Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

The Numerical Solution of Integral Equations on the Half-Line

When considering systems of differential equations with very general boundary conditions, exact solution methods encounter great difficulties, which become insurmountable in the study of nonlinear problems. In this case it is necessary to apply to certain numerical methods. It is important to note that the use of numerical methods often allows you to abandon the simplified interpretation of the mathematical model of the process. The problems of numerical solution of nonlinear Volterra integral equations of the first kind with a differentiable kernel, which degenerates at the initial point of the diagonal, are studied in the paper. This equation is reduced to the Volterra integral equation of the third kind and a numerical method is developed on the basis of that regularized equation. The convergence of the numerical solution to the exact solution of the Volterra integral equation of the first kind is proved, an estimate of the permissible error and a recursive formula of the computational process are obtained. Keywords: nonlinear integral equation, system of nonlinear algebraic equations, error vectors, the Volterra equation, small parameter, numerical methods.

Discrete collocation method for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

In this paper, we consider a bivariate nonlinear Fredholm integral equation of the second kind. Then an approximate solution for some of these problems is investigated. To determine the aimed solution, the hybrid of 2D block-pulse functions with Chebyshev polynomials basis with the operational matrices is applied. In this work, we generalize the operational matrices stated by Behbahani (J Basic Appl 4:131–141, 2015), from one-dimensional to 2D space. Comparing this technique with other works, we can distinguish the advantage of developing the solution with fewer runtime and computations with more satisfactory approximations. In this procedure, using operational matrices, the nonlinear Fredholm integral equations are reduced to a system of nonlinear algebraic equations. Furthermore, the convergence analysis for this numerical method is investigated. Numerical examples are presented to describe the performance and rectitude of this method.

Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

Numerial Methods for Volterra Integral Equations with Singular Kernels

The purpose of this paper is to provide an efficient numerical technique for solving two-dimensional nonlinear fuzzy Fredholm integral equations. The fuzzy Gauss quadrature rule is applied to the above mentioned equation, which reduces the fuzzy integral equations to the nonlinear approximate equations that can be easily solved by a numerical iterative algorithm. The convergence of the presented numerical method is investigated under several mild conditions. Finally, some illustrative numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method.

In the present study, first, we introduce an iterative method based on quadrature formula for solving two-dimensional nonlinear fuzzy Fredholm integral equations (2DNFFIE). Then, we present error estimation and the numerical stability analysis for the proposed method. Finally, to show the efficiency of the proposed method, supporting examples are also provided.

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

HE simplified Oswatitsch equation is a one-dimensional nonlinear singular integral equation, occurring in transonic aerodynamics.l It is an approximate version of Oswatitsch's integral equation, whose kernel has a dipole singularity. According to transonic small perturbation theory, the Oswatitsch equation is the basic equation governing the steady inviscid irrotational flow of a perfect gas past a thin symmetric profile at zero incidence, with subsonic freestream Mach number Mx rstrud 4 and Nixon5 deserve special mention. In the present work, the simplified Oswatitsch equation has been solved by two different numerical procedures, viz., the direct iteration scheme (DIS) proposed by Niyogi and Chakraborty6 and by the recent perturbed iterative scheme (PIS) put forward by Dey7 for solving a system of nonlinear algebraic equations. Further, this nonlinear integral equation was used as a test case for studying the global convergence behavior of PIS. From computational results, it has been found that for a parabolic arc profile there exists a range of values for the reduced thickness ratio r (which is a transonic similarity parameter), where both the procedures lead to the same shock-free supercritical solution, and that in this range PIS converges much faster than DIS. However, for higher values of r beyond this range, there exists another range where, contrary to expectations, DIS converges but PIS fails to converge, indicating that DIS has a wider range of convergence.

Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions

In this paper, we study the quadratic rule for the numerical solution of linear and nonlinear two-dimensional Fredholm integral equations based on spline quasi-interpolant. Also the convergence analysis of the method is given. We show that the order of the method is $$O(h_{x}^{m+1})+O(h_{y}^{m^{\prime }+1})$$. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

An efficient implicit spectral element method for time-dependent nonlinear diffusion equations by evaluating integrals at one quadrature point