In this paper, efficient numerical methods are given to solve linear Volterra integral (VI) equations and Volterra Integro differential (VID) equations of the first and second types with exponential, singular, regular and convolution kernels. These methods based on Laguerre polynomials (LPs) and Touchard polynomials (TPs) that convert these equations into a system of linear algebraic equations. The results are compared with one another method and with each other. The results show that these methods are applicable and efficient. In addition, the accuracy of solution is presented and also the calculations and Graphs are done by using matlab2018 program.

Haar wavelet collocation points method is developed to the computational solution for nonlinear Fredholm integral and integro-differential equations on interval [0, tf ] using Leibnitz-Haar wavelet collocation points method. Essential principle is transmutation of the integral equation to equivalent higher order differential equation together with initial conditions. The transmutation is carried out using the Leibniz law. Haar wavelet collocation points and its operational matrix is employed to transform the higher order differential equation to a set of algebraic equations, then resolving these equations usage MATLAB program to calculate the demanded Haar coefficients. The computational results of the proposed approach is presented in four problems and make a simulation against the accurate solution. In addition, Error analysis is exhibited the proficiency of the proposed technique and when Haar wavelet resolutions increases the results are close to the accurate solutions.

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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 vec 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.