Solving One-Dimensional Partial Differential Equations

Solving One-Dimensional Partial Differential Equations

Copyright: © 2021 |Pages: 25
DOI: 10.4018/978-1-7998-7078-4.ch007
OnDemand:
(Individual Chapters)
Available
$37.50
Current Special Offers
No Current Special Offers
TOTAL SAVINGS: $37.50

Abstract

This chapter describes the pdepe command, which is used to solve spatially one-dimensional partial differential equations (PDEs). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the pdepe solver uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics are presented in the final part of the chapter. They illustrate how to solve: heat transfer PDE with temperature dependent material properties, startup velocities of the fluid flow in a pipe, Burger's PDE, and coupled FitzHugh-Nagumo PDE.
Chapter Preview
Top

7.1. Standard Forms Of The Pde, Initial And Boundary Conditions Required By The Pde-Solver

7.1.1. PDE in Standard Form

To solve one-dimensional PDEs, the pdepe command is provided by MATLAB®. The command is designed to solve 1D PDE that can be presented in the following standard form:

978-1-7998-7078-4.ch007.m01
where u is the function that should be defined as a result of the PDE solution; t is the solution time range t0 (initial) ... tf (final) and the coordinate x is the coordinate range that is varied between x=a and x=b; m can be 0, 1 or 2 corresponding to Cartesian, cylindrical or spherical coordinates respectively;

978-1-7998-7078-4.ch007.m02 is called a flux term while 978-1-7998-7078-4.ch007.m03 − a source term.

General form of this equation solution is u as function of coordinate and time - u(x,t).

In accordance with the presented standard equation form, the pdepe command is intended to solve PDEs of the first or second order with respect to coordinate x. The solving PDE can be an equation of the elliptical or parabolic type, which are usually solved in mechanics. Some examples of PDEs and their adaptation to the standard equation are presented in Table 1.

Table 1.
Some PDEs in traditional and pdepe-adopted forms
PDEPDE –Adopted, Standard FormMatching Terms of PDE and the pdepe Standard Form
mcfs
978-1-7998-7078-4.ch007.m04978-1-7998-7078-4.ch007.m050978-1-7998-7078-4.ch007.m06978-1-7998-7078-4.ch007.m070
978-1-7998-7078-4.ch007.m08978-1-7998-7078-4.ch007.m0901978-1-7998-7078-4.ch007.m10978-1-7998-7078-4.ch007.m11
978-1-7998-7078-4.ch007.m12978-1-7998-7078-4.ch007.m1301978-1-7998-7078-4.ch007.m14978-1-7998-7078-4.ch007.m15
978-1-7998-7078-4.ch007.m16978-1-7998-7078-4.ch007.m171978-1-7998-7078-4.ch007.m18978-1-7998-7078-4.ch007.m190
978-1-7998-7078-4.ch007.m20978-1-7998-7078-4.ch007.m2101978-1-7998-7078-4.ch007.m220
978-1-7998-7078-4.ch007.m23978-1-7998-7078-4.ch007.m240π2978-1-7998-7078-4.ch007.m250
978-1-7998-7078-4.ch007.m26978-1-7998-7078-4.ch007.m2701978-1-7998-7078-4.ch007.m28978-1-7998-7078-4.ch007.m29
978-1-7998-7078-4.ch007.m30978-1-7998-7078-4.ch007.m3121978-1-7998-7078-4.ch007.m32978-1-7998-7078-4.ch007.m33

Complete Chapter List

Search this Book:
Reset