Block Pulse Research Articles

Numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using two-dimensional block pulse operational matrix

In this paper, our purpose is to construct a two-dimensional fractional integral operational matrix and its use for the numerical solution of two-dimensional fractional integral equations. We use these operational matrices and properties of two-dimensional block pulse functions (2D-BPFs), to reduce two-dimensional fractional integral equations (2D-FIEs) to a system of algebraic equations. Obtained algebraic system based on the original problem can be linear or nonlinear. Then we show convergence of the proposed methods and we find the error bounds. To show the accuracy, efficiency and speed of the proposed method linear and nonlinear examples are presented.

On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation

AbstractThis paper applied the idea of block pulse (BP) transform in the equivalent linearization of a nonlinear system. The BP transform gives effective tools to approximate complex problems. The main goal of this work is on using BP transform properties in process of linearization. The accuracy of the proposed method compared with the other equivalent linearization including the stochastic equivalent linearization and the regulation linearization methods. Numerical simulations are applied to the nonlinear Van der Pol oscillator system under Gaussian white noise excitation to demonstrate the feasibility of the present method. Different values of nonlinearity are considered to show the effectiveness of the present method. Besides, by comparing the mean-square responses for divers values of nonlinearity and excitation intensity depicted the present method is able to approximate the behavior of nonlinear system and is in agreement with other methods.

Solving the Interval Riccati differential equation by Wavelet operational matrix method

Riccati differential equation is an important equation, in many fields of engineering and applied sciences, so recently lots of methods have been proposed to solve this equation. Haar Wavelet operational matrix,is one of the effective methods to solve this equation, that is very simple and easy, compared to other orders. In this paper, we want to solve the nonlinear riccati differential equation in interval initial condition. first we simplify it by using the block pulse function to expand the Haar wavelet one. we have three cases for each interval, but now it can be solved for positive interval Haar coefficients. The results reveal that the proposed method is very effective and simple.

Numerical solutions of stochastic Lotka-Volterra equations via operational matrices

In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.

Finite‐Element Analysis of a Varying‐Width Bistable Piezoelectric Energy Harvester

AbstractIn this study, a finite‐element analysis of a varying‐width piezoelectric energy harvester (PEH) has been presented. A varying‐width piezoelectric energy harvester (cantilever type) has variation in the width at definite intervals along its length. To harvest the energy over the wide frequency range of environmental vibrations, nonlinearity is introduced in the stiffness by mean of two neodymium magnets. The width of the proposed varying‐width cantilever PEH is calculated using block pulse functions (BPFs). The objective of applying BPFs is to proximate a triangular PEH (with fixed base) into a proposed varying‐width PEH with three rectangular sections along length. The use of BPFs enable the use of rectangular patches of lead zirconate titanate (PZT‐5A). This leads to a reduction in cost as machining PZT at other than straight cuts results in an extensive increase in the production costs. The varying‐width piezoelectric cantilever beam is subjected to harmonic base excitations by applying a vertical acceleration of 0.2 g (g=9.81 m s−2). Numerical study indicates that the bistable varying‐width PEH generates at least two times the average power than that generated by a bistable uniform‐width PEH for the same volume of piezoelectric material and for the same linear natural frequency. Furthermore, the bistable varying‐width PEH is optimized using a genetic algorithm technique to maximize the mean power density.

Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel

In this paper, numerical solutions of the linear and nonlinear fractional integro- differential equations with weakly singular kernel where fractional derivatives are considered in the Caputo sense, have been obtained by Legendre wavelets method. The block pulse functions and their properties are employed to derive a general procedure for forming the operational matrix of fractional integration for Legendre wavelets. The application of this matrix for solving initial problem is explained. The mentioned equations are transformed into a system of algebraic equations. The error analysis of the proposed method is investigated. Finally, some numerical examples are shown to illustrate the efficiency of the approach.

Deterministic analysis of distributed order systems using operational matrix

Recently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines. However, only a few numerical methods are available for analyzing the distributed order systems. This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational matrix technique. The proposed method reduces different analysis problems to a system of algebraic equations by using block pulse functions, which makes it easy to handle an arbitrary input. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method. The proposed method was found to be an efficient tool for analyzing linear distributed order systems.

Wavelets Galerkin method for solving stochastic heat equation

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Approximation Algorithm for Multi-input and Multi-output System using Generalized Block Pulse Integration Operational Matrices

Approximation Algorithm for Multi-input and Multi-output System using Generalized Block Pulse Integration Operational Matrices

Analysis of time-varying delay systems by hybrid of block-pulse functions and biorthogonal multiscaling functions

This paper proposes new effective methods for analysis of time-varying delay systems using new hybrid functions. The excellent properties of the hybrid functions which consist of block pulse functions and biorthogonal multiscaling functions are presented. We utilise operational matrices of product, delay and integration to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

Numerical solution of Volterra integral-algebraic equations using block pulse functions

This paper presents a method for computing numerical solutions for linear Volterra integral-algebraic equations using block pulse functions. The problem is transformed to a linear lower triangular system of algebraic equations using the operational matrix associated with block pulse functions. Convergence result and numerical examples are presented to illustrate the efficiency and applicability of the method.

Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices

Most fractional differential equations do not have closed form solutions. Development of effective numerical techniques has been an interesting research topic for decades. In this context, this paper proposes a numerical technique, for solving linear and nonlinear multi-order fractional differential equations, based on newly computed generalized triangular function operational matrices for Riemann–Liouville fractional order integral. The orthogonal triangular functions are evolved from a simple dissection of piecewise constant orthogonal block pulse functions. Theoretical error analysis is performed to estimate the upper bound of absolute error between the exact Riemann–Liouville fractional order integral and its approximation in the triangular functions domain. Numerical examples are considered for investigating the applicability and effectiveness of proposed technique to solve multi-order fractional differential equations. The results encourage the use of orthogonal TFs for analysis of real processes exhibiting fractional dynamics.

Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method

Numerical methods for weakly singular Volterra integral equations are rarely considered in the literature. The solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation. In this paper, we present a numerical solution of weakly singular Volterra integral equations including the Abels equations by the second Chebyshev wavelet (SCW). We give the SCW operational matrix of fractional integration, and combine with the block pulse functions (BPFs) to derive the procedure of solving this kind integral equations. The proposed method is illustrated with numerical examples. The results reveal that the method is accurate and easy to implement.

A LINEARIZATION TECHNIQUE FOR NONLINEAR SDOF SYSTEMS BASED ON BLOCK PULSE TRANSFORM

This paper addresses a linearization technique for a single degree of freedom (SDOF) linearizablenonlinear system based on block pulse (BP) transform. The BP transform gives effective tools tosimplify control and system problems. The main goal of this work is on using BP transformproperties in process of linearization of nonlinear problems. It is necessary to compare the resultsobtained using this method with other traditional methods to verify the effectiveness of the proposedmethod. The efficiency of BP transform method compared with the traditional equivalentlinearization (EL) method. Both numerical simulations are applied to the Duffing nonlinearoscillator system to demonstrate the feasibility of the proposed method based on BP transform.Finally, the results of the comparison between the approaches depicted the proposed method is ableto approximate the nonlinear systems behavior. The results showed the superiority of the proposedapproach in the sense that it is more accurate by computational advantageous.

Block pulse transform method for linearization of nonlinear SDOF systems

AbstractThis paper addresses a linearization technique for a linearizable nonlinear SDOF system based on block pulse (BP) transform. BP transform method is a useful tool for giving a solution of difference equations with less computational costs. The main goal of this work is on reducing the costs of linearizing computation. It is necessary to compare the results obtained using this method with other traditional methods to verify the effectiveness of the proposed method. Therefore, the efficiency of BP transform method is compared with the traditional equivalent linearization (EL) method, used to linearize nonlinear systems. Two numerical simulations are applied to the Duffing oscillator system to demonstrate the feasibility of the proposed method based on BP transform. Finally, the results of comparison between existing approaches that have applied to the considered problem depicted the proposed method base on orthogonal functions is able to approximate the nonlinear system’s behavior and shows the superiority of the proposed approach in the sense that it is more accurate by computational advantageous.

Haar Wavelet Operational Matrix Method for the Numerical Solution of Fractional Order Differential Equations

AbstractIn this paper, we propose a new operational matrix method of fractional order integration based on Haar wavelets to solve fractional order differential equations numerically. The properties of Haar wavelets are first presented. The properties of Haar wavelets are used to reduce the system of fractional order differential equations to a systemof algebraic equationswhich can be solved numerically byNewton’s method.Moreover, the proposed method is derived without using the block pulse functions considered in open literature and does not require the inverse of the Haar matrices. Numerical examples are included to demonstrate the validity and applicability of the present method.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

FPGA Based Heartbeats Monitor with Fingertip Optical Sensor

The heart is an organ of human body which has a vital function, small abnormalities can have a big impact on the performance of the body. Heart disease is the number one cause of death in the world. Examination of the heart can be detected from blood flow in the fingertips, in order to obtain information about the number and rhythm of the heartbeat. This research aims to design and implement the FPGA board to monitor the heart rate with optical sensors.The results of this study are expected to facilitate the patient's medical team or independently in detecting heart health. The series is composed of blocks of sensors, signal conditioning block, the block pulse counter, block timer 10 seconds and blocks the viewer. Based on the test results of the 10 respondents with a variety of age and gender, has built a tool that the percentage error of 3.94%.

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.

PIECEWISE CONSTANT SOLUTION OF NON LINEAR VOLTERRA INTEGRAL EQUATION

In this paper, modification in the computational methods, for solving Non-linear Volterra integral equations, is presented. Here, two piecewise constant methods are considered for obtaining the solutions. The first method is based on Walsh Functions (WF) and the second method is via Block Pulse Functions (BPF). Comparison between the two methods is presented by calculating the errors vis-à-vis exact solution. Computational efficiency of BPF is established by profiling the computations for two examples with MATLAB 7.10 Profiler.