Finite Element Solution of the
Time Dependent 1D Heat Equation
using Implicit Time Stepping

FD1D_HEAT_IMPLICIT is a MATLAB library which solves the time-dependent 1D heat equation, using the finite element method in space, and an implicit version of the method of lines, using the backward Euler method, to handle integration in time.

This program solves

        dUdT - k * d2UdX2 = F(X,T)
over the interval [A,B] with boundary conditions
        U(A,T) = UA(T),
        U(B,T) = UB(T),
over the time interval [T0,T1] with initial conditions
        U(X,T0) = U0(X)

The spatial derivatives are approximated using the finite difference method, with piecewise linear elements.

The solver applies an implicit backward Euler approximation to the first derivative in time.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


FEM1D_HEAT_IMPLICIT is available in a MATLAB version.

Related Data and Programs:

FD1D_HEAT_IMPLICIT, a MATLAB library which uses the finite difference method and implicit time stepping to solve the time dependent heat equation in 1D.

FEM1D_HEAT_EXPLICIT, a MATLAB library which uses the finite element method and explicit time stepping with the forward Euler method to solve the time dependent heat equation in 1D.

FEM1D_HEAT_STEADY, a MATLAB program which uses the finite element method to solve the steady (time independent) heat equation in 1D.


  1. George Lindfield, John Penny,
    Numerical Methods Using MATLAB,
    Second Edition,
    Prentice Hall, 1999,
    ISBN: 0-13-012641-1,
    LC: QA297.P45.

Source Code:

Examples and Tests:

TEST01 runs with initial condition 50 everywhere, boundary conditions of 90 on the left and 70 on the right, and no right hand side source term.

TEST02 uses an exact solution of g(x,t) = exp ( - t ) .* sin ( sqrt ( k ) * x ).

TEST03 runs on the interval -5 <= X <= 5, with initial condition 15 on the entire left and 25 on the entire right. The solution should settle down to a straight line from the left boundary to the right.

You can go up one level to the MATLAB source codes.

Last revised on 31 January 2012.