The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

An effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix. This approach needs no integration, so all calculations can be easily implemented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.

Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions

Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations

Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.

Two-dimensional triangular functions and their applications to nonlinear 2D Volterra-Fredholm integral equations

Using triangular orthogonal functions for solving Fredholm integral equations of the second kind

The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.

A new set of orthogonal functions and its application to the analysis of dynamic systems

In the present work, Dirac delta function (DF) set and sample-and-hold functions (SHF) set are used for microprocessor-based simulation of discrete time as well as sample-and-hold systems. Such simulations are useful for identification of control systems with known input and output sequence. The presented method utilizes operational matrices of different orders in the DF and SHF domain to develop different operational transfer functions. A few open-loop as well as closed-loop systems have been studied and the simulation results obtained are compared with exact solutions derived with the help of z -transform analysis. Experiments have also been carried out to establish the validity of the proposal.

Anindita Dasgupta

Papers

A new set of orthogonal functions and its application to the analysis of dynamic systems

Microprocessor-based simulation of sampled data systems with/without a hold device using a set of sample-and-hold functions and Dirac delta functions