Abstract:
Higher-order numerical techniques are developed for the solution of
(i) homogeneous heat equation u t = u xx
and
(ii) inhomogeneous heat equation u t = u xx + s(x, t)
subject to initial condition u(x, 0) = f (x), 0 < x < 1, boundary condition
u(0, t) = g(t)0 < t ≤ T and with non-local boundary condition(s)
(i)
b
0
u(x, t)dx = M (t) 0 < t ≤ T, 0 < b < 1
(ii) u(0, t) =
(iii) u(1, t) =
1
0
φ(x, t)u(x, t)dx + g 1 (t), 0 < t ≤ T and
1
0
ψ(x, t)u(x, t)dx + g 2 (t), 0 < t ≤ T
as appropriate.
The integral conditions are approximated using Simpson’s
1
3
rule while the
space derivatives are approximated by higher-order finite difference approxi-
mations. Then method of lines, semidiscritization approach, is used to trans-
form the model partial differential equations into systems of first-order linear
ordinary differential equations whose solutions satisfy recurrence relations in-
volving matrix exponential functions. The methods are higher-order accurate
in space and time and do not require the use of complex arithmetic. Parallel
algorithms are also developed and implemented on several problems from lit-
erature and are found to be highly accurate. Solutions of these problems are
compared with the exact solutions and the solutions obtained by alternative
techniques where available.
