scholarly journals An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Mohammad Alaroud ◽  
Nedal Tahat ◽  
Shrideh Al-Omari ◽  
D. L. Suthar ◽  
Selma Gulyaz-Ozyurt
Keyword(s):  
Approximate Solutions ◽  
Caputo Fractional Derivatives ◽  
Approximate Analytical Solutions ◽  
The Laplace Transform ◽  
Fractional Models ◽  
Residual Power Series ◽  
Linear And Nonlinear ◽  
The Impact ◽  
Generalized Taylor’S Formula ◽  
Residual Power

Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

scholarly journals A Novel Attractive Algorithm for Handling Systems of Fractional Partial Differential Equations

2021 ◽  
Vol 20 ◽  
pp. 524-539
Author(s):  
Mohammad Alaroud ◽  
Yousef Al-Qudah
Keyword(s):  
Fractional Derivatives ◽  
Applied Mathematics ◽  
Convergent Series ◽  
Initial Value ◽  
Caputo Fractional Derivatives ◽  
Approximate Analytical Solutions ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
Analysis Of Results ◽  
Residual Power

The purpose of this work is to provide and analyzed the approximate analytical solutions for certain systems of fractional initial value problems (FIVPs) under the time-Caputo fractional derivatives by means of a novel attractive algorithm, called the Laplace residual power series (LRPS) algorithm. It combines the Laplace transform operator and the RPS algorithm. The proposed algorithm produces the fractional series solutions in the Laplace space based upon basically on the limit concept and then transforming bake them to original spaces to get a rapidly convergent series approximate solution. To validate the efficiency, accuracy, and applicability of the proposed algorithm, two illustrative examples are performed. Obtained solutions are simulated graphically and numerically. The analysis of results reached shows that the proposed algorithm is applicable, effective, and very fast in determining the solutions for many fractional problems arising in the various areas of applied mathematics

scholarly journals Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems

2021 ◽  
Vol 10 (1) ◽  
pp. 282-292
Author(s):  
Marwan Alquran ◽  
Maysa Alsukhour ◽  
Mohammed Ali ◽  
Imad Jaradat
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Laplace Transform ◽  
Iterative Algorithm ◽  
Autonomous Systems ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

Abstract In this work, a new iterative algorithm is presented to solve autonomous n-dimensional fractional nonlinear systems analytically. The suggested scheme is combination of two methods; the Laplace transform and the residual power series. The methodology of this algorithm is presented in details. For the accuracy and effectiveness purposes, two numerical examples are discussed. Finally, the impact of the fractional order acting on these autonomous systems is investigated using graphs and tables.

scholarly journals Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2868
Author(s):  
Hussam Aljarrah ◽  
Mohammad Alaroud ◽  
Anuar Ishak ◽  
Maslina Darus
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

scholarly journals Solving Boundary-Layer Problems by Residual-Power-Series Method

2016 ◽  
Vol 8 (3) ◽  
pp. 68
Author(s):  
Mohd Taib Shatnawi
Keyword(s):  
Boundary Layer ◽  
Power Series ◽  
Nonlinear Boundary ◽  
Power Series Method ◽  
Boundary Layer Equations ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Linear And Nonlinear ◽  
Boundary Layer Problems ◽  
Residual Power

<p><span lang="EN-US">In this paper, the so-called residual-power-series (RPS) method is presented for solving nonlinear boundary-layer equations. The RPS method provides a single unified treatment for the linear and nonlinear terms in the equations. The accuracy and efficiency of the RPS method is demonstrated for both a single and a system of two coupled boundary-layer equations on an unbounded domain.</span></p>

scholarly journals Approximate Solutions of Two-Dimensional Burgers' and Coupled Burgers' Equations by Residual Power Series Method

2018 ◽  
Vol 22 ◽  
pp. 01044
Author(s):  
Selahattin Gulsen ◽  
Mustafa Inc ◽  
Harivan R. Nabi
Keyword(s):  
Power Series ◽  
Taylor Series Expansion ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Series Method ◽  
Power Series Method ◽  
Burgers Equations ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

In this study, two-dimensional Burgers' and coupled Burgers' equations are examined by the residual power series method. This method provides series solutions which are rapidly convergent and their components are easily calculable by Mathematica. When the solution is polynomial, the method gives the exact solution using Taylor series expansion. The results display that the method is more efficient, applicable and accuracy and the graphical consequences clearly present the reliability of the method.

scholarly journals Fractional residual power series method for the analytical and approximate studies of fractional physical phenomena

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 799-805
Author(s):  
Gamal Mohamed Ismail ◽  
Hamdy Ragab Abdl-Rahim ◽  
Hijaz Ahmad ◽  
Yu-Ming Chu
Keyword(s):  
Power Series ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Series Method ◽  
Fractional Partial Differential Equation ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Physical Phenomena ◽  
Residual Power

AbstractIn this article, analytical exact and approximate solutions for fractional physical equations are obtained successfully via efficient analytical method called fractional residual power series method (FRPSM). The fractional derivatives are described in the Caputo sense. Three applications are discussed, showing the validity, accuracy and efficiency of the present method. The solution via FRPSM shows excellent agreement in comparison with the solutions obtained from other established methods. Also, the FRPSM can be used to solve other nonlinear fractional partial differential equation problems. The final results are presented in graphs and tables, which show the effectiveness, quality and strength of the presented method.

scholarly journals Approximate solutions and conservation laws of the periodic base temperature of convective longitudinal fins in thermal conductivity

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 267-273
Author(s):  
Isa Aliyu ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Dumitru Baleanu
Keyword(s):  
Thermal Conductivity ◽  
Power Series ◽  
Linear Models ◽  
Approximate Solutions ◽  
Base Temperature ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

In this paper, the residual power series method is used to study the numerical approximations of a model of oscillating base temperature processes occurring in a convective rectangular fin with variable thermal conductivity. It is shown that the residual power series method is efficient for examining numerical behavior of non-linear models. Further, the conservation of heat is studied using the multiplier method.

scholarly journals Residual power series method for solving nonlinear reaction-diffusion-convection problems

2021 ◽  
Vol 39 (3) ◽  
pp. 177-188
Author(s):  
Maisa Khader ◽  
Mahmoud H. DarAssi
Keyword(s):  
Power Series ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Nonlinear Reaction ◽  
Residual Power Series ◽  
Diffusion Convection ◽  
Residual Power

In this paper, the residual power series method (RPSM) is applied to one of the most frequently used models in engineering and science, a nonlinear reaction diffusion convection initial value problems. The approximate solutions using the RPSM were compared to the exact solutions and to the approximate solutions using the homotopy analysis method.

scholarly journals Residual Power Series Method for Solving Klein-Gordon Schrödinger Equation

2021 ◽  
Vol 9 (2) ◽  
pp. 123-127
Author(s):  
Ssaad A. Manaa ◽  
Fadhil H. Easif ◽  
Jomaa J. Murad
Keyword(s):  
Power Series ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Approximate Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Klein Gordon ◽  
Residual Power

In this work, the   Residual Power Series Method(RPSM) is used to find the approximate solutions of Klein Gordon Schrödinger (KGS) Equation. Furthermore, to show the accuracy and the efficiency of the presented method, we compare the obtained approximate solution of Klein Gordon Schrödinger equation by Residual Power Series Method(RPSM) numerically and graphically with the exact solution.

scholarly journals Approximate and generalized solutions of conformable type Coudrey–Dodd–Gibbon–Sawada–Kotera equation

2020 ◽  
pp. 2150021
Author(s):  
Mehmet Senol ◽  
Lanre Akinyemi ◽  
Ayşe Ata ◽  
Olaniyi S. Iyiola
Keyword(s):  
Fractional Order ◽  
Approximate Solutions ◽  
Generalized Solutions ◽  
Analysis Method ◽  
Approximate Methods ◽  
Power Series Method ◽  
Closed Form Solutions ◽  
Homotopy Analysis ◽  
Residual Power Series ◽  
Residual Power

In this study, we consider conformable type Coudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation. Three powerful analytical methods are employed to obtain generalized solutions of the nonlinear equation of interest. First, the sub-equation method is used as baseline where generalized closed form solutions are obtained and are exact for any fractional order [Formula: see text]. Furthermore, residual power series method (RPSM) and [Formula: see text]-homotopy analysis method ([Formula: see text]-HAM) are then applied to obtain approximate solutions. These are possible using some properties of conformable derivative. These approximate methods are very powerful and efficient due to the absence of the need for linearization, discretization and perturbation. Numerical simulations are carried out showing error values, [Formula: see text]-curve for [Formula: see text]-HAM and the effects of fractional order on the solution profiles.

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