Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.

Two-dimensional Volterra–Fredholm integral equations of Hammerstein type are studied. Using the Banach Fixed Point Theorem, the existence and uniqueness of a solution to these equations in the space $$L^\infty ([0,1]\times [0,1])$$ is proved. Then, the operational matrices of integration and product for two-variable Bernoulli polynomials are derived and utilized to reduce the solution of the considered problem to the solution of a system of nonlinear algebraic equations that can be solved by Newton’s method. The error analysis is given and some examples are provided to illustrate the efficiency and accuracy of the method.

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.

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.

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results.

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

A computational method for solving nonlinear stochastic Volterra integral equations

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.

The Numerical Solution of Integral Equations on the Half-Line

An effective scheme for solving system of fractional Volterra–Fredholm integro-differential equations based on the Müntz–Legendre wavelets

In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra partial integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain $$\left( [0, 1]\times [0, 1]\right) $$ . Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, $$l_{2}\hbox {-}norm$$ errors and $$l_{\infty }\hbox {-}norm$$ errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too.

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.

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.

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