Finding exact solutions of nonlinear PDEs using the natural decomposition method

2016 ◽  
Vol 40 (1) ◽  
pp. 223-236 ◽  
Author(s):  
Mahmoud Rawashdeh ◽  
Shehu Maitama
Keyword(s):  
Exact Solutions ◽  
Decomposition Method ◽  
Nonlinear Pdes ◽  
Natural Decomposition

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

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1178-1187
Author(s):  
Hajira ◽  
Hassan Khan ◽  
Yu-Ming Chu ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Decomposition Method ◽  
Graphical Representation ◽  
Simple Procedure ◽  
Boundary Value ◽  
Decomposition Technique ◽  
Approximate Analytical Solutions ◽  
Natural Decomposition ◽  
Fractional Boundary Value Problems

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.

scholarly journals Solution for fractional forced KdV equation using fractional natural decomposition method

2020 ◽  
Vol 5 (2) ◽  
pp. 798-810 ◽  
Author(s):  
P. Veeresha ◽  
◽  
D. G. Prakasha ◽  
Jagdev Singh ◽  
◽  
...  
Keyword(s):  
Decomposition Method ◽  
Kdv Equation ◽  
Natural Decomposition

scholarly journals Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Nemat Dalir
Keyword(s):  
Fluid Mechanics ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Second Order ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Exact Analytical Solutions ◽  
First Order

Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

scholarly journals Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 386 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Boundary Layer Equations ◽  
Compatibility Analysis ◽  
Klein Gordon

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

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