzaki transform we get:
u (x , m + 1) = −E −1 {v 2 E [ A m ]} , u (x , 0) = 0 , u (x ,1) = t cos x (11)
By using equation (11), we find that:
t3
A1 = −t cos x ⇒ u (x , 2) = E −1 {v 2 E [t cos x ]} =
cos x
3!
t3
t 3
t 5
A 2 = − cos x ⇒ u (x ,3) = E −1 v 2 E cos x = cos x
3!
3! 5!
.
.
.
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t3 t5
Then the solution is u (x , t ) = cos x t + + + ..... = cos x sinh t
3! 5!
Example 2.3:
Consider the following second order nonlinear partial differential equation
∂u ∂u ∂ 2u
2
= +u , u (x .0) = x 2 (12)
∂t ∂x ∂x 2
To find the solution take Elzaki transform of (12) and using the condition we get:
E [u (x , t ) ] = v 2 x 2 + vE [ A m + B m ]
h
∂u (x , m ) ∂u (x , h − m ) h
∂ 2u (h − m )
Where A m =∑ , B m = ∑ u (x , m ) , are projected
m =0 ∂x ∂x m =0 ∂x 2
∂u ∂ 2u
2
differential transform of , u
∂x ∂x 2
Take the inverse Elzaki transform to get:
u (x , m + 1) = E −1 {vE [ A m + B m ]} , u (x , 0) = x 2 (13)
From equation (13) we find that:
A 0 = 4x 2 , B 0 = 2x 2 ⇒ u (x ,1) = E −1 6x 2v 3 = 6x 2t
A1 = 48x 2t , B 1 = 24x 2t ⇒ u (x , 2) = E −1 72x 2v 4 = 36x 2t 2
.
.
.
Then the solution of equation (12) is
x2
u (x , t ) = x 2 + 6x 2t + 36x 2t 2 + ..... = x 2 (1 − 6t ) −1 =
1 − 6t
Which is an exactly the same solution obtained by the Adomian decomposition method.
Example 2.4:
Let us consider the nonlinear partial differential equation
∂u ∂u ∂ 2u x +1
2
= 2u +u 2 2 , u (x , 0) = (14)
∂t ∂x ∂x 2
Taking Elzaki transform of equation (14) and using the condition leads to:
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x +1
E [u (x , t ) ] = v 2 + vE [ 2A m + B m ]
2
Taking the inverse Elzaki transform to find:
x +1
u (x , m + 1) = E −1 {vE [ 2A m + B m ]} , u (x , 0) = (15)
2
h k
∂u (x , h − m ) ∂u (x , h − k )
A m = ∑ ∑ u (x , m )
k =0 m = 0 ∂x ∂x
Where
h k
∂ 2u (x , h − k )
B m = ∑ ∑ u (x , m ) u (x , h − m )
k =0 m =0 ∂x 2
From equation (15) we have:
x +1 x +1 3(x + 1) 2
u (x , 0) = , u (x ,1) = t , u (x , 2) = t ,...........
2 2 8
Then the solution of equation (14) is
x +1 1
x +1
(1 − t ) 2 =
−
u (x , t ) =
2 2 1−t
3. Conclusion
The solution of linear and nonlinear partial differential equations can be obtained using Elzaki transform
and projected differential transform method without any discretization of the variables. The results for all
examples can be obtained in series form, and all calculations in the method are very easy. This technique is
useful to solve linear and nonlinear partial differential equations.
References
[1] S. lslam, Yasir Khan, Naeem Faraz and Francis Austin (2010), Numerical Solution of Logistic
Differential Equations by using the Laplace Decomposition Method, World Applied Sciences Journal 8
(9):1100-1105.
[2] Nuran Guzel and Muhammet Nurulay (2008), Solution of Shiff Systems By using Differential
Transform Method, Dunlupinar universities Fen Bilimleri Enstitusu Dergisi, ISSN 1302-3055, PP. 49-59.
[3] Shin- Hsiang Chang , I. Ling Chang (2008), A new algorithm for calculating one – dimensional
differential transform of nonlinear functions, Applied Mathematics and Computation 195, 799-808.
[4] Tarig M. Elzaki (2011), The New Integral Transform “Elzaki Transform” Global Journal of Pure and
Applied Mathematics, ISSN 0973-1768, Number 1, pp. 57-64.
[5] Tarig M. Elzaki & Salih M. Elzaki (2011), Application of New Transform “Elzaki Transform” to Partial
Differential Equations, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768,Number 1, pp.
65-70.
[6] Tarig M. Elzaki & Salih M. Elzaki (2011), On the Connections Between Laplace and Elzaki transforms,
Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1, pp. 1-11.
[7] Tarig M. Elzaki & Salih M. Elzaki (2011), On the Elzaki Transform and Ordinary Differential Equation
With Variable Coefficients, Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 6,
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Number 1, pp. 13-18.
[8] Lokenath Debnath and D. Bhatta (2006). Integral transform and their Application second Edition,
Chapman & Hall /CRC.
[9] A.Kilicman and H.E.Gadain (2009). An application of double Laplace transform and Sumudu transform,
Lobachevskii J. Math.30 (3), pp.214-223.
[10] J. Zhang (2007), A Sumudu based algorithm m for solving differential equations, Comp. Sci. J.
Moldova 15(3), pp – 303-313.
[11] Hassan Eltayeb and Adem kilicman (2010), A Note on the Sumudu Transforms and
differential Equations, Applied Mathematical Sciences, VOL, 4, no.22,1089-1098
[12] Kilicman A. & H. ELtayeb (2010). A note on Integral transform and Partial Differential Equation,
Applied Mathematical Sciences, 4(3), PP.109-118.
[13] Hassan ELtayeh and Adem kilicman (2010), on Some Applications of a new Integral Transform, Int.
Journal of Math. Analysis, Vol, 4, no.3, 123-132.
[14] N.H. Sweilam, M.M. Khader (2009). Exact Solutions of some capled nonlinear partial differential
equations using the homotopy perturbation method. Computers and Mathematics with Applications 58
2134-2141.
[15] P.R. Sharma and Giriraj Methi (2011). Applications of Homotopy Perturbation method to Partial
differential equations. Asian Journal of Mathematics and Statistics 4 (3): 140-150.
[16] M.A. Jafari, A. Aminataei (2010). Improved Homotopy Perturbation Method. International
Mathematical Forum, 5, no, 32, 1567-1579.
[17] Jagdev Singh, Devendra, Sushila (2011). Homotopy Perturbation Sumudu Transform Method for
Nonlinear Equations. Adv. Theor. Appl. Mech., Vol. 4, no. 4, 165-175.
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Vol.2, No.4, 2012
Table 1. Elzaki transform of some Functions
f (t ) E [ f (t )] = T (u )
1
v2
t
v3
tn n ! v n +2
e at v2
1 − av
sinat
av 3
1 + a 2v 2
cosat
v2
1 + a 2v 2
59
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