scholarly journals On the numerical solution of the diffusion equation with a nonlocal boundary condition

2003 ◽  
Vol 2003 (2) ◽  
pp. 81-92 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Recurrence Relation ◽  
Exponential Function ◽  
Nonlocal Boundary ◽  
Matrix Exponential ◽  
Finite Difference Approximation ◽  
Partial Differential ◽  
First Order ◽  
Matrix Exponential Function

Parabolic partial differential equations with nonlocal boundary specifications feature in the mathematical modeling of many phenomena. In this paper, numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one-dimensional diffusion equation with a nonlocal constraint in place of one of the standard boundary conditions. The method of lines (MOL) semidiscretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The partial derivative with respect to the space variable is approximated by a second-order finite-difference approximation. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. We use a partial fraction expansion to compute the matrix exponential function via Pade approximations, which is particularly useful in parallel processing. The algorithm is tested on a model problem from the literature.

scholarly journals Vector Additive Decomposition for 2D Fractional Diffusion Equation

2008 ◽  
Vol 13 (2) ◽  
pp. 137-143
Author(s):  
N. Abrashina-Zhadaeva ◽  
N. Romanova
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Analytical Solution ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Finite Difference Approximation ◽  
Partial Differential

Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations.  An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients.  In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

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