Series Solutions of Delay Integral Equations via a Modified Approach of Homotopy Analysis Method (original) (raw)
Numerical Algorithms, 2013
The paper presents an application of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind. In this method a series is created, sum of which (if the series is convergent) gives the solution of discussed equation. Conditions ensuring convergence of this series are presented in the paper. Error of approximate solution, obtained by considering only partial sum of the series, is also estimated. Examples illustrating usage of the investigated method are presented as well, including the example having practical application for calculating the charge in supply circuit of flash lamps used in cameras.
Solution of the Quadratic Integral Equation by Homotopy Analysis Method
Annals of Pure and Applied Mathematics
In the present paper, we derive an approximate solution of the quadratic integral equation by using the homotopy analysis method (HAM). This approach provides a solution in the form of a rapidly converging series, and it includes an auxiliary parameter that controls the series solution's convergence. We compare the HAM solution with the exact solution graphically. Additionally, an absolute error comparison between the exact and HAM solutions is performed. The findings indicate that HAM is a very straightforward and attractive approach for computation.
Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations
2010
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.
Note on new homotopy perturbation method for solving non-linear integral equations
2016
In this paper, exact solution for the second kind of nonlinear integral equations are presented. An application of modified new homotopy perturbation method is applied to solve the second kind of non-linear integral equations such that Voltrra and Fredholm integral equations. The results reveal that the modified new homotopy perturbation method is very effective and simple and gives the exact solution. Also the comparison of the results of applying this method with those of applying the homotopy perturbation method reveals the effectiveness and convenience of the new technique.
International Journal of Mathematical Archive EISSN 2229-5046, 2021
In this paper, the analytical solution of Abel's integral equation of the first kind is investigated by using the q-homotopy analysis transform method (q-HATM). The q-HATM is a hybrid method that combines the q-homotopy analysis method (q-HAM) and the Laplace transform method (LTM). The analytical results obtained by the proposed method are in series form, indicating that the approach is simple to implement and computationally appealing.
2012
Abstract Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful tools for solving many functional equations such as ordinary and partial differential and integral equations. In this paper (HAM) is applied to solve linear Fredholm and Volterra first and second kind integral equations, the deformation equations are solved analytically by using MATLAB integration functions.
Mathematical Modeling and Computing
Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied in many different problems of mathematical physics and engineering. In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions. Practical investigations revealed that ND-HAM leads an easy way how to find initial guess and it approaches the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM). Uniqueness solution of the problem and convergence of ND-HAM are proved in the Banach space. Finally, two examples are illustrated to show the accuracy and validity of the proposed method. Five methods are compared in each example.
Applied Mathematics and Computation, 2006
In this paper, a homotopy perturbation method is proposed to solve non-singular integral equations. Comparisons are made between AdomianÕs decomposition method and the proposed method. It is shown, AdomianÕs decomposition method is a homotopy, only. Finally, by using homotopy perturbation method, a new iterative scheme, like AdomianÕs decomposition method, is proposed for solving the non-singular integral equations of the first kind. The results reveal that the proposed method is very effective and simple.
A new analytical technique to solve Fredholm’s integral equations
Numerical Algorithms, 2011
This paper shows that the homotopy analysis method, the wellknown method to solve ODEs and PDEs, can be applied as well as to solve linear and nonlinear integral equations with high accuracy. Comparison of the present method with Adomian decomposition method (ADM), which is well-known in solving integral equations, reveals that the ADM is only special case of the present method. Also, some linear and nonlinear examples are presented to show high efficiency and illustrate the steps of the problem resolution.
arXiv (Cornell University), 2021
Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied it in many different problems of mathematical-physics and engineering. In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions. Practical investigations revealed that ND-HAM leads easy way how to find initial guess and it approaches to the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM). Two examples are illustrated to show the accuracy and validity of the proposed method. Five methods are compared in each example.