Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Moreover, the convergence of the expansion coefficients and an upper estimate for the truncation error are included. The obtained results for several case studies illustrate the accuracy and reliability of the proposed approach.

Approximation of function belonging to generalized Hölder’s class by first and second kind Chebyshev wavelets and their applications in the solutions of Abel’s integral equations

In this paper, first and second kind Chebyshev wavelets are studied. New estimators $$E_{2^{k-1},0}^{(1)}$$ E 2 k - 1 , 0 ( 1 ) , $$E_{2^{k-1},M}^{(2)}$$ E 2 k - 1 , M ( 2 ) , $$E_{2^{k-1},0}^{(3)}$$ E 2 k - 1 , 0 ( 3 ) , $$E_{2^{k-1},M}^{(4)}$$ E 2 k - 1 , M ( 4 ) for first kind Chebyshev wavelets and estimators $$E_{2^{k},0}^{(5)}$$ E 2 k , 0 ( 5 ) , $$E_{2^{k},M}^{(6)}$$ E 2 k , M ( 6 ) , $$E_{2^{k},0}^{(7)}$$ E 2 k , 0 ( 7 ) and $$E_{2^{k},M}^{(8)}$$ E 2 k , M ( 8 ) for second kind Chebyshev wavelets for a function f belonging to generalized H$$\ddot{o}$$ o ¨ lder’s class have been obtained. Also, a method based on first and second kind Chebyshev wavelet approximations has been presented for solving integral equations. Comparison of solutions obtained by both wavelets method has been studied. It is found that second kind Chebyshev wavelet method gives better and accurate solutions as compared to first kind Chebyshev wavelet method. This is a significant achievement of this research paper in wavelet analysis.