Space-time Dependent Coefficients Research Articles

Version [1.1] - [2D-xtDCR-LTBEM: A Fortran code for numerical solutions to the unsteady anisotropic-DCR equation of spatio-temporal coefficients]

Variable coefficient equations are usually used as governing equations for modeling problems which are associated with Functionally Graded Materials (FGM). Version 1.1 - 2D-xtDCR-LTBEM is a Fortran code that can be used to find numerical solutions of two-dimensional (2D) initial–boundary value problems which are governed by the unsteady anisotropic-diffusion convection reaction (DCR)-type and the anisotropic-diffusion reaction (DR)-type equations with a class of space–time dependent coefficients, by utilizing a combined Laplace Transform-Boundary Element Method (LTBEM). This code may provide valuable help for simulation of problems of FGM.

Nonparametric Estimation for Independent and Identically Distributed Stochastic Differential Equations with Space-Time Dependent Coefficients

Nonparametric Estimation for Independent and Identically Distributed Stochastic Differential Equations with Space-Time Dependent Coefficients

A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

Two‐dimensional transient problems for anisotropic functionally graded materials governed by a diffusion reaction equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time‐dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation, which can be written in forms of boundary‐only integral equations and solved using a standard boundary element method for the solutions. These boundary element method solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials are solved. The numerical solutions obtained are quite accurate.

A numerical investigation of unsteady space–time dependent coefficients anisotropic-diffusion convection reaction equation

Two dimensional transient problems for anisotropic FGMs governed by a diffusion convection reaction (DCR) equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation which can be written in forms of boundary-only integral equations and solved using a standard BEM for the solutions. These BEM solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials with compressible or incompressible flow are solved. The numerical solutions obtained are quite accurate.

A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order -,k. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for 2times 2 and 3times 3 systems, complemented by several examples.

Transport-collapse scheme for heterogeneous scalar conservation laws

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

The diffusion phenomenon for damped wave equations with space-time dependent coefficients

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

Stochastic solution of fractional Fokker–Planck equations with space–time-dependent coefficients

This paper develops solutions of fractional Fokker–Planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non-Gaussian Lévy processes, with space–time-dependent drift, diffusion and jump coefficients, thus significantly extends Magdziarz and Zorawik's result in [14]. Fractional Fokker–Planck equation describing subdiffusion is solved by our result in full generality from perspective of stochastic representation.

Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method

Recently the unified method for finding traveling wave solutions of nonlinear evolution equations was proposed by one of the authors. It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries equation with space-time dependent coefficients.

Decay of energy for wave equations with strong damping terms and space–time-dependent coefficients

We study the asymptotic behaviour of solutions to the wave equations with strong damping terms and space–time-dependent coefficients by establishing decay estimates for the total energy of solutions.

A semilinear wave equation with space–time dependent coefficients and a memory boundarylike antiperiodic condition: A low-frequency asymptotic expansion

This paper studies a low-frequency asymptotic expansion for a unique strong solution to an initial-boundary value problem of a semilinear wave equation. This equation admits space–time dependent coefficients and a memory boundarylike antiperiodic condition. For some small parameters from coefficients of this semilinear wave equation and of boundary conditions, we approximate a unique strong solution to this problem by a polynomial of these parameters, and coefficients of this polynomial are strong solutions of a sequence of well-defined linear initial-boundary value problems.

A semilinear wave equation with space-time dependent coefficients and a memory boundarylike antiperiodic condition: Regularity and stability

This paper deals with regularity and stability of solutions to an initial-boundary value problem of a semilinear wave equation. This equation admits space-time dependent coefficients and a memory boundarylike antiperiodic condition. For regularity or existence of a unique strong solution, the Faedo–Galerkin method and the energy method, associated with the maximal solution of a nonlinear Volterra integral equation, are applied. Then, as an application of the regularity result, stability of this unique strong solution is obtained by the energy method.

Regularity of the solution of a nonlinear wave equation

The paper deals with the regularity of the weak solution of anonlinear initial-boundary value problem given by a semi-linearwave equation with space-time dependent coefficients and aboundary-like antiperiodic condition.

Liapunov functionals application to dynamic stability analysis of continuous systems

Using the appropriate energy-like Liapunov functional sufficient conditions for the uniform stability of undeflected form of structures are derived. The structures are described by partial differential equations and integro-partial differential equations with time and space–time-dependent coefficients. Stability domains obtained by applying the linearized equations of motion are compared with those employing the typical nonlinearity e.g. Kármán nonlinear effects and Brazier's nonlinearity. A relation between the stability of nonlinear equations and linearized ones is analyzed. An influence of geometrical, and material parameters as well as constant components of axial and in-plane forces for different classes of parametric excitation on stability regions is shown.