Abstract
The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.
Highlights
The application of integral equations can be found in various fields which include mathematics, physics and engineering
This study describes new techniques using a combination of Newton–Kantrovich and Haar wavelets to solve the second kind of nonlinear Fredholm and Volterra integral equations
When applying large values of N such as N = 16 to the approximation of the Haar wavelets x16 (t), we recommend using the previous approximate solution x8 (t) or x4 (t) which considers the smaller value of N as an initial condition
Summary
The application of integral equations can be found in various fields which include mathematics, physics and engineering. The process of solving the integral equations analytically is very complicated and for application purposes, it will be sufficient to solve the latter numerically. Many methods have been established to find numerical solutions for integral equations. These methods include the polynomial approximation [1,2], linear multistep methods [3], modified homotopy perturbation [4], wavelets [5,6,7,8,9], triangular functions [10] and Newton–Kantorovich method [11,12,13,14]. Finding the numerical solutions for integral equations are often a complicated process and requires a large number of arithmetic computations
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