Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

In this paper, two new estimators $$ E_{2^{k-1},0}^{(1)}(f) $$ E 2 k - 1 , 0 ( 1 ) ( f ) and $$ E_{2^{k-1},M}^{(1)}(f) $$ E 2 k - 1 , M ( 1 ) ( f ) of characteristic function and an estimator $$ E_{2^{k-1},M}^{(2)}(f) $$ E 2 k - 1 , M ( 2 ) ( f ) of function of H$$\ddot{\text {o}}$$ o ¨ lder’s class $$H^{\alpha } [0,1)$$ H α [ 0 , 1 ) of order $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.

Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itô-Volterra integral equations

Purpose The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically. Design/methodology/approach For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time. Findings By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated. Originality/value Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

Solving Fuzzy Volterra Integrodifferential Equations of Fractional Order by Bernoulli Wavelet Method

A matrix method called the Bernoulli wavelet method is presented for numerically solving the fuzzy fractional integrodifferential equations. Using the collocation points, this method transforms the fuzzy fractional integrodifferential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown coefficients. To illustrate the method, it is applied to certain fuzzy fractional integrodifferential equations, and the results are compared.

A New Bernoulli Wavelet Method for Numerical Solutions of Nonlinear Weakly Singular Volterra Integro-Differential Equations

In this paper, Bernoulli wavelet method has been developed to solve nonlinear weakly singular Volterra integro-differential equations. Bernoulli wavelets are generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton’s method. Convergence analysis of the present method has been discussed in this paper. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.