The present paper is devoted to the application of local polynomial and non-polynomial
interpolation splines of the third order of approximation for the numerical solution of the Volterra integral
equation of the second kind. Computational schemes based on the use of the splines include the ability to
calculate the integrals over the kernel multiplied by the basis function which are present in the computational
methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra
integral equations is also discussed. The results of the numerical experiments are presented.

/pdf/application-local-polynomial-and-non-polynomial-splines-of-2p32bv42vd.pdf

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

In this paper, an artificial intelligence approach using neural networks is described to solve a class of delay optimal control problems of fractional order with equality and inequality constraints. In the proposed method, a functional link neural network based on the Muntz–Legendre polynomial is developed. The problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Pade approximation. According to the Pontryagin’s minimum principle for optimal control problems of fractional order and by constructing an error function, the authors then define an unconstrained minimization problem. The authors use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by a single-layer Muntz–Legendre neural network model. The authors then exploit an unconstrained optimization scheme for adjusting the network parameters (weights and bias) and to minimize the computed error function. Some numerical examples are given to illustrate the effectiveness of the proposed method.

Müntz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

https://www.mdpi.com/2504-3110/5/3/70/pdf

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

Applying the three‐dimensional block‐pulse functions to solve system of Volterra–Hammerstein integral equations

This article proposes a hybrid technique for finding approximation solutions of the fractional 2D Sobolev equation. In the proposed approach, the Müntz‐Legender functions and Müntz‐Legender wavelets are, respectively, utilized to approximate the solution of the problem under consideration in the time and spatial directions. By implementing the presented technique, solving the 2D fractional Sobolev equation is converted into solving a system of algebraic equations. Three examples are solved to examine the validity of the proposed method.

Neda Negarchi

Papers

Numerical solution of Volterra–Fredholm integral equations using the collocation method based on a special form of the Müntz–Legendre polynomials