In this note, a general form of Fredholm-Volterra integro-differential equation order one is considered. The truncated Legendre series is used as bases function to approximate unknown function and Gauss-Legendre quadrature formula is applied for kernel integrals. Reduced algebraic equations are solved by using collocation method with roots of Legendre polynomials as collocation points. Three numerical examples with the comparisons are provided to show the validity and accuracy of the suggested method. Numerical results reveal that proposed method is dominated with repeated trapezoidal rule and differential transform method. � 2018 Author(s).

An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given. � 2017, American Institute of Mathematical Sciences. All rights reserved.

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed. � 2017 Author(s).

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods. � 2017 Author(s).

In this note, we consider a supersingular integral equations (SuperSIEs) of the first kind on the interval [-1,1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain D = [-1,1]×[-1,1]. Projection method together with Chebyshev polynomials of the first, second, third and fourth kinds are used to find bounded, unbounded and semi-bounded solutions of SuperSIEs respectively. Exact calculations of singular integrals for Chebyshev polynomials allow us to obtain high accurate approximate solution. Gauss- Chebyshev quadrature formulas are used for high accurate computations of regular kernel integrals. Two examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveal that approximate solutions are exact if solution of SuperSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points and errors are drastically decreases. © 2020 Author(s).