Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations

In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work

Modified Homotopy Perturbation Method For Solving High-Order Integro-Differential Equation

In this work, a new modification of homotopy perturbation method was proposed to find analytical solution of high-order integro-differential equations. The Modification process yields the Taylor series of the exact solution. Canonical polynomials are used as basis function. The assumed approximate solution was substituted into the problem considered in which the coefficients of the homotopy perturbation parameter p were compared, and then solved, resulting to a single algebraic equation. Thus, algebraic linear system of equations were obtained by equating the coefficients of various powers of the independent variables in the equation to zero,  which are then solved simultaneously using MAPLE 18 software to obtain the values of the unknown constants in the equations. The values of the unknown constants were substituted back to get the initial approximation which yield the final solution. Some examples were given to illustrate the effectiveness of the method. Keywords: Homotopy perturbation, Integro-differential equation, Canonical polynomial, Basis functio

On the Regularization-Homotopy Analysis Method for Linear and Nonlinear Fredholm Integral Equations of the First Kind

Fredholm integral equations of the first kind are considered by applying regularization method and the homotopy analysis method. This kind of integral equations are considered as an ill-posed problem and for this reason needs an effective method in solving them. This method first transforms a given Fredholm integral equation of the first kind to the second kind by the regularization method and then solves the transformed equation using homotopy analysis method. Approximation of the solution will be of much concern since it is not always the case to get the solution to converge and the existence of the solution is not always guaranteed as this kind of Fredholm integral equation is not well-posed

An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations

We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique

Optimal homotopy asymptotic and homotopy perturbation methods for linear mixed volterra-fredholm ıntegral equations

Bu çalışmada, karma Volterra-Fredholm integral denklemleri optimal homotopi asimptotik metod (OHAM) ve Homotopi Perturbasyon metodu (HPM) vasıtasıyla irdelenmiştir. Yaklaşımımız zamandan bağımsız ve basit hesaplamalar yolu ile tam çözüme oldukça yaklaşık çözümler veren bir yöntemdir. Bu iki yöntemin karşılaştırılması tartışılmıştır. OHAM yaklaşımının doğruluğu ve etkinliği HPM çözümleri ile dört örnek kullanılarak karşılaştırılmıştır. Sonuçlar OHAM ın HPM ye göre daha verimli ve esnek bir yöntem olduğunu göstermektedir.In this paper, we study the mixed Volterra-Fredholm integral equations of the second kind by means of optimal homotopy asymptotic method (OHAM) and Homotopy Perturbation method (HPM).Our approach is independent of time and contains simple computations with quite acceptable approximate solutions in which approximate solutions obtained by these methods are close to exact solutions. Comparison of these methods have been discussed. The accuracy and efficiency of OHAM approach with respect to Homotopy Perturbation method (HPM) is illustrated by presenting four test examples. The results indicate that the OHAM is very effective and flexible to use with respect to HPM

a collocation method via the quasi affine biorthogonal systems for solving weakly singular type of volterra fredholm integral equations

Abstract Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples

A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

In this article, the Elzaki homotopy perturbation method is applied to solve fractional stiff systems. The Elzaki homotopy perturbation method (EHPM) is a combination of a modified Laplace integral transform called the Elzaki transform and the homotopy perturbation method. The proposed method is applied for some examples of linear and nonlinear fractional stiff systems. The results obtained by the current method were compared with the results obtained by the kernel Hilbert space KHSM method. The obtained result reveals that the Elzaki homotopy perturbation method is an effective and accurate technique to solve the systems of differential equations of fractional order