The construction of a hierarchy of high-frequency microlocal artificial boundary conditions for the two-dimensional linear Schrödinger equation is proposed. These conditions are derived for a circular boundary and are next extended to a general arbitrarily-shaped boundary. They present the features of being differential in space and non-local in time since their definition involves some tempora...

Over the past years mathematical models, based on experimental data from MRI and CT scans, have been well developed to simulate the growth of aggressive forms of malignant brain tumours. The tumour growth model we are considering here, apart from proliferation and diffusion, is being characterized by a discontinuous diffusion coefficient to incorporate the heterogeneity of the brain tissue. For...

This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, time and space derivatives are considered in terms of Caputo Conformable derivatives, respectively. The present discussion is following ways: we first approximate derivative proposed equation by a s...

We present a paradigm for developing thermodynamical consistent numerical algorithms thermodynamically partial differential equation (TCPDE) systems, called the supplementary variable method (SVM). add proper number of variables to TCPDE system coupled with its energy and other deduced equations through perturbations arrive at consistent, well-determined, solvable structurally stable system. Th...

An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem $H^2$ estimates discretized hyperbolic--parabolic system. For practical computation, further in space, resulting a fully discrete fini...

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, focus are on both nonlinear and linear (BS) equations with numerical approximations. We aim to find an effective approximations for model. Several models from most relevant class European option analyzed study. The problem is approached by transforming into convection-diffusion equation later ...