Variational Methods: The Ritz Method
The Ritz Method is a precursor to the finite element method. Here is a code that provides a solution to an ODE whose solution is known.
The Ritz Method
Problem: Consider the differential equation
-d^2u/dx^2 - u + x^2 = 0 for 0<x<1;
with the boundary conditions:
u(o) = 0, u(1) = 0.
The Ritz method yield an N-parameter Ritz solution U_N = C_1phi_1 + C_2phi_2 + ... C_iphi_i + ... C_Nphi_N, where phi_i = x^i - x^(i+1)
and C_is are obtained from
K_ijC_i = F_i. In this case, K_ij = B(phi_i, phi_j) = int_0^1(dphi_i/dx *dphi_j/dx - phi_iphi_j)dx is the bilinear part of the variational problem and and F_i = -int_0^1(x^2phi_i)dx is the linear part.
引用格式
Angwenyi David (2026). Variational Methods: The Ritz Method (https://ww2.mathworks.cn/matlabcentral/fileexchange/102599-variational-methods-the-ritz-method), MATLAB Central File Exchange. 检索时间: .
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