General Block-pulse Functions Research Articles

OPERATIONAL MATRIX OF GENERALIZED BLOCK PULSE FUNCTIONS FOR SOLVING FRACTIONAL VOLTERRA–FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

This article aims to introduce two effective numerical approaches for solving the mixed linear fractional Volterra-Fredholm integro-differential equation (FVIDE) and multi-higher order with initial conditions. Even though these approaches transfer the integro-differential equations into a system of algebraic equations through operational matrices of generalized block pulse functions, these problems can be transferred to a system of algebraic equations by expanding the solution's multi-highest order derivative through the block pulse functions (BPFs). The block pulse operational matrices for fractional-order integration are derived by expanding the Riemann-Liouville fractional integral for repeated fractional integration in block pulse functions (BPFs). Also, the fractional derivative is done by using the generalized operational matrices of BPFs with the fractional calculus properties in the first scheme. The methods are attractive in the calculation, and examples are supplied to explain their use. The estimated and exact outcomes are compared in a table. In addition, they decrease the error terms in the specified domain using the least squares-error technique. For this, most general programs are written in Python V.3.8.8 software (2021).

Study on the existence and approximate solution of fractional differential equations with delay and its applications to financial models

This paper devoted to investigation on the system of fractional differential equations with delay. We use the Schauder fixed point theorem to prove the existence of solution for this system. In addition, we employ the hybrid functions of general block-pulse functions and the Legendre polynomials to solve our problem. Finally, we study the fractional order of the financial delay system, to illustrate the efficiency of the method.

Numerical approximation of the system of fractional differential equations with delay and its applications

In this paper, the system of fractional differential equations with delay is studied. The computational method based on the hybrid functions of the general block-pulse functions and Legendre polynomials is presented to obtain the numerical results for the solution of our problem. In fact, the system under study is converted to the corresponding nonlinear algebraic system of equations, which is resolved by using the collocation technique. Also, the error estimate for the suggested computational method that verifies the convergence is presented. Finally, we examine two models arising in engineering and applied science, including the Chen and human immunodeficiency viruses systems, to illustrate the efficiency of the method.

Numerical Algorithm to Solve FractionalIntegro-differential Equations Based on OperationalMatrix of Generalized Block Pulse Functions

Numerical Algorithm to Solve FractionalIntegro-differential Equations Based on OperationalMatrix of Generalized Block Pulse Functions

Numerical solutions of optimal control for linear Volterra integrodifferential systems via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid.

Numerical solutions of integrodifferential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials

In this paper, integrodifferential systems are converted into a system of algebraic equations by hybrid of general block-pulse functions and the second Chebyshev polynomials. The approximate solutions of integrodifferential systems are derived. The numerical examples illustrate that the algorithm is valid.

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, the linear-quadratic problem of linear time-varying systems with delays are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples illustrate that the algorithms are valid.

Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials

Using the operational properties of general block-pulse functions and Legendre polynomials, the linear inverse time systems are transformed into a system of algebraic equations. The numerical solutions of the systems are derived. Moreover, applying the results to the linear quadratic optimal control problems, the approximate solutions of optimal control of time delay systems are derived.

Numerical solution of delay systems containing inverse time by hybrid functions

The hybrid functions consisting of general block-pulse functions and Legendre polynomials are presented to solve delay systems containing inverse time. The direct algorithm for a product of a matrix function and a vector function is given. The general operational matrix is introduced. The delay function and inverse time function are expanded by hybrid functions. The approximate solution of delay systems containing inverse time is derived. Numerical examples illustrate that the algorithms are applicable.

A new set of pulse-width modulated generalized block pulse functions (PWM-GBPF) and its application to cross-/auto-correlation of time-varying functions

The present work proposes a significant modification to conventional block pulse functions (BPF) by describing a novel set of pulse-width modulated generalized block pulse functions (PWM-GBPF) which has been utilized to develop generalized correlation matrices of operational nature. These matrices are used to determine cross-/auto-correlation of time-varying functions. Numerical examples are treated to establish the validity of the proposal. Also, a representational error analysis has been carried out for the PWM-GBPF to show that this kind of BPF set introduces a smaller error than the conventional equal width BPF.

Analysis of systems with multiple time-varying delays via generalized block pulse functions

A generalized block pulse function (GBPF) is employed to solve the state equations of systems with multiple time-varying delays. The operational matrix for the integration and the general functional operational matrix of the GBPF are derived. These two operational matrices are introduced to solve the state equation in order to simplify the calculation procedure. A computational algorithm for solving the multi-delay state equation is described. The biggest advantage of using the GBPF is that the time interval of calculation can be adjusted arbitrarily. A small time interval is chosen for a steep rate of change of the state variable, and a large time interval for shower changes. The number of expansion coefficients is thereby reduced and computer time minimized. Two illustrative examples of constant delay and time-varying delay are given and the computational results are satisfactory.

Analysis of linear distributed systems via the single step method of generalized block-pulse functions

Abstract The single step method of generalized block-pulse functions (GBPF) is successfully applied to the analysis of linear distributed system governed by simultaneous first-order linear partial differential equations with variable coefficients. At first, a set of generalized block-pulse functions of varying, adjustable subinterval length is introduced and the GBPFs expansion of the product of two functions is given. Based on the GBPFs, a simple numerical algorithm is developed with adjustable step size to satisfy accuracy considerations. Both discrete and piecewise constant solutions are obtained. A numerical example is given to compare the solution obtained by this method with the exact solution, showing that the result is excellent.

Solution of functional differential equations via generalized block pulse functions

A generalized block pulse function (GBPF) is employed to solve the functional differential equation. The operational matrix for the integration of the GBPF and the stretch matrix are introduced in order to simplify the calculation procedure. The most important advantages of using the present GBPF are the reduction of the number of expansion coefficients to be calculated as well as the minimization of computer time. With the proposed method, the system can be calculated through a large value of final time. In addition, the accuracy of the computational results is greatly increased within an arbitrary desired time interval by increasing the number of expansion coefficients to be calculated. Two illustrative examples are given. Compared with the conventional block pulse function (BPF) method, the proposed approximation is effective and the results are very satisfactory.

Solutions of stiff differential equations via generalized block pulse functions

In this paper, the ordinary block pulse functions with constant subinterval length are modified to generalized block pulse functions (GBPFs) with time-