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.

A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method

Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations.

Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

NUMERICAL SOLUTION OF MIXED LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY MODIFIED BLOCK PULSE FUNCTIONS

A numerical method based on modified block pulse functions is proposed for solving the mixed linear Volterra-Fredholm integral equations. We obtain an integration operational matrix of modified block pulse functions on interval [0,T). A modified block pulse functions and their operational matrix of integration, the mixed linear Volterra-Fredholm integral equations can be reduced to a linear system of algebraic equations. The rate of convergence is O(h) and error analysis of the proposed method are discussed. Some examples are provided to show that the proposed method have a good degree of accuracy.

An efficient algorithm to solve damped forced oscillator problems by Bernoulli operational matrix of integration

An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives was obtained in Yang and Srivastava (Commun Nonlinear Sci Numer Simul 29(1–3):499–504, 2015). In this paper, we obtain the numerical solution of damped forced oscillator problems by employing the operational matrix of integration of Bernoulli orthonormal polynomials. The operational matrix of integration is determined with the help of the integral operator on Bernoulli orthonormal polynomials. Numerical examples of two different problems of spring are given to delineate the performance and perfection of this approach and compared the results with the exact solution.

Новый численный метод решения нелинейных стохастических интегральных уравнений

The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.