Shahid Saeed Siddiqi

In this paper, the Laguerre polynomials have been manipulated to develop the approximate solutions of the linear system of Fredholm integral equations (SFIEs). The method converted the system of integral equations (IEs) into a system of linear algebraic equations which can be solved using any standard method. The accuracy of the proposed method has, also, been discussed and explored via different examples which shows that the method is more efficient than the existing techniques.

The induction motor behaviour is represented by a fifth order differential equation model. Addition of a torque correction factor to the model accurately reproduces the transient torques and instantaneous real and reactive power flows of the full seventh order differential equation model. The variational iteration homotopy perturbation method (VIHPM) is employed to solve the seventh order boundary value problems. The approximate solutions of the problems are obtained in terms of a rapidly convergent series. Several numerical examples are given to illustrate the implementation and efficiency of the method.

Homotopy perturbation method is used for solving the multi-point boundary value problems. The approximate solution is found in the form of a rapidly convergent series. Several numerical examples have been considered to illustrate the efficiency and implementation of the method and the results are compared with the other methods in the literature.

The paper proposes, an algorithm to produce novel m-point (for any integer m>=2) binary non-stationary subdivision scheme. It has been developed using uniform trigonometric B-spline basis functions and smoothness is being analyzed using the theory of asymptotically equivalence. The results show that the most of well-known binary approximating schemes can be considered as the non-stationary counterpart of the proposed algorithm. Furthermore, the schemes developed by the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some examples are considered, by choosing an appropriate tension parameter 0<alpha<pi/3, to show the usefulness.

Adomian decomposition method is used for solving the seventh order boundary value problems. The approximate solutions of the problems are calculated in the form of a rapid convergent series and not at grid points. Two numerical examples have been considered to illustrate the efficiency and implementation of the method.

A new binary four-point approximating subdivision scheme has been presented that generates the limiting curve of C ¹ continuity. A global tension parameter has been introduced to improve the performance of the binary four-point approximating subdivision scheme that generates a family of C ¹ limiting curves. The ternary four-point approximating subdivision scheme has also been introduced that generates a limiting curve of C ² continuity. The proposed schemes are close to being interpolating. The Laurent polynomial method has been used to investigate the order of derivative continuity of the schemes and Hölder exponents of the schemes have also been calculated. Performances of the subdivision schemes have been exposed by considering several examples.