Tariq Aziz

Spline methods for the solution of fourth-order parabolic partial differential equations

In this paper a fourth-order non-homogeneous parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time. Stability analysis of the method has been carried out. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our method. We also obtain two new high accuracy schemes of O(k^4,h^6) and O(k^4,h^8) and two new schemes which are analogues of Jains formula for the non-homogeneous case. Comparison of our results with those of some known methods show the superiority of the present approach.

A survey on parametric spline function approximation

This survey paper contains a large amount of material and indeed can serve as an introduction to some of the ideas and methods for the solution of ordinary and partial differential equations starting from Schoenbergs work [Quart. Appl. Math. 4 (1946) 345-369]. The parametric spline function which depends on a parameter @w>0, is reduces to the ordinary cubic or quintic spline for @w=0. A note on parametric spline function approximation, which is special case of this work has been published in [Comp. Math. Applics. 29 (1995) 67-73]. This article deals with the odd-order parametric spline relations.

A fourth-order finite-difference method based on non-uniform mesh for a class of singular two-point boundary value problems

Abstract We present a three-point finite-difference method based on non-uniform mesh for a class of singular two-point boundary value problem (x α y′)′=f(x,y) y(0)=A, y(1)=B, 0⩽α We show that the method, based on non-uniform mesh, provides O(h4)-convergent approximations. This method is illustrated by two numerical examples, one is linear and the other is non-linear.

Sextic spline solution of a singularly perturbed boundary-value problems

Abstract A sixth-order uniform mesh difference scheme using sextic splines for solving a self-adjoint singularly perturbed two-point boundary-value problem arising in the study of chemical reactor theory, of the form - e u ″ + p ( x ) u = f ( x ) , p ( x ) > 0 , u ( 0 ) = α o , u ( 1 ) = α 1 is derived. Our scheme leads to a pentadiagonal linear system. The convergence analysis is given and the method is shown to have fifth-order convergence. Numerical illustrations are given to confirm the theoretical analysis of our method.

Sextic spline solution for solving a fourth-order parabolic partial differential equation

A fourth-order parabolic partial differential equation in one space variable which arises in the study of transverse vibrations of a uniform flexible beam is solved. The numerical solution is obtained by using a new three-level method based on a sextic spline in space and finite-difference discretization in time. Stability analysis of the method has been carried out. It is shown that we obtain a scheme of O(k 2+h 4) and O(h 4+h 2 k 2). The method is tested on a problem which has appeared in physical applications. Comparison with some known methods shows the superiority of the present method.

A non-uniform mesh finite difference method and its convergence for a class of singular two-point boundary value problems

In the present paper, we discuss the construction of second-order three-point finite difference method based on non-uniform mesh for a class of singular two-point boundary value (BV) problem This is the method and its second-order convergence for various α ∈ (0, 1) are illustrated by a numerical example.

Tension spline method for second-order singularly perturbed boundary-value problems

We consider a difference scheme based on cubic spline in tension for second-order singularly perturbed boundary-value problem of the form The method is shown to have second- and fourth-order convergence depending on the choice of parameters λ1 and λ2 involved in the method. The method is tested on an example and the results found to be in agreement with the theory.

A variable-mesh approximation method for singularly perturbed boundary-value problems using cubic spline in tension

A non-uniform mesh difference scheme using cubic spline in tension is presented to solve a class of non-turning point singularly perturbed two point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative subject to Dirichlet-type boundary conditions. To demonstrate the applicability of the proposed method, two numerical examples have been solved and the results are presented along with their comparison with those obtained with and without variable mesh. This paper is a continuation of the previous work [Aziz, T. and Khan, A. (2002). A spline method for second order singularly-perturbed boundary-value problems. J. Comput. Appl. Math., 147(2), 445–452.] given for uniform mesh case.

Quintic splines method for second-order boundary value problems

Abstract We develop a numerical method for computing smooth approximations to the solution of a system of second-order boundary value problems associated with obstacle, unilateral and contact problems based on uniform mesh quintic splines. It is shown that this method gives better approximations than those produced by other collocation, finite-difference and spline methods. A numerical example is given to illustrate the applicability of the new method.

The Turn of the Month Effect in Asia-Pacific Markets: New Evidence

A predictable pattern in equity returns based on the calendar time is dubbed as calendar anomaly. The prevalence of calendar anomalies is considered evidence against the efficient market hypothesis. This article examines one of the most important calendar anomalies, the turn-of-the-month (TOM) effect, in 12 major Asia-Pacific markets during the period January 2000 to April 2015, using both parametric and non-parametric tests. Under investigation, 11 out of 12 markets exhibit significant TOM effects that are independent of the turn-of-the-year (TOY) effect. Moreover, these effects are not present during the period of financial crisis. The persistence of the TOM effect in these markets, even after a quarter of a century of its initial reporting, is a puzzle which needs an explanation.