A new efficient method for cases of the singular integral equation of the first kind

Abstract

Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution u(x) is represented in the form of a series in the reproducing kernel space, afterwards the n-term approximate solution u n (x) is obtained and it is proved to converge to the exact solution u(x). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.

Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

“…Table 3 presents the absolute error for the above three cases. Clearly, Table 3 shows that obtained absolute errors are significantly good for really small values of N, N = 5, that can never be achieved in [10,26]. The exact value for u(x) in the above examples is obtained through Mathematica 11, while the approximated results are calculated using Matlab R2018a on a 4 GHz personal laptop with 8 GB of RAM.…”

“…Eshkuvatov [10] introduced the method taking Chebyshev polynomials of all four kinds for all four different solution cases of the CSIE. Reproducing the kernel Hilbert space (RKHS) method has been proposed by A. Dezhbord et al [26]. The representation of solution u(x) is in the form of a series in reproducing kernel spaces.…”

“…depending on the κ. Furthermore, the results of the numerical example are compared with [10,26] for k = 0. Comparison reveals that the addressed method gives a more accurate approximation than these methods, Section 4 provides this phenomena.…”

Reproducing kernel method for a class of weakly singular Fredholm integral equations

“…Alvandi and Paripour solved nonlinear Abel's integral equations with weakly singular kernel in the reproducing kernel space and removed the singularity of the equation considered [13]. The same authors presented a simple and efficient method to solve linear Volterra integro-differential equations [14], nonlinear Volterra-Fredholm integro-differential equations [15], and see [16][17][18][19][20][21].…”

A numerical approach for solving the high-order nonlinear singular Emden–Fowler type equations

“…This theory has been successfully applied to integral equations [14,15], partial differential equations [16], boundary value problems [17][18][19][20][21][22], fractional differential equations [23], and so on [24][25][26][27][28][29].…”