One-dimensional parabolic equation; Nonlocal boundary conditions; Taylor approximation; Operational matrices; Krylov subspace iterative methods; Restarted GMRES Abstract This article contributes a matrix approach by using Taylor approximation to obtain the numerical solution of one-dimensional time-dependent parabolic partial differential equations (PDEs) subject to nonlocal boundary integral c...

A computational framework for the construction of solutions to linear homogenous partial differential equations (PDEs) with variable coefficients is developed in this paper. The considered class PDEs reads: ∂p∂t−∑j=0m∑r=0njajrtxr∂jp∂xj=0 F-operators are introduced and used transform original PDE into image PDE. Factorization solution rational exponential parts enables us construct analytic with...

In this paper, we prove a convergence theorem for singular perturbations problems class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to degenerate PDEs. Our approach mainly based on G-stochastic analysis argument. As byproduct, also establish averaging principle stochastic driven by G-Brow...

We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations. A subclass of the latter comprises all the systems of linear equations to which the area of linear algebra is devoted. In fact, many a problem in numerical analysis can be reduced to one of solving a system of linear equations. We already witnessed this in the use of Newto...

this paper presents a new numerical method for solution of eikonal equation in two dimensions.in contrast to the previously developed methods which try to define the solution surface by its level sets(contour curves), the developed methodology identifies the solution surface by resorting to its characteristics. the suggested procedure is based on the geometric properties of the solution surface...

Cancer is not just one disease, but rather a complicated interaction of many abnormal features and many different cell types, which are situated in a heterogeneous habitat of normal tissue. The mathematical modelling of cancer growth and treatment is at full swing, and a significant challenge arises due to the interactions of cancer with a complicated and structured microenvironment of healthy ...

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. T...

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The spatial discretization of nonlinear partial differential equations (PDEs) results in large systems of nonlinear ordinary differential equations (ODEs). The discretization of the Brusselator equation is a characteristic example. For the parallel numerical solution of the Brusselator equation we use an iterated Runge-Kutta method. We propose modifications of the original method that exploit t...

The solution of ordinary differential equations (ODEs) is intrinsically bound up with the solution of partial differential equations (PDEs). In this chapter the basic integration methods are introduced, covering: Euler, Runge-Kutta, variable step, extrapolation, BDFs, NDFs, and Adams. Various test examples for the different integration methods are provided along with annotated code. The Newton ...