Abstract
M.Sc.
There have been an extensive study on solutions of differential equations
modeling physical phenomena that blows up in finite time. The blow-up
time often represents an important change in the properties of such models
and hence it is very important to compute it as accurate as possible. In
this work, an adaptive in time numerical method for computing blow-up
solutions for nonlinear ODEs is introduced. The method is named implicit
midpoint-implicit Euler method (IMIE) and is based on the implicit Euler
and the implicit midpoint method. The method is used to compute blow-up
time for different examples of ODEs, PDEs and VIDEs. The PDEs studied
are reaction-diffusion equations whereby the method of lines is first used to
discretize the equation in space to obtain a system of ODEs.
Quadrature rules are used to approximate the integral in the VIDE to
get a system of ODEs. The IMIE method is then used then to solve the
system of ODEs. The results are compared to results obtained by the PECEIE
method and Matlab solvers ode45 and ode15s. The results show that
the IMIE method gives better results than the PECE-IE and ode15s and
compares quite remarkably with the 4th order ode45 yet it is of order 1 with
order 2 superconvergence at the mesh points.