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.

scholarly journals A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
F. Z. Geng ◽  
X. M. Li
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Initial Position ◽  
New Method ◽  
Riccati Differential Equation ◽  
Reproducing Kernel Method ◽  
Riccati Differential Equations ◽  
Linear Problems ◽  
Quasilinearization Technique

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

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.

Application of Homotopy Perturbation Method with Chebyshev Polynomials to Nonlinear Problems

2010 ◽  
Vol 65 (1-2) ◽  
pp. 65-70
Author(s):  
Changbum Chun
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Chebyshev Polynomials ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Efficiency And Reliability

AbstractIn this paper, we present an efficient modification of the homotopy perturbation method by using Chebyshev’s polynomials and He’s polynomials to solve some nonlinear differential equations. Some illustrative examples are given to demonstrate the efficiency and reliability of the modified homotopy perturbation method.

scholarly journals A high order approach for nonlinear Volterra-Hammerstein integral equations

2021 ◽  
Vol 7 (1) ◽  
pp. 1460-1469
Author(s):  
Jian Zhang ◽  
◽  
Jinjiao Hou ◽  
Jing Niu ◽  
Ruifeng Xie ◽  
...  
Keyword(s):  
Integral Equation ◽  
Nonlinear Integral Equation ◽  
Homotopy Perturbation Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Linear Equations ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Hammerstein Integral Equations ◽  
Homotopy Perturbation

<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

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 Nonlinearities Distribution Homotopy Perturbation Method Applied to Solve Nonlinear Problems: Thomas-Fermi Equation as a Case Study

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
K. Boubaker ◽  
A. Sarmiento-Reyes ◽  
A. Perez-Sesma ◽  
...  
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Thomas Fermi ◽  
Homotopy Perturbation

We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results.

Homotopy Perturbation Method for Solving Partial Differential Equations

2009 ◽  
Vol 64 (3-4) ◽  
pp. 157-170 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Muhammad Aslam Noor
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Problems ◽  
Partial Differential ◽  
Homotopy Perturbation ◽  
Adomian’S Polynomials

Abstract We apply a relatively new technique which is called the homotopy perturbation method (HPM) for solving linear and nonlinear partial differential equations. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the HPM solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this technique over the decomposition method

HOMOTOPY PERTURBATION METHOD FOR SOLVING SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 02 (04) ◽  
pp. 657-665 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ahmet Yildirim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Problems ◽  
Partial Differential ◽  
Homotopy Perturbation ◽  
Suggested Technique ◽  
Adomian’S Polynomials

In this paper, we apply the homotopy perturbation method (HPM) for solving systems of partial differential equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the suggested technique solves nonlinear problems without using the Adomian's polynomials is an advantage of this algorithm over the decomposition method.

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