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

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

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

Journal of Function Spaces ◽

10.1155/2021/3230272 ◽

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.

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Analytical solutions forfuzzysystem using power series approach

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i8.5973 ◽

2016 ◽

Vol 12 (8) ◽

pp. 6553-6559

Author(s):

Sana Abughurra

Keyword(s):

Power Series ◽

Analytical Solutions ◽

Initial Value Problems ◽

Convergent Series ◽

Initial Value ◽

Generalized Differentiability ◽

Residual Power Series ◽

Strongly Generalized Differentiability ◽

Fuzzy Equations ◽

Residual Power

The aim of the present paper is present a relatively new analytical method, called residual power series (RPS) method, for solving system of fuzzy initial value problems under strongly generalized differentiability. The technique methodology provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. Several computational experiments are given to show the good performance and potentiality of the proposed procedure. The results reveal that the present simulated method is very effective, straightforward and powerful methodology to solve such fuzzy equations.

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Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent–Miodek system with energy-dependent Schrödinger potential

Advances in Difference Equations ◽

10.1186/s13662-019-2397-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 7

Author(s):

Mehmet Şenol ◽

Olaniyi S. Iyiola ◽

Hamed Daei Kasmaei ◽

Lanre Akinyemi

Keyword(s):

Analytical Techniques ◽

Convergent Series ◽

Analysis Method ◽

Power Series Method ◽

Strongly Nonlinear ◽

Homotopy Analysis ◽

Residual Power Series ◽

Fractional Differential ◽

Energy Dependent ◽

Residual Power

Abstract In this paper, we present analytical-approximate solution to the time-fractional nonlinear coupled Jaulent–Miodek system of equations which comes with an energy-dependent Schrödinger potential by means of a residual power series method (RSPM) and a q-homotopy analysis method (q-HAM). These methods produce convergent series solutions with easily computable components. Using a specific example, a comparison analysis is done between these methods and the exact solution. The numerical results show that present methods are competitive, powerful, reliable, and easy to implement for strongly nonlinear fractional differential equations.

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

Open Physics ◽

10.1515/phys-2020-0190 ◽

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.

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

Nonlinear Engineering ◽

10.1515/nleng-2021-0022 ◽

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.

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Multiple Fractional Solutions for Magnetic Bio-Nanofluid Using Oldroyd-B Model in a Porous Medium with Ramped Wall Heating and Variable Velocity

Applied Sciences ◽

10.3390/app10113886 ◽

2020 ◽

Vol 10 (11) ◽

pp. 3886 ◽

Cited By ~ 3

Author(s):

Muhammad Saqib ◽

Ilyas Khan ◽

Yu-Ming Chu ◽

Ahmad Qushairi ◽

Sharidan Shafie ◽

...

Keyword(s):

Porous Medium ◽

Temperature Field ◽

Velocity Field ◽

Fractional Derivatives ◽

Fluid Velocity ◽

Fractional Operators ◽

Caputo Fractional Derivatives ◽

Wall Heating ◽

The Laplace Transform ◽

Fractional Analysis

Three different fractional models of Oldroyd-B fluid are considered in this work. Blood is taken as a special example of Oldroyd-B fluid (base fluid) with the suspension of gold nanoparticles, making the solution a biomagnetic non-Newtonian nanofluid. Based on three different definitions of fractional operators, three different models of the resulting nanofluid are developed. These three operators are based on the definitions of Caputo (C), Caputo–Fabrizio (CF), and Atnagana–Baleanu in the Caputo sense (ABC). Nanofluid is taken over an upright plate with ramped wall heating and time-dependent fluid velocity at the sidewall. The effects of magnetohydrodynamic (MHD) and porous medium are also considered. Triple fractional analysis is performed to solve the resulting three models, based on three different fractional operators. The Laplace transform is applied to each problem separately, and Zakian’s numerical algorithm is used for the Laplace inversion. The solutions are presented in various graphs with physical arguments. Results are computed and shown in various plots. The empirical results indicate that, for ramped temperature, the temperature field is highest for the ABC derivative, followed by the CF and Caputo fractional derivatives. In contrast, for isothermal temperature, the temperature field of C-derivative is higher than the CF and ABC derivatives, respectively. It was noticed that the velocity field for the ABC derivative is higher than the CF and Caputo fractional derivatives for ramped velocity. However, the velocity field for the Caputo fractional derivative is lower than the ABC and CF for isothermal velocity.

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A numerical approach for the nonlinear temporal conformable fractional foam drainage equation

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500893 ◽

2020 ◽

pp. 2150089 ◽

Cited By ~ 1

Author(s):

Hira Tariq ◽

Hatıra Günerhan ◽

Hadi Rezazadeh ◽

Waleed adel

Keyword(s):

Convergent Series ◽

Numerical Approach ◽

Power Series Method ◽

Residual Power Series ◽

Novel Method ◽

Uniformly Convergent ◽

Foam Drainage Equation ◽

Residual Power ◽

Straightforward Approach ◽

Foam Drainage

In this paper, we examine a novel method called the residual power series method (RPSM) which is used in finding an analytic approximate solution to the nonlinear temporal foam drainage equation of fractional order. The solution is obtained in a uniformly convergent series form. Also, 3D graphs for the solution are provided for different values of [Formula: see text] which proves that the method is an effective and straightforward approach of providing excellent results for similar problems.

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The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform

Journal of Function Spaces ◽

10.1155/2021/1029196 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

Pongsakorn Sunthrayuth ◽

Rasool Shah ◽

A. M. Zidan ◽

Shahbaz Khan ◽

Jeevan Kafle

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Stokes Equations ◽

Three Dimensional ◽

Nonlinear Problems ◽

Navier Stokes ◽

Power Series Method ◽

Navier Stokes Equations ◽

Residual Power Series ◽

Residual Power

This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.

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Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Ordern-1<β

Mathematical Problems in Engineering ◽

10.1155/2016/6380978 ◽

2016 ◽

Vol 2016 ◽

pp. 1-13

Author(s):

Amal Khalaf Haydar ◽

Ruaa Hameed Hassan

Keyword(s):

Fractional Order ◽

Initial Value Problems ◽

Fractional Derivatives ◽

Laplace Transforms ◽

Initial Value ◽

Caputo Fractional Derivatives

The main aim in this paper is to use all the possible arrangements of objects such thatr1of them are equal to 1 andr2(the others) of them are equal to 2, in order to generalize the definitions of Riemann-Liouville and Caputo fractional derivatives (about order0<β<n) for a fuzzy-valued function. Also, we find fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional ordern-1<β<nunder H-differentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations.

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