Implementing Reproducing Kernel Method to Solve Singularly Perturbed Convection-Diffusion Parabolic Problems

Abstract

In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.

Uncertain M-fractional differential problems: existence, uniqueness, and approximations using Hilbert reproducing technique provisioner with the case application: series resistor-inductor circuit

“…Despite the extensive study and use of the HRT tools in various engineering and scientific fields [10][11][12][13][14][15][16][17][18][19][20][21][22], there is still ongoing research on the principle of reproducing kernels. Nevertheless, the HRT has demonstrated its usefulness as a scheme for solving a wide range of stochastic and nonlinear problems fractionally, offering a versatile numerical approach to address solution performances.…”

“…The HRT, a powerful mathematical tool, has made significant discoveries in various stochastic physics and nonlinear frameworks [10][11][12]. Built upon the concept of Hilbert space, which provides a framework for analyzing functions and their properties, this algorithm has found applications in signal processing, probability, modeling, and control theory [13][14][15][16][17][18][19][20][21][22]. It has proven particularly useful in solving problems related to function approximation, interpolation, and regression.…”