To obtain the approximate solution to Riccati matrix differential equations, a new variational iteration approach was ‎proposed, which is suggested to improve the accuracy and increase the convergence rate of the approximate solutons to the ‎exact solution. This technique was found to give very accurate results in a few number of iterations. In this paper, the ‎modified approaches were derived to give modified solutions of proposed and used and the convergence analysis to the exact ‎solution of the derived sequence of approximate solutions is also stated and proved. Two examples were also solved, which ‎shows the reliability and applicability of the proposed approach. ‎

In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.

In this paper, a method based on modified adomian decomposition method for solving Seventh order integro-differential equations (MADM). The distinctive feature of the method is that it can be used to find the analytic solution without transformation of boundary value problems. To test the efficiency of the method presented two examples are solved by proposed method.

An approximate solution of the liner system of ntegral cquations fot both fredholm(SFIEs)and Volterra(SIES)types has been derived using taylor series expansion.The solusion is essentailly

In this paper the modified trapezoidal rule is presented for solving Volterra linear Integral Equations (V.I.E) of the second kind and we noticed that this procedure is effective in solving the equations. Two examples are given with their comparison tables to answer the validity of the procedure.

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

In this paper, an approximate solution of nonlinear two points boundary variational problem is presented. Boubaker polynomials have been utilized to reduce these problems into quadratic programming problem. The convergence of this polynomial has been verified; also different numerical examples were given to show the applicability and validity of this method.