scholarly journals The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 987 ◽  
Author(s):  
A. A. Alderremy ◽  
Hassan Khan ◽  
Rasool Shah ◽  
Shaban Aly ◽  
Dumitru Baleanu
Keyword(s):  
Mathematical Modeling ◽  
Analytical Solution ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Analytical Techniques ◽  
Whitham Equations ◽  
Suggested Technique ◽  
Approximate Analytical Solutions ◽  
Physical Phenomena

This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

scholarly journals APPLICATION OF ABOODH TRANSFORM ON SOME FRACTIONAL ORDER MATHEMATICAL MODELS

2020 ◽  
Vol 20 (3) ◽  
pp. 661-672
Author(s):  
JAWARIA TARIQ ◽  
JAMSHAD AHMAD
Keyword(s):  
Mathematical Models ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Iteration Method ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Physical Phenomena ◽  
Convergent Solution

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo’s sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order.

scholarly journals Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 597 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Wave Equations ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
High Rate ◽  
Numerical Examples ◽  
Linear And Nonlinear

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

scholarly journals An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 426 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Poom Kumam ◽  
Dumitru Baleanu ◽  
Muhammad Arif
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytical Techniques ◽  
Telegraph Equation ◽  
High Rate ◽  
Analytical Technique ◽  
Decomposition Procedure ◽  
Adomian Decomposition ◽  
Telegraph Equations

In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.

scholarly journals Solution of Fractional Autonomous Ordinary Differential Equations

2021 ◽  
Author(s):  
Rami AlAhmad ◽  
Qusai AlAhmad ◽  
Ahmad Abdelhadi
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Singular Kernel ◽  
Autonomous Differential Equations ◽  
Implicit Analytical Solution

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

scholarly journals Approximate analytical solutions of non-linear local fractional heat equations

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 837-841 ◽  
Author(s):  
Shuxian Deng
Keyword(s):  
Analytical Solutions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Heat Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractional Heat Equation ◽  
Approximate Analytical Solutions ◽  
Adomian Decomposition ◽  
Non Linear

Consider the non-linear local fractional heat equation. The fractional complex transform method and the Adomian decomposition method are used to solve the equation. The approximate analytical solutions are obtained.

Exact Analytical Solution of Fractional Relaxation-Oscillation Equation by Adomian Decomposition and He’s Variational Technique

2009 ◽  
Author(s):  
K. C. Basak ◽  
P. C. Ray ◽  
R. K. Bera
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Order ◽  
Decomposition Method ◽  
Exact Analytical Solution ◽  
Relaxation Oscillation ◽  
Variational Technique ◽  
Adomian Decomposition ◽  
Fractional Relaxation ◽  
Oscillation Equation

Exact solution of linear fractional relaxation-oscillation equation is obtained by the decomposition method of Adomian and also by He’s variational method for fractional order α, for 1 < α ≤ 2. Surface plots of the above solution are drawn for different values of fractional order α and time t. Amplitude of the oscillation increases with α but it decreases as time increases.

scholarly journals Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

2019 ◽  
Vol 10 (1) ◽  
pp. 122 ◽  
Author(s):  
Hassan Khan ◽  
Umar Farooq ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
...  
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytical Techniques ◽  
Form Solution ◽  
Physical Models ◽  
Fractional Partial Differential Equations ◽  
Analytical Technique ◽  
Adomian Decomposition ◽  
Closed Contact

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

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