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 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 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 A New Analytical Technique to Solve System of Fractional-Order Partial Differential Equations

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 150037-150050 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Umar Farooq ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Partial Differential ◽  
Analytical Technique

scholarly journals Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method

Electronics ◽  
2019 ◽  
Vol 8 (9) ◽  
pp. 1015 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Applied Sciences ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Telegraph Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Present Technique ◽  
Telegraph Equations

In the current paper, fractional-order hyperbolic telegraph equations are considered for analytical solutions, using the decomposition method based on natural transformation. The fractional derivative is defined by the Caputo operator. The present technique is implemented for both fractional- and integer-order equations, showing that the current technique is an accurate analytical instrument for the solution of partial differential equations of fractional-order arising in all branches of applied sciences. For this purpose, several examples related to hyperbolic telegraph models are presented to explain the procedure of the suggested method. It is noted that the procedure of the present technique is simple, straightforward, accurate, and found to be a better mathematical technique to solve non-linear fractional partial differential equations.

Invariant solutions of fractional-order spatio-temporal partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nkosingiphile Mnguni ◽  
Sameerah Jamal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Growth Models ◽  
Spatial Patterning ◽  
Partial Differential ◽  
Symmetry Approach ◽  
Time Component ◽  
Spatio Temporal ◽  
Lie Symmetry Approach

Abstract This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.

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