Adem Kilicman

Kronecker operational matrices for fractional calculus and some applications

The problems of systems identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haar and many other functions. In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann-Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.

A new addition formula for elliptic curves over GF(2/sup n/)

We propose an addition formula in projective coordinates for elliptic curves over GF(2/sup n/). The new formula speeds up the elliptic curve scalar multiplication by reducing the number of field multiplications. This was achieved by rewriting the elliptic curve addition formula. The complexity analysis shows that the new addition formula speeds up the addition in projective coordinates by about 10-2 percent, which leads to enhanced scalar multiplication methods for random and Koblitz curves.

On the applications of Laplace and Sumudu transforms

In this paper, we study the properties of Sumudu transform and relationship between Laplace and Sumudu transforms. Further, we also provide an example of the double Sumudu transform in order to solve the wave equation in one dimension which is having singularity at initial conditions.

A new integral transform and associated distributions

In this paper, we generalize the concepts of a new integral transform, namely the Sumudu transform, to distributions and study some of their properties. Further, we also apply this transform to solve one-dimensional wave equation having a singularity at the initial conditions.

Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform

A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

An application of double Laplace transform and double Sumudu transform

In this paper, we produce some properties and relationship between double Laplace and double Sumudu transforms. Further, we use the double Sumudu transform to solve wave equation in one dimension having singularity at initial conditions.

On Sumudu Transform and System of Differential Equations

The regular system of differential equations with convolution terms solved by Sumudu transform.

Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform

The homotopy perturbation method, Sumudu transform, and He’s polynomials are combined to obtain the solution of fractional Black-Scholes equation. The fractional derivative is considered in Caputo sense. Further, the same equation is solved by homotopy Laplace transform perturbation method. The results obtained by the two methods are in agreement. The approximate analytical solution of Black-Scholes is calculated in the form of a convergence power series with easily computable components. Some illustrative examples are presented to explain the efficiency and simplicity of the proposed method.

A note on solutions of wave, Laplace's and heat equations with convolution terms by using a double Laplace transform

In this study we consider general linear second-order partial differential equations and we solve three fundamental equations by replacing the non-homogeneous terms with double convolution functions and data by a single convolution.

A note on defining singular integral as distribution and partial differential equations with convolution term

In this study first we consider the singular integrals as generalized functions in two dimensions and then we solve the non-homogeneous wave equation with convolutional term by using the generalized functions as boundary conditions.