scholarly journals Analytical Analysis of Fractional-Order Multi-Dimensional Dispersive Partial Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 939
Author(s):  
Shuang-Shuang Zhou ◽  
Mounirah Areshi ◽  
Praveen Agarwal ◽  
Nehad Ali Shah ◽  
Jae Dong Chung ◽  
...  
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Suggested Technique ◽  
In Series ◽  
Fractional Orders

In this paper, a novel technique called the Elzaki decomposition method has been using to solve fractional-order multi-dimensional dispersive partial differential equations. Elzaki decomposition method results for both integer and fractional orders are achieved in series form, providing a higher convergence rate to the suggested technique. Illustrative problems are defined to confirm the validity of the current technique. It is also researched that the conclusions of the fractional-order are convergent to an integer-order result. Moreover, the proposed method results are compared with the exact solution of the problems, which has confirmed that approximate solutions are convergent to the exact solution of each problem as the terms of the series increase. The accuracy of the method is examined with the help of some examples. It is shown that the proposed method is found to be reliable, efficient and easy to use for various related problems of applied science.

scholarly journals Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 335 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Muhammad Arif ◽  
Poom Kumam
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Third Order ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
In Series

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

scholarly journals Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 532 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Poom Kumam ◽  
Muhammad Arif ◽  
Dumitru Baleanu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Present Article ◽  
Fractional Order ◽  
Present Method ◽  
Decomposition Method ◽  
Proportional Delay ◽  
Partial Differential ◽  
Analytical Technique ◽  
In Series

In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. Therefore, Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear Partial deferential equations particularly fractional-order partial differential equations with proportional delay.

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

2014 ◽  
Vol 6 (01) ◽  
pp. 107-119 ◽  
Author(s):  
D. B. Dhaigude ◽  
Gunvant A. Birajdar
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Discrete Schrödinger Equation ◽  
Adomian Decomposition

AbstractIn this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution whenα= 1.

scholarly journals Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Iteration Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Variational Iteration

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

scholarly journals Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan Eltayeb ◽  
Adem Kılıçman
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Adomian Decomposition ◽  
Sumudu Transform

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

scholarly journals Haar wavelet method for solving coupled system of fractional order partial differential equations

2021 ◽  
Vol 21 (3) ◽  
pp. 1444
Author(s):  
Abbas AL-Shimmary ◽  
Sajeda Kareem Radhi ◽  
Amina Kassim Hussain
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Coupled System ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Coupled Systems ◽  
Operational Matrices ◽  
Orthogonal Matrices ◽  
Partial Differential

<p><span><span><br /></span></span></p><p><span id="docs-internal-guid-231c3b14-7fff-77a8-4252-ec4aa31140d7"><span>This paper deal with the numerical method, based on the operational matrices of the Haar wavelet orthonormal functions approach to approximate solutions to a class of coupled systems of time-fractional order partial differential equations (FPDEs.). By introducing the fractional derivative of the Caputo sense, to avoid the tedious calculations and to promote the study of wavelets to beginners, we use the integration property of this method with the aid of the aforesaid orthogonal matrices which convert the coupled system under some consideration into an easily algebraic system of Lyapunov or Sylvester equation type. The advantage of the present method, including the simple computation, computer-oriented, which requires less space to store, time-efficient, and it can be applied for solving integer (fractional) order partial differential equations. Some specific and illustrating examples have been given; figures are used to show the efficiency, as well as the accuracy of the, achieved approximated results. All numerical calculations in this paper have been carried out with MATLAB.</span></span></p>

scholarly journals Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Ordinary and Partial Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Ahmed Farooq Qasim ◽  
Ekhlass S. AL-Rawi
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Productive System ◽  
Linear And Nonlinear

In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.

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