Title of Thesis
Numerical Solution Of Boundary Value And Initial-Boundary-value
Problems Using Spline Functions
Faculty of Engineering Sciences / GIK Institute
of Engineering Sciences and Technology, Topi
|Number of Pages|
|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
The following two types of problems in
differential equations are investigated:
(i) Second and sixth-order linear and nonlinear
boundary-value problems inordinary differential equations using
non-polynomial spline functions.
(ii) One dimensional nonlinear Initial-boundary-value problems in
partialdifferential equations using B-spline collocation method.
Polynomial splines, non-polynomial splines and B-splines are
introduced. Somewell known results and preliminary discussion about
convergence analysis of boundary-value problems and stability theory
are described.Quartic non-polynomial spline functions are used to
develop numerical methods for computing approximations to the
solution of linear, nonlinear and system of secondorderboundary-value
problems and singularly perturbed boundary-value
problems.Convergence analysis of the method is discussed.Numerical
methods for computing approximations to the solution of linear
andnonlinear sixth-order boundary-value problems with two-point
boundary conditions are developed using septic non-polynomial
splines. Second-, Fourth- and Sixth-orderconvergence is
obtained.Numerical method based on collocation method using quartic
B-spline functions forthe numerical solution of one-dimensional
modified equal width (MEW) wave equationis developed. The scheme is
shown to be unconditionally stable using Von-Neumannapproach.
Propagation of a single wave, interaction of two waves and
Maxwellian initial condition are discussed.
Algorithms based on quartic and Quintic B-spline 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
Quartic and quintic B-spline functions have been used to develop
collocationmethods for the numerical solution of
Kuramoto-Sivashinsky (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.