Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
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
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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
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
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
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
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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
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
Adomian Decomposition Method for Solving the Nonlinear Heat EquationIJERA Editor
This paper studies the application of the Adomian Decomposition Method to find the exact and approximate solutions of the heat equation with power nonlinearity. First, the relevant literature is studied in understanding the importance and extent of applicability of the method in the applied science. The literature review has been incorporated in the introduction of the paper. The rest of the paper is divided in three further sections. The first part Adomian Decomposition Method provides a step-by-step guide of applying the method on any heat equation with nonlinearity. The second section is labeled as Applications. It considers two examples from the previous works of Pumak (2005) and Hetmaniok et al. (2010) to find the exact and approximate solutions of the equations respectively.
Analytical Solutions of simultaneous Linear Differential Equations in Chemica...IJMERJOURNAL
ABSTRACT: Analytical method for solving homogeneous linear differential equations in chemical kinetics and pharmacokinetics using homotopy perturbation method has been proposed. The mathematical model that depicts the pharmacokinetics is solved. Herein, we report the closed form of an analytical expression for concentrations species for all values of kinetic parameters. These results are compared with numerical results and are found to be in satisfactory agreement. The obtained results are valid for the whole solution domain.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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.
Galerkin’s indirect variational method in elastic stability analysis of all e...eSAT Publishing House
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
Adomian Decomposition Method for Solving the Nonlinear Heat EquationIJERA Editor
This paper studies the application of the Adomian Decomposition Method to find the exact and approximate solutions of the heat equation with power nonlinearity. First, the relevant literature is studied in understanding the importance and extent of applicability of the method in the applied science. The literature review has been incorporated in the introduction of the paper. The rest of the paper is divided in three further sections. The first part Adomian Decomposition Method provides a step-by-step guide of applying the method on any heat equation with nonlinearity. The second section is labeled as Applications. It considers two examples from the previous works of Pumak (2005) and Hetmaniok et al. (2010) to find the exact and approximate solutions of the equations respectively.
Analytical Solutions of simultaneous Linear Differential Equations in Chemica...IJMERJOURNAL
ABSTRACT: Analytical method for solving homogeneous linear differential equations in chemical kinetics and pharmacokinetics using homotopy perturbation method has been proposed. The mathematical model that depicts the pharmacokinetics is solved. Herein, we report the closed form of an analytical expression for concentrations species for all values of kinetic parameters. These results are compared with numerical results and are found to be in satisfactory agreement. The obtained results are valid for the whole solution domain.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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.
Galerkin’s indirect variational method in elastic stability analysis of all e...eSAT Publishing House
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
Building Positive Employment Relations in India: The Way Forwardchikatisrinu
There has been a persistent and pervasive incidence of poverty, unemployment and social exclusion and the resultant social turbulence all over the developing world even after their following neo-liberal policies. The employment relations and labor market institutions have been undergoing sweeping changes since last two decades all over the world and more particularly in the developing countries like India due to the ongoing globalization and the resultant hyper-competition, rapid technological and product innovations and the ICT revolution. Under the influence of new world order, the employers in India both pub-lic and private sector have resorted to massive restructuring of their businesses in order to gain competitive advantage and to survive and grow in the competitive global markets. This has also resulted in restructuring of their internal labor markets.
Whitepaper Mobile Solutions for the Education IndustryCygnet Infotech
The education industry finds itself on the tip of a revolution as conventional methods of learning and teaching are rapidly replaced by high-tech learning & training. Today, teachers and students are dynamic and technology savvy and they love to access learning resources from anywhere,anytime.This trend has redefined the entire training and learning process and the way universities, colleges and educational organizations deliver learning solutions.
The variational iteration method for calculating carbon dioxide absorbed into...iosrjce
In this paper, the variational iteration method (VIM) is used to get an approximate solution for a
system of two coupled nonlinear ordinary differential equations which represent the concentrations of carbon
dioxide 퐶푂2 and phenyl glycidyl ether. In this system there are boundary conditions of Dirichlet type and the
other is a mixed set of Neumann and Dirichlet type. Our calculations evidenced by tables and figures for the
analysis of the maximal error remainder values. The variational iteration method gives approximate solutions
with fast convergence. Comparison with the results obtained by the Adomian decomposition method (ADM)
reveals that the numerical solutions obtained by the VIM converge faster than those of Adomian's method. The
software we used in our study of these calculations is Mathematica®
9.
Efficient approximate analytical methods for nonlinear fuzzy boundary value ...IJECEIAES
This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
A Regularization Approach to the Reconciliation of Constrained Data SetsAlkis Vazacopoulos
A new iterative solution to the statistical adjustment of constrained data sets is derived in this paper. The method is general and may be applied to any weighted least squares problem containing nonlinear equality constraints. Other methods are available to solve this class of problem, but are complicated when unmeasured variables and model parameters are not all observable and the model constraints are not all independent. Of notable exception however are the methods of Crowe (1986) and Pai and Fisher (1988), although these implementations require the determination of a matrix projection at each iteration which may be computationally expensive. An alternative solution is proposed which makes the pragmatic assumption that the unmeasured variables and model parameters are known with a finite but equal uncertainty. We then re-formulate the well known data reconciliation solution in the absence of these unknowns to arrive at our new solution; hence the regularization approach. Another procedure for the classification of observable and redundant variables is also given which does not require the explicit computation of the matrix projection. The new algorithm is demonstrated using three illustrative examples previously used in other studies.
It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dynamics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed (envelope/general) solution of the multidimensional Clairaut equation. The equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given.
Sinc collocation linked with finite differences for Korteweg-de Vries Fraction...IJECEIAES
A novel numerical method is proposed for Korteweg-de Vries Fractional Equation. The fractional derivatives are described based on the Caputo sense. We construct the solution using different approach, that is based on using collocation techniques. The method combining a finite difference approach in the time-fractional direction, and the Sinc-Collocation in the space direction, where the derivatives are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Easy and economical implementation is the strength of this method.
Continuous models of economy adopted to either define evolutions of economic
systems or investigate economic dynamics, amongst its other areas of applications, are
known to be related to differential equations. The considered model in this article takes
the form of a second order non-linear ordinary differential equation (ODE) which is
conventional solved by reducing to the system of first order. This approach is
computationally tasking, unlike block methods which bypasses reduction by directly
solving the model. A new approach is introduced named Modified Taylor Series
Approach (MTSA). Hence, the resultant MTSA-derived block method is implemented to
solve the second order non-linear model of economy under consideration.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
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This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
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Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
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Securing your Kubernetes cluster_ a step-by-step guide to success !
Some numerical methods for Schnackenberg model
1. International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISBN: 2319-6491
Volume 2, Issue 2 (January 2013) PP: 71-78
www.ijeijournal.com P a g e | 71
Some numerical methods for Schnackenberg model
Saad A. Manaa
Department of Mathematics, Faculty of Science, University of Zakho,
Duhok, Kurdistan Region, Iraq
Abstract:- In this paper, Schnackenberg model has been solved numerically for finding an approximate solution
by Finite difference method and Adomain decomposition method. Example showed that ADM more accurate
than FDM and more efficient for this kind of problems as shown in tables (1,2) and figures (1-4).
Keywords:- Adomain Decomposition method, Partial Differential Equations, Schnackenberg equations.
I. INTRODUCTION
Many physical, chemical and engineering problems mathematically can be modeled in the form of
system of partial differential equations or system of ordinary differential equations. Finding the exact solution
for the above problems which involve partial differential equations is difficult in some cases. Here we have to
find the numerical solution of these problems using computers which came into existence [11].
For nonlinear partial differential equations, however, the linear superposition principle cannot be applied to
generate a new solution. So, because most solution methods for linear equations cannot be applied to nonlinear
equations, there is no general method of finding analytical solutions of nonlinear partial differential equation,
and numerical techniques are usually required for their solution[6].
1.1 MATHEMATICAL MODEL:
A general class of nonlinear-diffusion system is in the form
)1(
)(),(
)(),(
2222
1111
xgvufvbuavd
t
v
xgvufvbuaud
t
u
With homogenous dirchlet or neumann boundary condition on a bounded domain Ω , n≤3, with locally
lipschitz continuous boundary. It is well known that reaction and diffusion of chemical or biochemical species
can produce a variety of spatial patterns. This class of reaction diffusion systems includes some significant
pattern formation equations arising from the modeling of kinetics of chemical or biochemical reactions and from
the biological pattern formation theory. In this group, the following four systems are typically important and
serve as mathematical models in physical chemistry and in biology:
Brusselator model:
02,1,
2
,02,2,01),1(1 gagvufbbabba
where a and b are positive constants.
Gray-Scott model:
FggvufFbabkFa 2,01,
2
,2,02,01),(1
where F and k are positive constants.
Glycolysis model:
2,1,
2
,2,02,1,11 gpgvufkbakba
where k, p and are positive constants.
Schnackenberg model:
bgagvufbabka 2,1,
2
,0221,1
Where k, a and b are positive constants [14].
Then one obtains the following system of two nonlinearly coupled reaction-diffusion equations,
2. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 72
)2(
2
2
2
1
bvuvd
t
v
avukuud
t
u
With initial and boundary conditions:
𝑢 𝑡, 𝑥 = 𝑣 𝑡, 𝑥 = 0, 𝑡 > 0, 𝑥 ∈ 𝜕𝛺
𝑢 0, 𝑥 = 𝑢0 𝑥 , 𝑣 0, 𝑥 = 𝑣0(𝑥) 𝑥 ∈ 𝜕𝛺
(3)
And with Neumann boundary conditions:
)4(00 Lxandxat
x
v
x
u
Where Landbakdd ,,,2,1 are positive constants [14].
Reaction-diffusion (RD) systems arise frequently in the study of chemical and biological phenomena
and are naturally modeled by parabolic partial differential equations (PDEs). The dynamics of RD systems has
been the subject of intense research activity over the past decades. The reason is that RD system exhibit very
rich dynamic behavior including periodic and quasi-periodic solutions [4, 13].
Various orders are self-organized far from the chemical equilibrium. The theoretical procedures and
notions to describe the dynamics of patterns formation have been developed for the last three decades [10].
Attempts have also been made to understand morphological orders in biology [5]. Clarification of the
mechanisms of the formation of orders and the relationship among them has been one of the fundamental
problems in non-equilibrium statistical physics [9].
Various finite difference algorithms or schemes have been presented for the solution of hyperbolic-
parabolic problem or its simpler derivatives, such as the classical diffusion equation. It is well-known that many
of these schemes are partially unsatisfactory due to the formation of oscillations and numerical diffusion within
the solution [12].
Solution by the finite difference method, although more general, will involve stability and convergence
problems, may require special handling of boundary conditions, and may require large computer storage and
execution time. The problem of numerical dispersion for finite difference solutions is also difficult to overcome
[7].
Adomian decomposition has been applied to solve many functional equations so far. In this article, we
have used this method to solve the heat equation, which governs on numerous scientific and engineering
experimentations. Some special cases of the equation are solved as examples to illustrate ability and reliability
of the method. Restrictions on applying Adomian decomposition method for these equations are discussed [3].
The decomposition method can be an inactive procedure for analytical solution of a wide class of dynamical
systems without linearization or weak nonlinearity assumptions, closure approximations, perturbation theory, or
restrictive assumptions on stochasticitiy [1].
II. MATERIALS AND METHODS
2.1 FINITE DIFFERENCE APPROXIMATIONS
The finite difference Scheme, generally reduces a linear, nonlinear partial differential equations into
system of linear, nonlinear equations and various methods were developed to find the numerical solution and
acceleration the convergence [8]. Assume that the rectangle }0,0:),{( btaxtxR is subdivided
into n-1 by m-1 rectangle with sides hx and kt . Start at the bottom row, where 𝑡 = 𝑡1 = 0, and the
solution is )()1,( ixftixu . The grid spacing is uniform in every row: 𝑥𝑖+1 = 𝑥𝑖 + ℎ and (𝑥𝑖−1 = 𝑥𝑖 − ℎ), and
it is uniform in every column: 𝑡𝑗+1 = 𝑡𝑗 + 𝑘 and (𝑡𝑗−1 = 𝑡𝑗 − 𝑘). And use the approximation jiu , for ),( ji txu
to obtain [8].
𝜕𝑢
𝜕𝑡
=
𝑢 𝑖,𝑗+1−𝑢 𝑖,𝑗
∆𝑡
(5)
𝜕𝑣
𝜕𝑡
=
𝑣 𝑖,𝑗+1−𝑣 𝑖,𝑗
∆𝑡
(6)
𝜕2 𝑢
𝜕𝑥2 =
𝑢 𝑖−1,𝑗 −2𝑢 𝑖,𝑗 +𝑢 𝑖+1,𝑗
(∆𝑥)2 (7)
𝜕2 𝑣
𝜕𝑥2 =
𝑣 𝑖−1,𝑗 −2𝑣 𝑖,𝑗 +𝑣 𝑖+1,𝑗
(∆𝑥)2 (8)
Substitute (5) – (8) in the Schnackenberg model (1) to get
3. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 73
bjivjiu
x
jivjivjiv
d
t
jivjiv
ajivjiujiku
x
jiujiujiu
d
t
jiujiu
,
2
),(
2)(
,1,2,1
2
,1,
,
2
),(,2)(
,1,2,1
1
,1,
tbjivjiutjivjivjiv
x
td
jivjiv
tajivjiutjitkujiujiujiu
x
td
jiujiu
,
2),(],1,2,1[
2
)(
2
,1,
,
2),(,],1,2,1[
2)(
1
,1,
Let
2)(
1
1
x
td
r
and
2)(
2
2
x
td
r
then
tbjivjiutjivrjivrjivrjivjiv
tajivjiutjitkujiurjiurjiurjiujiu
,
2
),(,12,22,12,1,
,
2
),(,,11,12,11,1,
)9(
,
2),(],1,1[2,]221[1,
,
2),(],1,1[1,]121[1,
tbjivjiutjivjivrjivrjiv
tajivjiutjiujiurjiutkrjiu
From the boundary conditions (4) we have
𝜕𝑢
𝜕𝑥
=
𝑢 𝑖,𝑗 −𝑢 𝑖−1,𝑗
2∆𝑥
= 0 , 𝑠𝑜 𝑢𝑖−1,𝑗 = 𝑢𝑖,𝑗 𝑎𝑛𝑑 𝑢1,𝑗 = 𝑢2,𝑗
And
𝜕𝑢
𝜕𝑥
=
𝑢 𝑖+1,𝑗 −𝑢 𝑖,𝑗
2∆𝑥
= 0 , 𝑠𝑜 𝑢𝑖+1,𝑗 = 𝑢𝑖,𝑗 𝑎𝑛𝑑 𝑢11,𝑗 = 𝑢10,𝑗
And also for v
𝜕𝑣
𝜕𝑥
=
𝑣 𝑖,𝑗 −𝑣 𝑖−1,𝑗
2∆𝑥
= 0 , 𝑠𝑜 𝑣𝑖−1,𝑗 = 𝑣𝑖,𝑗 𝑎𝑛𝑑 𝑣1,𝑗 = 𝑣2,𝑗
And
𝜕𝑣
𝜕𝑥
=
𝑣 𝑖+1,𝑗 −𝑣 𝑖,𝑗
2∆𝑥
= 0 , 𝑠𝑜 𝑣𝑖+1,𝑗 = 𝑣𝑖,𝑗 𝑎𝑛𝑑 𝑣11,𝑗 = 𝑣10,𝑗
And from the initial condition:
)(01,111,101,91,81,71,61,51,41,31,21,1
)(01,111,101,91,81,71,61,51,41,31,21,1
xvvvvvvvvvvvv
xuuuuuuuuuuuu
The result equation (8) is the finite difference method for the Schnackenber model.
2.2 ADOMIAN DECOMPOSITION METHOD
Nonlinear differential equations are usually arising from mathematical modeling of many frontier
physical systems. In most cases, analytic solutions of these differential equations are very difficult to achieve.
Common analytic procedures linearize the system or assume the nonlinearities are relatively insignificant. Such
procedures change the actual problem to make it tractable by the conventional methods. This changes, some
times seriously, the solution. The above drawbacks of linearization and numerical methods arise the need to
search for alternative techniques to solve the nonlinear differential equations, namely, the analytic solution
methods, such as the perturbation method, the iteration variational method [11–14] and the Adomian
decomposition method.
The decomposition method was first introduced by Adomian since the beginning of the 1980s.The
Adomian decomposition method (ADM) is used to solve a wide range of physical problems. This method
provides a direct scheme for solving linear and nonlinear deterministic and stochastic equations without the need
for linearization this yields convergent series solutions rapidly. An advantage of this method is that, it can
provide analytical approximation or an approximated solution to a rather wide class of nonlinear (and stochastic)
equations without linearization, perturbation, closure approximation, or discretization methods. Unlike the
common methods, i.e., weak nonlinearity and small perturbation which change the physics of the problem due
to simplification, ADM gives the approximated solution of the problem without any simplification. Thus, its
results are more realistic [1,2].
We define the operator
t
tL
t
dttL
0
(.)1
and
2
2
x
xxL
then
4. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 74
system (1) can be written as:
)10(
2
1 avukuuxxLdutL
)11(
2
2 bvuvxxLdvtL
By applying the inverse of operator tL on (10,11) we get:
aLvuLukLuLLdxuxtu tttxxt
12111
1 )()(),0(),(
bLvuLvLLdxvxtv ttxxt
1211
2 )()(),0(),(
By initial conditions in system (3) then (10,11) can be written as:
atLvutLutkLuxxLtLdxuxtu
1
)
2
(
11
)(
1
1)(0),(
btLvutLvxxLtLdxvxtv
1
)
2
(
1
)(
1
2)(0),(
By using Adomian decomposition method :
0
),(),(
n
xtnuxtu and
0
),(),(
n
xtnvxtv
)12(
1
0
1
0
1
)
0
(
1
1)(0
0
),( atL
n
nAtL
n
nutkL
n
nuxxLtLdxu
n
xtnu
)13(
1
0
1
)
0
(
1
2)(0
0
),( btL
n
nBtL
n
nvxxLtLdxv
n
xtnv
Where nA and nA are Adomian polynomials. But nA = nB because both non-linear terms are vu2
Where
00
)(
!
1
n
i
iu
i
Fn
d
n
d
n
nA
But here
00
)
0
,(
!
1
n
i
n
i
iv
i
iu
i
Fn
d
n
d
n
nA because non-linear term have two functions ),( xtu and ),( xtv
then by equation (12):
0111)(1
11
)(00
katLkAtLkutkLkuxxLtLdku
xuu
By equation (13):
011)(1
21
)(00
kbtLkBtLkvxxLtLdkv
xvv
)14(0
2
0)0,0(
0
0
0
)
0
0
,(
0
0
!0
1
0
BvuvuF
i i
ivi
iuiF
d
d
A
0k
)15(11
0
2
00011:
]0
2
0001[0
2
0001
1)0
2
0(1
0
1)0(1
1
1
0
1
0
1)0(1
11
tUu
avukuuxxLdULet
tavukuuxxLdattvutkutuxxLd
atLvutLutkLuxxLtLdatLAtLutkLuxxLtLdu
)16(11
0
2
0021:
]0
2
002[0
2
002
1)0
2
0(1)0(1
2
1
0
1)0(1
21
tVv
bvuvxxLdVLet
tbvuvxxLdbttvutvxxLd
btLvutLvxxLtLdbtLBtLvxxLtLdv
0)]10,10([
0
1
0
)
1
0
,(
1
1
!1
1
1
vvuuF
d
d
i i
iv
i
iu
i
F
d
d
A
5. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 75
0)]10)(
2
1
2
102
2
0[(0)]10(
2
)10[(
vvuuuu
d
d
vvuu
d
d
0]1
2
1
3
110
2
21
2
00
2
1
2
01020
2
0[
vuvuuvuvuvuuvu
d
d
tVuvtUuvuvuu 1
2
00)1(021
2
00102
)17(1]1
2
01002[1 BtVuUvuA
K=1
)]1
2
01002([
1
)1(
1
)]1([
1
1
1
1
1
1
1
)1(
1
12 tVuUvutLtUtkLtUxxLtLdatLAtLutkLuxxLtLdu
at
t
VuUvukUUxxLd
t
VuUvu
t
kU
t
UxxLd
2
2
]12
010021)1(1[
2
2
]12
01002[
2
2
1
2
2
)1(1
1
2
010021)1(12: VuUvukUUxxLdULet
)18(
2
2
22 at
t
Uu
1
1
)11(
1
)1(
1
22 BtLvtFLvxxLtLdv
)]1
2
01002([
1
)11(
1
)]1([
1
2 tVuUvutLtVtFLtVxxLtLd
2
2
]12
01002[)
2
2
1(
2
2
)1(2
t
VuUvu
t
VtF
t
VxxLd
bt
t
VuUvuFVVxxLdFtv
2
2
]12
010021)1(2[2
)19(
2
2
22
12
010021)1(22:
bt
t
VFtv
VuUvuFVVxxLdVLet
0)]2
2
10,2
2
10([
2
2
!2
1
0
2
0
)
2
0
,(
2
2
!2
1
2
vvvuuuF
d
d
i i
iv
i
iu
i
F
d
d
A
0)]2
2
10(
2
)2
2
10[(
2
2
!2
1
vvvuuu
d
d
0)]2
2
10](
2
)2
2
1()2
2
1(02
2
0[[
2
2
!2
1
vvvuuuuuu
d
d
0)]2
2
10)(
2
2
4
21
3
2
2
1
2
20
2
2102
2
0[(
2
2
!2
1
vvvuuuuuuuuu
d
d
0
2
2
4
021
3
20
2
1
2
020
2
201020
2
0[
2
2
!2
1
vuvuuvuvuuvuuvu
d
d
1
2
2
5
121
4
21
2
1
3
120
3
2110
2
21
2
0 vuvuuvuvuuvuuvu
0]2
2
2
6
221
5
22
2
1
4
220
4
2210
3
22
2
0
2
vuvuuvuvuuvuuvu
)20(0
2
12
2
011020202)2
2
0211040
2
120204(
2
1
vuvuvuuvuuvuvuuvuvuu
0
2)1()
2
2
2(2
0)1)(1(020)
2
2
2(022 vattU
t
VFtubttVattUuvat
t
UuA
taubtutvU
t
VutVUutUvuA )022
0(2
0
2)1(
2
2
22
0
2110222002
(20)and(19),(17),(16)eqsBy
)21(2)02
2
0(
2
]0
2
)1(2
2
0
2
1
1102200[2 BtaubtutvUVuVUuUvuA
6. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 76
atLAtLutkLuxxLtLdu
k
1
2
1
2
1)2(1
13
2
By equations (18) and (21)
)22()
2
2
)02
2
0(
3
3
]0
2
)1(2
2
0
2
1
11022002
2
1
)2(1
2
1
[3
3
3
]0
2
)1(2
2
0
2
1
1102200[
3
3
2
2
1
3
3
)2(1
2
1
)
2
]0
2
)1(
2
2
0
2
1
1102200([
1
)
2
2
2(
1
)]
2
2
2([
1
13
at
t
aubua
t
vUVuVUuUvukUUxxLdu
t
vUVuVUuUvu
t
kU
t
UxxLd
tvU
VuVUuUvutL
t
UtkL
t
UxxLtLdu
)23(
2
2
)02
2
0(
3
3
]0
2
)1(2
2
0
2
1
1102200)2(2
2
1
[3
2
2
)022
0(
3
3
]0
2)1(22
0
2
1
1102200[
3
3
)2(2
2
1
))022
0)2]0
2)1(22
0
2
1
1102200([1)]
2
2
2([1
23
(21)and(19)eqsBy
1
2
1)2(1
23
bt
t
aubu
t
vUVuVUuUvuVxxLdv
bt
t
aubu
t
vUVuVUuUvu
t
VxxLd
bttaubutvUVuVUuUvutL
t
VxxLtLdv
btLBtLvxxLtLdv
III. APPLICATION (NUMERICAL EXAMPLE)
We solved the following example numerically to illustrate efficiency of the presented methods.
∂u
∂t
= d1∆u − ku + u2
v + a , t > 0, 𝑥 ∈ 𝛺
𝜕𝑣
𝜕𝑡
= 𝑑1∆𝑣 − 𝑢2
𝑣 + 𝑏 , 𝑡 > 0, 𝑥 ∈ 𝛺
We the initial conditions
u(x, 0) = Us + 0.01 sin(x/ L) for 0 ≤x ≤ L
v (x, 0) = Vs – 0.12 sin(x/ L) for 0 ≤x ≤ L
u(0, t) = Us , u(L, t) = Us and v (0, t) = Vs, v (L, t) = Vs
We will take
d1=d2=0.01 , a=b= 0.09 , k=-0.004, Us=0, Vs=1
IV. FIGURES AND TABLES
Table 1 Comparison between the FDM and ADM for the values of concentration V.
t = 1 t=2 t=3
x ADM FDM ADM FDM ADM FDM
0 1.0001 1.0000 1.0001 1.0000 1.0035 1.0000
0.1 1.0014 1.0008 1.0022 0.9993 1.0048 0.9971
0.2 1.0020 1.0010 1.0022 0.9976 1.0043 0.9864
0.3 1.0021 1.0019 1.0012 0.9946 1.0021 0.9787
0.4 1.0018 1.0012 0.9994 0.9906 0.9984 0.9704
0.5 1.0011 1.0005 0.9971 0.9861 0.9936 0.9624
0.6 1.0003 0.9996 0.9945 0.9817 0.9881 0.9553
0.7 0.9995 0.9982 0.9921 0.9777 0.9826 0.9499
0.8 0.9988 0.9968 0.9900 0.9746 0.9781 0.9464
0.9 0.9984 0.9954 0.9887 0.9726 0.9750 0.9452
1.0 0.9982 0.9942 0.9882 0.9719 0.9739 0.9464
1.1 0.9984 0.9954 0.9887 0.9726 0.9750 0.9499
7. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 77
1.2 0.9988 0.9968 0.9900 0.9746 0.9781 0.9553
1.3 0.9995 0.9982 0.9921 0.9777 0.9827 0.9624
1.4 1.0003 0.9996 0.9945 0.9817 0.9881 0.9704
1.5 1.0011 1.0005 0.9971 0.9861 0.9936 0.9787
1.6 1.0018 1.0012 0.9994 0.9906 0.9984 0.9864
1.7 1.0021 1.0019 1.0012 0.9946 1.0021 0.9927
1.8 1.0020 1.0010 1.0022 0.9976 1.0043 0.9971
1.9 1.0014 1.0008 1.0022 0.9993 1.0048 0.9993
2.0 1.0001 1.0000 1.0010 1.0000 1.0035 1.0000
Table 2 Comparison between the FDM and ADM for the values of concentration U.
t =1 t=2 t=3
x ADM FDM ADM FDM ADM FDM
0 0.0015 0 0.0048 0 0.0081 0
0.1 -0.0150 -0.0232 -0.0102 -0.0276 -0.0108 -0.0319
0.2 -0.0306 -0.0456 -0.0241 -0.0539 -0.0221 -0.0623
0.3 -0.0450 -0.0664 -0.0367 -0.0783 -0.0316 -0.0902
0.4 -0.0579 -0.0851 -0.0476 -0.0999 -0.0390 -0.1149
0.5 -0.0692 -0.1014 -0.0568 0.1185 -0.0444 -0.1359
0.6 -0.0786 -0.1151 -0.0643 -0.1339 -0.0480 -0.1532
0.7 -0.0861 -0.1258 -0.0700 -0.1459 -0.0502 -0.1666
0.8 -0.0915 -0.1336 -0.0741 -0.1545 -0.0514 -0.1761
0.9 -0.0948 -0.1383 -0.0766 -0.1596 -0.0520 -0.1818
1.0 -0.0959 -0.1398 -0.0776 -0.1613 -0.0522 -0.1837
1.1 -0.0949 -0.1383 -0.0770 -0.1596 -0.0523 -0.1818
1.2 -0.0916 -0.1336 -0.0749 -0.1545 -0.0519 -0.1761
1.3 -0.0863 -0.1258 -0.0710 -0.1459 -0.0509 -0.1666
1.4 -0.0788 -0.1151 -0.0654 -0.1339 -0.0488 -0.1532
1.5 -0.0693 --0.1014 -0.0578 0.1185 -0.0451 -0.1359
1.6 -0.0580 -0.0851 -0.0484 0.1185 -0.0395 -0.1149
1.7 -0.0450 -0.0664 -0.0372 -0.0999 -0.0319 -0.0902
1.8 -0.0306 -0.0456 -0.0244 -0.0783 -0.0223 -0.0623
1.9 -0.0150 -0.0232 -0.0103 -0.0539 -0.0108 -0.0319
2.0 0.0015 -0.0000 0.0048 -0.0000 0.0023 -0.0000
Fig. 1 ADM for the values of concentration V Fig. 2 FDM for the values of concentration V
with 0<x<2 and 0<t<3 with 0<x<2 and 0<t<3
Fig. 3 ADM for the values of concentration U Fig. 4 FDM for the values of concentration U
with 0<x<2 and 0<t<3 with 0<x<2 and 0<t<3
8. Some numerical methods for Schnackenberg model
www.ijeijournal.com P a g e | 78
V. CONCLUSION
The Schnackenberg model solved Numerically using finite difference method and Adomain
decomposition method, and we found that’s finite difference method is earlier that ADM but ADM is more
accurate than FDM and more efficient as show in tables ( 1-2 ) and figures (1-4 ).
ACKNOWLEDGEMENTS
My deep feeling of gratitude and immense indebtedness to my wife for her continuous support and
encouragement on this work.
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