Haar wavelet operational matrix based numerical inversion of Laplace transform for irrational and transcendental transfer functions

Wavelet is relatively new but an emerging theory in the field of mathematics and signal processing which is being used in different engineering and mathematical tasks. From many existing wavelets, the Haar wavelet is one of the popular in mathematics and engineering due to its simplicity and compact support. In my study, the Haar wavelet operational matrix method is used to solve 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> order ordinary differential equations as well as two-dimensional partial differential equations. Comparing to the conventional methods Haar wavelet method is easy to find the required integral to solve differential equations. By making the block pulse operational matrix and Haar wavelet matrix, Haar wavelet operational matrix can be formed. Using the Haar wavelet the differential equation can be decomposed in the Haar series, and the integral can be calculated to get the numerical solution. The numerical solution of the Haar wavelet-based method is compared to the exact solution, and it gives very little error.

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.

The partial differential equations describing distributed parameter systems may often be reduced to transcendental transfer functions with the aid of appropriate boundary conditions. In the analysis and synthesis of closed loop systems, the transcendental transfer functions have to be approximated in a suitable manner. In this paper, discrete-time model of distributed parameter systems is obtained. The model employs a sample and hold circuit in the loop. The response of the model system is compared with the response obtained by approximating the transcendental transfer function by root factor and other approximations. The stability of linear and nonlinear systems with distributed parameters is investigated by employing the Mikhailov stability criterion.

The technological process of expanded clay cooling in a drum cooler is considered as a control object with distributed parameters. Taking into account reasonable assumptions and simplifications, the dynamics of the process under consideration is described by a system of non-homogeneous differential equations in partial derivatives of the first order, the solution of which made it possible to obtain operators linking the control action and the main disturbances with the temperature of expanded clay in the drum cooler. The obtained operators of the mathematical model, which are transcendental transfer functions, are approximated by typical forms of transfer functions for the possibility of their further practical application in the synthesis of automation systems. A structural diagram of the mathematical model of the process of expanded clay cooling in a drum cooler as a control object is synthesized. Computational experiments are carried out to study the dynamic and static operating modes of the control object.

Block-pulse functions method of the inverse laplace transform for irrational and transcendental transfer functions

The aim of this paper is to present a modified block-pulse functions (BPFs) technique of the inverse Laplace transform of one- and two-dimensional irrational and transcendental transfer functions. This method seems to be attractive in the case of the numerical computation of a large number of BPFs. Some illustrative examples are given

This paper proposes a new numerical technique for solving a specific class of fractional differential equations, which includes the <italic>ψ</italic>-Caputo fractional derivative. The class under consideration is nonlinear <italic>ψ</italic>-fractional Riccati differential equations (<italic>ψ</italic>-FRDEs). Our approach relies on the <italic>ψ</italic>-Haar wavelet (<italic>ψ</italic>-HW) operational matrix, which is a novel type of operational matrix of fractional integration. We derive an explicit formula for the <italic>ψ</italic>-fractional integral of the HW. This operational matrix has been used successfully to solve nonlinear <italic>ψ</italic>-FRDEs.The Quasi-linearization technique is employed to linearize the non-linear <italic>ψ</italic>-FRDEs. This technique reduces the problem to an algebraic equation that can be easily solved. The technique is a useful and straightforward mathematical tool for solving nonlinear <italic>ψ</italic>-FRDEs. The computational complexity of the operational matrix technique is minimal. The error analysis of the proposed method is thoroughly investigated. To justify the method's accuracy and efficiency, numerical results are given.

In this paper, a new operational matrix method based on Haar wavelets is proposed to solve linear and non-linear differential equations of fractional order. Contrary to wavelet operational methods available in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The main characteristics of our approach is that it converts fractional differential equations to system of algebraic equations and does not require the inverse of the Haar matrices. Illustrative examples are included to demonstrate the validity and applicability of the present method. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Haar wavelet method for solving fractional partial differential equations numerically

Fractional Bloch equation is a generalized form of the integer order Bloch equation. It governs the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance (NMR). Scale-3 (S-3) Haar wavelet operational matrix along with quasi-linearization is applied first time to detect the spin flow of fractional Bloch equations. A comparative analysis of performance of classical scale-2 (S-2) and novel scale-3 Haar wavelets (S-3 HW) has been carried out. The analysis shows that scale-3 Haar wavelets give better solutions on coarser grid point in less computation time. Error analysis shows that as we increase the level of the S-3 Haar wavelets, error goes to zero. Numerical experiments have been conducted on five test problems to illustrate the merits of the proposed novel scheme. Maximum absolute errors, comparison of exact solutions, and S-2 Haar wavelet and S-3 Haar wavelet solutions, are reported. The physical behaviors of computed solutions are also depicted graphically.

Purpose The objective of this study is to introduce a more efficient method than classical wavelet methods, for solving linear and nonlinear Caputo fractional variable-order diffusion-type equations. Design/methodology/approach We first construct the Haar wavelet operational matrices for Caputo variable-order integration. Next, we integrate these matrices with the L2 − 1σ approximations to solve linear Caputo fractional variable-order diffusion-type equations. For nonlinear problems, we employ the quasilinearization technique in conjunction with the operational matrices and L2 − 1σapproximations. The proposed method is named “the modified Haar wavelet (mHw) method.” The efficiency of the mHw method is demonstrated through comparisons with the exact solution and the solution obtained using the classical Haar wavelet method. Findings We have derived the Haar wavelet operational matrix of variable-order integration and the variable-order integration matrix of Haar wavelet for boundary value problems. These matrices, in conjunction with the L2 − 1σapproximations and the quasilinearization technique, form the basis for the construction of the mHw method. We also provide the theoretical analysis of the mHw method. Additionally, the mHw method is shown to be second-order accurate in both time and space domains. We also performed the comparison with the classical Haar wavelet method. Numerical simulations are presented to validate and illustrate the theoretical results. Originality/value Many engineers and scientists can utilize the presented method for solving their linear and nonlinear Caputo fractional variable-order models.

Parameter identification of fractional order linear system based on Haar wavelet operational matrix

In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.

Most direct methods solve finite time horizon optimal control problems with nonlinear programming solver. In this paper, we propose a numerical method for solving nonlinear optimal control problem with state and control inequality constraints. This method used quasilinearization technique and Haar wavelet operational matrix to convert the nonlinear optimal control problem into a quadratic programming problem. The linear inequality constraints for trajectories variables are converted to quadratic programming constraint by using Haar wavelet collocation method. The proposed method has been applied to solve Optimal Control of Multi-Item Inventory Model. The accuracy of the states, controls and cost can be improved by increasing the Haar wavelet resolution.

Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients