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Numerical Solution of Boundary-Value and Initial-Boundary-Value Problems using Spline Functions.

Haq, Fazal-i- (2009) Numerical Solution of Boundary-Value and Initial-Boundary-Value Problems using Spline Functions. PhD thesis, Ghulam Ishaq Khan Institute of Engineering Sciences & Technology, Swabi.

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Abstract

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 investigated. 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.

Item Type:Thesis (PhD)
Uncontrolled Keywords:Quartic, Quintic, Error, Solution, Numerical, Propagation, Nonlinear, approximations, Boundary, Initial, Spline, Benchmark, Functions, Problems
Subjects:Engineering & Technology (e)
ID Code:5238
Deposited By:Mr. Javed Memon
Deposited On:01 Oct 2010 13:26
Last Modified:30 May 2011 09:38

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