scholarly journals Numerical Investigation of Fractional-Order Swift–Hohenberg Equations via a Novel Transform

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1263
Author(s):  
Kamsing Nonlaopon ◽  
Abdullah M. Alsharif ◽  
Ahmed M. Zidan ◽  
Adnan Khan ◽  
Yasser S. Hamed ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Investigation ◽  
Fractional Order ◽  
Thermal Convection ◽  
Formation Process ◽  
Decomposition Method ◽  
Numerical Examples

In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

scholarly journals Solving Some Derivative Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method

2021 ◽  
Vol 13 (2) ◽  
pp. 101
Author(s):  
Abdoul wassiha NEBIE ◽  
Frederic BERE ◽  
Bakari ABBO ◽  
Youssouf PARE
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Adomian Decomposition ◽  
Diffusion Convection

In this paper, we propose the solution of some nonlinear partial differential equations of  fractional order that modeled diffusion, convection and reaction problems. For the solution of these equations we will use the SBA method which is a method based on the combination of the Adomian Decomposition Method (ADM), the Picard's principle  and the method of successive approximations.   

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 An accelerated solution for some classes of nonlinear partial differential equations

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
I. L. El-Kalla ◽  
E. M. Mohamed ◽  
H. A. A. El-Saka
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Numerical Examples ◽  
Adomian Decomposition

AbstractIn this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.

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 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.

scholarly journals New Iterative Method: An Application for Solving Fractional Physical Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Fractional Order ◽  
Small Perturbation ◽  
The Other ◽  
Numerical Examples ◽  
Linear And Nonlinear ◽  
The Given ◽  
New Iterative Method

The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which con…firm the power, accuracy, and simplicity of the given method compared with some of the other methods.

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.

Sign in / Sign up

Export Citation Format

Share Document