15 research outputs found
B-spline differential quadrature method for the modified burgers’ equation
In this study, the Quintic B-spline Differential Quadrature method (QBDQM) is applied to find the numerical solution of the modified Burgers’ equation (MBE). The efficiency and accuracy of the method are measured by calculating the maximum error norm L∞ and the discrete root mean square error L2. The obtained numerical results are compared with published numerical results and the comparison shows that the method is an effective numerical scheme to solve the MBE. A rate of convergence analysis is also give
Two different methods for numerical solution of the modified burgers’ equation
A numerical solution of the modified Burgers’ equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM)
method. The accuracy and efficiency of the methods are discussed by computing \u1d43f����2 and \u1d43f����∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve
the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM
Numerical investigations of shallow water waves via generalized equal width (GEW) equation
In this article, a mathematical model representing solution of the nonlinear generalized
equal width (GEW) equation has been considered. Here we aim to investigate solutions
of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite
element method. At first Galerkin finite element method is proposed and a priori bound
has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element
approximation have been studied respectively. In addition to that a powerful Fourier series
analysis has been performed and indicated that our method is unconditionally stable.
Finally, proficiency and practicality of the method have been demonstrated by illustrating it
on two important problems of the GEW equation including propagation of single solitons
and collision of double solitary waves. The performance of the numerical algorithm has
been demonstrated for the motion of single soliton by computing L∞ and L2 norms and
for the other problem computing three invariant quantities I1, I2 and I3. The presented
numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid
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
Cubic B-spline collocation method for coupled system of ordinary differential equations with various boundary conditions
This paper is concerned with collocation approach using cubic B-spline to solve coupled system of boundary value problems with various boundary conditions. The collocation equations are methodically derived using cubic B splines, for problems with Dirichlet data and an iterative method with assured convergence is described to solve the resulting system of algebraic equations. Problems with Cauchy or mixed boundary condition have been converted into series of Dirichlet problems using the bisection method. Nonlinear problem is linearized using quasilinearization to be handled by our method. Fourth order equation is converted into a coupled second order equations and solved by the proposed method . Several illustrative examples are presented with their error norms and order of convergence.Publisher's Versio
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
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. Econ. 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.South Afric