scholarly journals Mathematical Modelling and Numerical Simulation of Diffusive Processes in Slow Changing Domains

2020 ◽  
Author(s):  
Dmytro V. Yevdokymov ◽  
Yuri L. Menshikov
Keyword(s):  
Heat Conduction ◽  
Time Scale ◽  
Numerical Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Elliptic Boundary Value ◽  
Initial Boundary Value

Nowadays, diffusion and heat conduction processes in slow changing domains attract great attention. Slow-phase transitions and growth of biological structures can be considered as examples of such processes. The main difficulty in numerical solutions of correspondent problems is connected with the presence of two time scales. The first one is time scale describing diffusion or heat conduction. The second time scale is connected with the mentioned slow domain evolution. If there is sufficient difference in order of the listed time scale, strong computational difficulties in application of time-stepping algorithms are observed. To overcome the mentioned difficulties, it is proposed to apply a small parameter method for obtaining a new mathematical model, in which the starting parabolic initial-boundary-value problem is replaced by a sequence of elliptic boundary-value problems. Application of the boundary element method for numerical solution of the obtained sequence of problems gives an opportunity to solve the whole considered problem in slow time with high accuracy specific to the mentioned algorithm. Besides that, questions about convergence of the obtained asymptotic expansion and correspondence between initial and obtained formulations of the problem are considered separately. The proposed numerical approach is illustrated by several examples of numerical calculations for relevant problems.

scholarly journals Regularization method for parabolic equation with variable operator

2005 ◽  
Vol 2005 (4) ◽  
pp. 383-392
Author(s):  
Valentina Burmistrova
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximation Solution ◽  
Elliptic Boundary Value ◽  
Dirichlet Data ◽  
Ill Posed ◽  
Initial Boundary Value

Consider the initial boundary value problem for the equationut=−L(t)u,u(1)=won an interval[0,1]fort>0, wherew(x)is a given function inL2(Ω)andΩis a bounded domain inℝnwith a smooth boundary∂Ω.Lis the unbounded, nonnegative operator inL2(Ω)corresponding to a selfadjoint, elliptic boundary value problem inΩwith zero Dirichlet data on∂Ω. The coefficients ofLare assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution att=0and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

Elliptic Partial Differential-Algebraic Multiphysics Models in Electrical Network Design

2003 ◽  
Vol 13 (09) ◽  
pp. 1261-1278 ◽  
Author(s):  
G. Alì ◽  
A. Bartel ◽  
M. Günther ◽  
C. Tischendorf
Keyword(s):  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Differential Algebraic Equations ◽  
Electrical Network ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Distributed Models ◽  
Initial Boundary Value

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differential-algebraic network equations of the circuit with elliptic boundary value problems modeling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given.

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