scholarly journals Convergence and Error Estimation of a New Formulation of Homotopy Perturbation Method for Classes of Nonlinear Integral/Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2244
Author(s):  
Mohamed M. Mousa ◽  
Fahad Alsharari
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Main Concept ◽  
Convergence Theorems ◽  
Homotopy Perturbation ◽  
New Formulation

In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution was proved as well. The proposed HPM convergence theorems were confirmed and the efficiency of the technique was explored by applying the HPM for solving several classes of nonlinear integral/integro-differential equations.

scholarly journals A NEW NUMERICAL METHOD TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 469-478
Author(s):  
Jinjiao Hou ◽  
Jing Niu ◽  
Welreach Ngolo
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Nonlinear Problems ◽  
Reproducing Kernel Method ◽  
Homotopy Perturbation ◽  
Linear Problems

In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.

The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Ji-Huan He
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Differential Equations ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Solid State Physics ◽  
Content Type ◽  
Homotopy Perturbation ◽  
Fractional Differential

Purpose The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems. Design/methodology/approach In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution. Findings The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models. Originality/value The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

scholarly journals Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 40 ◽  
Author(s):  
Shumaila Javeed ◽  
Dumitru Baleanu ◽  
Asif Waheed ◽  
Mansoor Shaukat Khan ◽  
Hira Affan
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Order Differential Equations ◽  
Homotopy Perturbation ◽  
The Given ◽  
Ordinary Derivative

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

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