Elliptic partial differential equations (PDEs) are frequently used to model a variety of engineering phenomena, such as steady-state heat conduction in a solid, or reaction-diffusion type problems. However, computing a solution can sometimes be difficult or inefficient using standard solvers. Techniques have been developed, including the method of lines (Schiesser, 1991), which can solve parabo...

in this article, we study the analytical solutions of different parabolic heat equations with different boundaryconditions in the form of multi-term fractional differential equations. then we compare these analytical solutions with numerical finite difference methods. this comparison demonstrates the accuracy of the analytical and numerical methods presented here.

we focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of it¨o type, in particular, parabolic equations. the main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal for example, one asks an Einstein metric in the class. have property that their geodesics are distinguished, as unparameterised curves, sense of geometry. This characterises metrics. this article, we initiate study corresponding phenomena other geometries, partic...

We focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of It¨o type, in particular, parabolic equations. The main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.