Trigonometric b-spline based approach for solving initial and boundary value problems of dispersive equations

Various type of numerical methods are developed by employing spline function for solving dispersive PDEs such as finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). Each method inherits certain drawback like complexity, high computational cost and required the trial function or limitation to certain cases. Due to those constraints, FDM incorporated with B-spline function is introduced to solve partial differential equation (PDE). The main aim of this thesis is to explore the accurate and reliable solution to dispersive PDEs. The cubic trigonometric B-spline method (CuTBS), cubic hybrid B-spline method (CuHBS) and two new methods namely quintic trigonometric B-spline method (QuTBS) and hybrid quintic B-spline (QuHBS) method are chosen with the finite difference scheme to solve the third order dispersive PDEs called Modified Regularised Long Wave equation (MRLW), Benjamin -Bona- Mahony-Burgers equation (BBM-Burgers) and Modified Equal Width equation (MEW). The proposed methods produce the numerical solutions that are found to be better or in good compliance with those present methods in literature. Comparison of the maximum error ( L ) and the Euclidean error ( 2 L ) from the literatures are also done for each example. The performance of the proposed methods are identified to be more accurate than CuTBS and QuTBS method. In order to analyze the stability of the proposed methods, Von-Neumann stability analysis is applied to the linearized schemes. The schemes have been identified to be unconditionally stable. The highlights of the proposed method can be counted as follows: The diagonal matrix obtained from these methods helps in computing accurate solution and can be employed to easily solve PDEs with certain conditions. These methods have an edge over various methods as it approximates the solution at all point in the domain rather than the grid points. The main contribution of this thesis are the development of the quantic splines methods and the applicability to solving dispersive partial differential equations.