Muhammad I. Bhatti

Numerical solution of KdV equation using modified Bernstein polynomials

Abstract Here we present an algorithm for approximating numerical solution of Korteweg–de Vries (KdV) equation in a modified B-polynomial basis. A set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable. Galerkin method is used to determine the expansion coefficients to construct initial trial functions. For the time variable, the system of equations is solved using fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. We have presented our numerical result with an exact analytical result. Excellent agreement is found between exact and approximate solutions. This procedure has a potential to be used in more complex system of differential equations where no exact solution is available.

Using an Inquiry Approach to Teach Science to Secondary School Science Teachers.

Leaders in science education have actively promoted inquiry science since the 1960s and continue to do so today. The US National Science Education Standards recommend that science instruction and learning should be well grounded in inquiry. In spite of these efforts, however, little has changed in the way science is taught. Teacher-talk and textbooks are still the primary providers of science information for students. The objective of this paper is to: (a) define inquiry as a strategy for teaching science, (b) review the history of inquiry science teaching, and (c) present the Physics by Inquiry model for in-servicing middle school science teachers in order to provide assistance for teachers to successfully implement an inquiry approach to teaching science.

The calculation of integrals involving B-splines by means of recursion relations

A procedure is given for constructing the exact integrals involving B-splines using recursion relations. The recursive integrals form the basic constituents for the exact evaluation of integrals that appear in the calculation of atomic and molecular properties. The method can be applied to solve differential equations as well as to produce a complete set of basis functions that may approximate a function arbitrarily well depending on the degree k and the number of B-splines that are employed in the approximation. The advantage of this method is that the recursions developed over a fixed interval represent exact results for the particular integral involved. Several examples are also provided to show how these recursion relations can be applied to evaluate integrals involving multiple B-splines of same or different degree. Closed forms of these integrals involve complicated recursion relations.

Solutions of the radial Dirac equation in a B-polynomial basis

An algorithm is given for constructing accurate solutions to the radial Dirac equation in a B-polynomial basis set. The B-polynomial Galerkin method has been applied to produce the spectrum of the Dirac equation for the bound states of hydrogenic systems. Matrix formulation is used throughout the entire procedure and boundary conditions are applied to generate finite discrete eigenvalues, which include both negative and positive energies as well as corresponding states. An excellent agreement is found between previously existing accurate calculations. To check the quality of the spectrum, the resulting basis sets are used to evaluate the TRK sum rules. The procedure can be readily extended to produce the spectrum of complex systems.