The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

Abstract

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

“…This section consists of the implementation of our method on some examples. The results obtained from our method show the accuracy and superiority of the method compared with those in [15,28]. All calculations are done in Maple 2018 software.…”

“…where G i (t), and K i j (t, s) are majorant of g i (t) and k i j (t, s), respectively which are obtained by taking absolute values of the coefficients in (15). Clearly, {Y i (t)} n i=1 is a majorant for {y i (t)} n i=1 , and all of it's coefficients Ȳi,µ are positive.…”

“…The numerical results of hybrid numerical method [28] are shown in Table 1 in which m is the number of subintervals of Ω . Example 2 [15] Consider the problem (12) with…”

“…During the implementation of the our method, similar to the previous example, the Tausolution coincide with exact solution. Table 2 represent the results reported in [15] for different n. The errors and behavior of Tau parameters for various values of N are reported in Table 4 and Figs. 5-8.…”

“…These type of integral equations arise in many scientific applications such as diffusion problems, spread of epidemics, the behavior of viscoelastic materials in mechanics (see [12,17,35] and references therein). There are several numerical methods for solving Abel-Volterra integral equations system such as waveform relaxation method [16], extrapolation method [29], block by block method [34], reproducing kernel Hilbert space method [15], parallel algorithm [11], Laplace transform method [26]. However, few research has been done on the Abel-Volterra integral equations system (1).…”