Exact solutions of the Laplace fractional boundary value problems via natural decomposition method

Hajira Hajira,Rasool Shah,Dumitru Baleanu,Muhammad Arif,Yu-Ming Chu,Hassan Khan

doi:10.1515/phys-2020-0196

Abstract

Abstract In this article, exact solutions of some Laplace-type fractional boundary value problems (FBVPs) are investigated via natural decomposition method. The fractional derivatives are described within Caputo operator. The natural decomposition technique is applied for the first time to boundary value problems (BVPs) and found to be an excellent tool to solve the suggested problems. The graphical representation of the exact and derived results is presented to show the reliability of the suggested technique. The present study is mainly concerned with the approximate analytical solutions of some FBVPs. Moreover, the solution graphs have shown that the actual and approximate solutions are very closed to each other. The comparison of the proposed and variational iteration methods is done for integer-order problems. The comparison, support strong relationship between the results of the suggested techniques. The overall analysis and the results obtained have confirmed the effectiveness and the simple procedure of natural decomposition technique for obtaining the solution of BVPs.

Highlights

In the last few centuries, fractional partial differential equations (FPDEs) have been effectively used to model several structures and processes that can be used to develop their mathematical models
For the study of FPDEs, the books cited in refs [5,6] and [7] are suggested for readers and for implementation we refer the book given in ref. [8], which is entirely devoted to the various applications of fractional calculus (FC) in physics, astrophysics, chemistry, etc
The book [8] describes the application of FPDEs in nuclear physics, classical mechanics, quantum mechanics, hadron spectroscopy, group theory and quantum field theory

Summary

Introduction

In the last few centuries, fractional partial differential equations (FPDEs) have been effectively used to model several structures and processes that can be used to develop their mathematical models. The Adomian decomposition method (ADM) [29,30,31,32] has been recently implemented to analyze BVPs. ADM provides the solution in the form of infinite series having a quick rate of convergence toward the actual solution. [35], the sinc-collocation approach is applied for the solution of a series of problems with second-order BVPs. Dehghan et al [36] have applied the Adomian–Pade method for the solution to solve Volterra functional systems of equations.

Results

Conclusion