Mehdi Ghasemi

A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations

This paper suggests a novel multi-parametric homotopy method for systems of Fredholm integral equations. This modified method contains three convergence-control parameters to form an improved homotopy. We present efficient error estimation for the approximate solution. The results of present method are compared with the Adomian decomposition method (ADM), the homotopy perturbation method (HPM) and standard homotopy analysis method results that provides confirmation for the validity of proposed approach. Some test examples are given to clarify the efficiency and high accuracy of the present method. The results reveal that the present method is very effective and convenient.

Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space

In this paper, approximate solutions to a class of fractional differential equations with delay are presented by using a semi-analytical approach in Hilbert function space. Further, the uniqueness of the solution is proved in the space of real-valued continuous functions, as well as the existence of the solution is proved in Hilbert function space. We also prove convergence and perform an analysis error for the proposed approach. Sophisticated delay differential equations of fractional order are considered as test examples. Numerical results illustrate the efficiency of the proposed approach computationally.

Solution of nonlinear fractional differential equations using an efficient approach

We present an efficient approach for solving nonlinear fractional differential equations. The convergence analysis of the approach is studied. To demonstrate the working of the presented approach, we consider three special cases of nonlinear fractional differential equations. The results of theses examples and comparison with different methods provide confirmation for the validity of the proposed approach.

Solution of system of the mixed Volterra–Fredholm integral equations by an analytical method

Abstract In this paper, we present an analytical method to solve systems of the mixed Volterra–Fredholm integral equations (VFIEs) of the second kind. By using the so called ħ - curves , we determine the convergence parameter ħ , which plays a key role to control convergence of approximation solution series. Further, we show that the homotopy perturbation method (HPM), which is a well-known tool for solving systems of integral equations, is only the special case of the presented method. Some test examples are given to clarify the efficiency and high accuracy of the method. An efficient error estimation for the approximate solution is also presented for the proposed method.

Periodic solution for strongly nonlinear vibration systems by using the homotopy analysis method

AbstractIn this paper, the periodic solutions for the oscillation of a mass attached to a stretched elastic wire are obtained using the homotopy analysis method (HAM). HAM helps us to obtain square root frequency (Ω=ω2) in the form of approximation series of the convergence control parameter ℏ. Finally, the so-called valid region of ℏ is determined by plotting the Ω-ℏ curve. Comparison of the obtained results with exact solutions provides confirmation for the validity of HAM.

The Reproducing Kernel Method for Some Variational Problems Depending on Indefinite Integrals

In this paper we introduce the reproducing kernel method to solve a class of variational problems (VPs) depending on indefinite integrals. We discuss an analysis of convergence and error for the proposed method. Some test examples are presented to demonstrate the validity and applicability of method. The results of numerical examples indicate that the proposed method is computationally very simple and attractive.

Option pricing using a computational method based on reproducing kernel

Abstract One of the most important subject in financial mathematics is the option pricing. The most famous result in this area is Black–Scholes formula for pricing European options. This paper is concerned with a method for solving a generalized Black–Scholes equation in a reproducing kernel Hilbert space. Subsequently, the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method. Furthermore, the error estimates for obtained approximation in reproducing kernel Hilbert space are presented. Finally, a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method.

A Family of Iterative Methods with Accelerated Eighth-Order Convergence

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.