1D Linear Differential Equations - Finite Element Method
1D - Boundary Value Problem
Boundary value problem to solve
where external force
for x in [0, L] and boundary conditions are: phi(0) = phiL, phi(L) = phiR.
The problem is trivial and can be easily solve analytically.
Analytical solution.
The solution reads:
Because of its simplicity the presented problem is a good example to test the numerical solution obtained with help of fem1D_lde.m program (written in Octave and Matlab code).
FEM solution.
FEM solution can be obtained from the Octave (MATLAB) program:
fem1D_lde.m consists of the function fem1D_lde(L, h1, NL, NI, NR, phiL, phiR, q, k, linear, varargin) where input arguments are:
- L - domain length
- h1 - length of edge compartments
- NL - number of edge compartments at the left side
- NI - number of internal compartments
- NR - number of the edge compartments at the right side
- phiL - left boundary value
- phiR - right boundary value
- q - force as anonymous function: @(x, L)(-2*x/L + 1)
- k - coefficient
- linear - can be set as 0 or 1; 0 means that QI vector is computed by integration whereas 1 means that it is obtained from linear approximation (see DEMO)
- varargin - optional input arguments; the first from a set of variable input arguments is the analytical solution given as anonymous function: @(x, k, L, phiL, phiR) (-1/(3*L*k)*x.^3 + 1/(2*k)*x.^2 - (1/(6*k)*L - (phiR - phiL)/L)*x + phiL)
To test the finite element method try to consider the following situations:
- NL = NR = 0, NI = 4;
How good is a numerical solution? How large is the element size h?
- When is it useful to set NL and NR different from 0?
- Is it a difference between QI vectors computed for different values of the parameter linear (0 or 1)?
Note! If do not feel quite familiar with the Octave (Matlab) language of programming and you do not want to go through them by your own you can always join the course:
Introduction to Octave and Matlab.
Let us see how easy and fast is programming using the Matrix Laboratory.
Last update: April 2, 2019
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