B-Spline collocation method for numerical solution of nonlinear kawahara and modified kawahara equations

In this paper, a collocation method is applied for solving the Kawahara and modified Kawahara equations. For the spatial discretization, we use the sextic B-spline collocation (SBSC) method on uniform meshes, finite difference scheme is employed for the time discretization. The stability analysis of the collocation methods are examined by the Von Neumann approach. Numerical results demonstrate the efficiency and accuracy of the proposed methods.Publisher's Versio

A Novel Numerical Approach for Odd Higher Order Boundary Value Problems

In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth, seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary conditions.  We exploit Galerkin weighted residual method with Legendre polynomials as basis functions. Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods. Keywords: Galerkin method, fifth, seventh and ninth order linear and nonlinear BVPs, Legendre Polynomials

A cubic spline collocation method for solving Bratu’s Problem

This paper, we develop a numerical method for solving a Bratu-type equations by using the cubic splinecollocation method (CSCM) and the generalized Newton method. This method converges quadratically if arelation-ship between the physical parameter ? and the discretization parameter h is satisfied. An error estimatebetween the exact solution and the discret solution is provided. To validate the theoretical results, Numericalresults are presented and compared with other collocation methods given in the literature.Keywords: Bratu-type equations, Boundary value problems, Cubic spline collocation method

An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature

Two Legendre-Dual-Petrov-Galerkin Algorithms for Solving the Integrated Forms of High Odd-Order Boundary Value Problems

Two numerical algorithms based on dual-Petrov-Galerkin method are developed for solving the integrated forms of high odd-order boundary value problems (BVPs) governed by homogeneous and nonhomogeneous boundary conditions. Two different choices of trial functions and test functions which satisfy the underlying boundary conditions of the differential equations and the dual boundary conditions are used for this purpose. These choices lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost. The various matrix systems resulting from these discretizations are carefully investigated, especially their complexities and their condition numbers. Numerical results are given to illustrate the efficiency of the proposed algorithms, and some comparisons with some other methods are made

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms

A fourth-order numerical scheme for solving the modified Burgers equation

AbstractA finite-difference scheme based on fourth-order rational approximants to the matrix–exponential term in a two-time level recurrence relation is proposed for the numerical solution of the modified Burgers equation. The resulting nonlinear system, which is analyzed for stability, is solved using an already known modified predictor–corrector scheme. The results arising from the experiments are compared with the corresponding ones known from the available literature

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature