Solution of the thermal conductivity problem
• The mathematical model of many problems consists of solving a differential equation (or
system of differential equations) with additional conditions.
• We will examine the solution of one-dimensional thermal conductivity problem with the
help of numerical methods.
• A temperature at the point x, i.e. function y(x), of a rod with a length 1 and constant
temperatures at the ends - degrees A and B, is a solution of the differential equation
with boundary conditions y(0) = A, y(1) = B.
• The function p(x) describes the heat conductor properties of a rod, and q(x) and f(x)
intensity of heat exchange through the surface with the outside.
• When p(x) > 0 and q(x) ≥ 0, this problem always has a solution (the unique). However,
we cannot always find it by analytical methods.
• Special numerical methods are then applied.
Finite differences method
• When solving a problem by the finite differences method, a discrete network of the
problem definition area is formed from separate points (in this case of the rod points).
• For example, let's form a smooth discrete network with step h:
• The finite differences method calculates the approximate values yi of the searched
function y(x) at points xi of the discrete network, i.e. we find yi ≈ y(xi), i = 0, .., n.
• The numerical algorithm is constructed in such a way that taking more and more points
(i.e. a smaller step h) gives more and more accurate values of yi.
• These values are found from a system of linear equations, which is formed as follows: at
each internal point xi of the discrete network, a linear equation is formed for the unknowns
yi, by approximating (approximately replacing) the derivatives of the differential equation
with linear algebraic formulas - finite differences.
• The boundary conditions are approximated at the edge points of the network.
Finite differences
• Remember the definition of a derivative:
• We can then approximate the value of the derivative (approximately change) with the help of
a finite difference:
• This derivative approximation is called the right-hand difference. Using Taylor series it can be
proved that its error
• Left difference and its error:
• Central difference and its error:
Finite differences scheme
• The approximation of the second order derivative is obtained as follows:
• Its error:
• We return to the solution of the thermal conductivity problem. For simplicity, let us first
take p(x) = 1, q(x) = 1.
• We approximate the differential problem at the network points by linear system of
equations (also called the finite differences scheme) for the unknowns yi, i = 0, .., n:
• Note that the system matrix is tridiagonal and can be quickly solved by the tridiagonal
matrix algorithm!
Finite differences scheme
• We approximate the term of the thermal conductivity equation (p(x)y')' at the network
points xi as follows:
• Then we approximate the differential problem at the network points regard to the
unknowns yi, i = 0, .., n, with such a finite differences scheme:
• Matrix of the system is tridiagonal. Notice that we approximate the derivative at the
internal points of the network, i.e. replace it with an approximate values. What errors |y(xi) -
yi|, i = 0, .., n, can we expect?
Example of solving the problem of thermal conductivity
• Let's take p(x) = 1 + x, q(x) = 1, f(x) = -(1+ x) and A = 1, B = e:
• According to the created finite differences scheme with steps h = 0.25 and h = 0.125 we get
a tridiagonal system of equations that we can solve by tridiagonal matrix algorithm. In the
following table we compare the approximate values yi obtained using steps h = 0.25 and h =
0.125 with the exact known (in this case) solution y(x) =
