

Title of Thesis
Numerical Solution Of Boundary Value And InitialBoundaryvalue
Problems Using Spline Functions 
Author(s)
Fazalihaq 
Institute/University/Department
Details Faculty of Engineering Sciences / GIK Institute
of Engineering Sciences and Technology, Topi 
Session 2009 
Subject Engineering Sciences 
Number of Pages 159 
Keywords (Extracted from title, table of contents and
abstract of thesis) Quartic, Quintic, Error, Solution,
Numerical, Propagation, Nonlinear, approximations, Boundary,
Initial, Spline, Benchmark, Functions, Problems 
Abstract
The following two types of problems in
differential equations are investigated:
(i) Second and sixthorder linear and nonlinear
boundaryvalue problems inordinary differential equations using
nonpolynomial spline functions.
(ii) One dimensional nonlinear Initialboundaryvalue problems in
partialdifferential equations using Bspline collocation method.
Polynomial splines, nonpolynomial splines and Bsplines are
introduced. Somewell known results and preliminary discussion about
convergence analysis of boundaryvalue problems and stability theory
are described.Quartic nonpolynomial spline functions are used to
develop numerical methods for computing approximations to the
solution of linear, nonlinear and system of secondorderboundaryvalue
problems and singularly perturbed boundaryvalue
problems.Convergence analysis of the method is discussed.Numerical
methods for computing approximations to the solution of linear
andnonlinear sixthorder boundaryvalue problems with twopoint
boundary conditions are developed using septic nonpolynomial
splines. Second, Fourth and Sixthorderconvergence is
obtained.Numerical method based on collocation method using quartic
Bspline functions forthe numerical solution of onedimensional
modified equal width (MEW) wave equationis developed. The scheme is
shown to be unconditionally stable using VonNeumannapproach.
Propagation of a single wave, interaction of two waves and
Maxwellian initial condition are discussed.
Algorithms based on quartic and Quintic Bspline collocation methods
are designed or the numerical solution of the modified regularized
long wave (MRLW) equation.Stability analysis is performed.
Propagation of a solitary wave, interaction of multiplesolitary
waves, and generation of train of solitary waves are also
investigated.
Quartic and quintic Bspline functions have been used to develop
collocationmethods for the numerical solution of
KuramotoSivashinsky (KS) equation. Also,using splitting technique,
the equation is reduced to a problem of second order in space.
Using error norms L2 and L∞ and conservative properties of mass,
momentum andenergy, accuracy and efficiency of the suggested methods
is established through comparison with the existing numerical
techniques. Performance of the algorithms is tested through
application of the methods on benchmark problems.

