

Title of Thesis
HigherOrder Techniques For Heat Equation Subject
To NonLocal Specifications 
Author(s)
MUHAMMAD
AZIZURREHMAN 
Institute/University/Department
Details Department of Mathematics / GC University,
Lahore 
Session 2009 
Subject Mathematics 
Number of Pages 130 
Keywords (Extracted from title, table of contents and
abstract of thesis) HigherOrder, Techniques, Heat,
Equation, Subject, NonLocal, Specifications, homogeneous, Numerical
Experiments 
Abstract Higherorder
numerical techniques are developed for the solution of
(i) homogeneous heat equation ut = u_{xx}
and
(ii) inhomogeneous heat equation ut = 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 nonlocal boundary
condition(s)
(i) Rb 0 u(x; t)dx = M(t) 0 < t · T; 0 < b < 1
(ii) u(0; t) = R1 0 Á(x; t)u(x; t)dx + g1(t); 0 < t <= T and
(iii) u(1; t) = R1 0 Ã(x; t)u(x; t)dx + g2(t); 0 < t <= T
as appropriate.
The integral conditions are approximated using Simpson's 1/3 rule
while the space derivatives are approximated by higherorder finite
difference approximations. Then method of lines, semidiscritization
approach, is used to transform the model partial differential
equations into systems of firstorder linear ordinary differential
equations whose solutions satisfy recurrence relations involving
matrix exponential functions. The methods are higherorder 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 literature 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.

