Haar Wavelet Operational Matrix Research Articles

Optimizing pantograph fractional differential equations: A Haar wavelet operational matrix method

In this study, we developed an operational matrix method of integration using Haar wavelets to solve both linear and nonlinear pantograph fractional differential equations by taking Atangana's beta derivative. The proposed method utilizes an operational matrix and truncated Haar wavelet series to transform linear and nonlinear problems into objective functions. These objective functions are then minimized using differential evolution optimization. Illustrative examples demonstrate convergence and validation of the proposed method. The proposed method yields exceptional numerical outcomes in terms of competitiveness and accuracy when compared to exact solutions. In addition, performance metrics, including the mean absolute deviation, root mean squared error, P2norm, P∞ norm, Theil's inequality coefficient, variance account for, coefficient of determination, and Nash-Sutcliffe efficiency, were calculated for a varying number of collocation points. The results indicate that the Haar wavelet operational matrix method is a straightforward and efficient method for solving pantograph fractional differential equations.

Legendre wavelet operational matrix method for the analysis of thermal radiation effect on natural convection of a vertical plate embedded in a saturated porous medium

In this paper, the heat transfer characteristics of natural convection around a vertical flat surface embedded in a saturated porous medium are studied by taking into account the thermal radiation effect. With the aid of the similarity transformations, the governing equations of the problem are reduced to the system of ordinary differential equations over the infinite domain. Moreover, the Legendre wavelet operational matrix method (LWOMM) is hired to find the numerical solution to the considered problem. Initially, to ensure the efficiency of the proposed strategy, a third-order nonlinear boundary value problem having the exact solution is considered a test problem. The numerical findings elucidate that the solutions obtained by the LWOMM are in good agreement with the exact solution as compared with the solution obtained by the Haar wavelet operational matrix method, with the consumption of less computational CPU time. Also, the temperature and velocity profiles are discussed graphically.

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

Irrational and transcendental functions can often be seen in signal processing or physical phenomena analysis as consequences of fractional-order and distributed order models that result in fractional or partial differential equations. In cases when finding solution in analytical form tends to be difficult or impossible, numerical calculations such as Haar wavelet operational matrix method can be used. Haar wavelet establishes a direct procedure for transfer function inversion using the wavelet operational matrix for orthogonal function set integration. In this paper an inverse Laplace transform of irrational and transcendental transfer functions using Haar wavelet operational matrix is proposed. Results for a number inverse Laplace transforms are numerically solved and compared with the analytical solutions and solutions provided by commonly used Invlap and NILT algorithms. This approach is useful when the original cannot be represented by an analytical formula and validity of the obtained result needs to be crosschecked and error estimated.

A NEW NUMERICAL TECHNIQUE FOR SOLVING <i>ψ</i>-FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

This paper proposes a new numerical technique for solving a specific class of fractional differential equations, which includes the ${\psi}$-Caputo fractional derivative. The class under consideration is nonlinear ${\psi}$-fractional Riccati differential equations (${\psi}$-FRDEs). Our approach relies on the ${\psi}$-Haar wavelet (${\psi}$-HW) operational matrix, which is a novel type of operational matrix of fractional integration. We derive an explicit formula for the ${\psi}$-fractional integral of the HW. This operational matrix has been used successfully to solve nonlinear ${\psi}$-FRDEs.The Quasi-linearization technique is employed to linearize the non-linear $\bm\psi$-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 ${\psi}$-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.

Numerical Solution for Linear State Space Systems using Haar Wavelets Method

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.

Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix

Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix

Generalized formulation to estimate the Supercapacitor’s R-C series impedance using fractional order model

The main objective of this paper is to develop a new technique for the supercapacitor’s parameter identification that can handle the issue of initial voltage. An effective approach is proposed using the fractional-order derivative for accurate identification of a well-known series Resistance-Capacitance (R-C) model. An expression is derived using the Caputo definition and the Haar wavelet operational matrix, which is sufficient for both charging and discharging phase data of supercapacitors. To extract impedance parameters, voltage stimulated step response is utilized and parameters are calculated despite random initial voltage stored in a supercapacitor. This operational matrix approach transforms complex fractional derivative terms into simple algebraic expressions and reduces the overall complexity. The proposed technique shows a very good agreement with experimental data that exhibit different initial voltages and different time-frames. Investigations and experiments with various supercapacitors clearly reveal the importance of the developed equation for a fractional R-C model.

A single-step identification strategy for the coupled TITO process using fractional calculus

The reliable performance of a complete control system depends on accurate model information being used to represent each subsystem. The identification and modelling of multivariable systems are complex and challenging due to cross-coupling. Such a system may require multiple steps and decentralized testing to obtain full system models effectively. In this paper, a direct identification strategy is proposed for the coupled two-input two-output (TITO) system with measurable input–output signals. A well-known closed-loop relay test is utilized to generate a set of inputs–outputs data from a single run. Based on the collected data, four individual fractional-order transfer functions, two for main paths and two for cross-paths, are estimated from single-run test signals. The orthogonal series-based algebraic approach is adopted, namely the Haar wavelet operational matrix, to handle the fractional derivatives of the signal in a simple manner. A single-step strategy yields faster identification with accurate estimation. The simulation and experimental studies depict the efficiency and applicability of the proposed identification technique. The demonstrated results on the twin rotor multiple-input multiple-output (MIMO) system (TRMS) clearly reveal that the presented idea works well with the highly coupled system even in the presence of measurement noise.

Fractional-order two-input two-output process identification based on Haar operational matrix

The identification of multi-input-multi-output process is always challenging due to significant loop interactions. This article proposes an identification technique for two-input-two-output processes with fractional order models via Haar wavelet feature. The presented technique can estimate independent four fractional single-pole time delay models without additional steps to decouple processes during identification. The method uses a well-known relay feedback test to produce the input-output data for measurements. Then, the Haar wavelet operational matrix based algebraic approach is utilised to identify the unknown process models with reduced complexity. Numerical analysis on various examples show the efficacy even in presence of noise and without additional filter or preprocessing of measured signals. The comparative study proves advantages of the presented approach.

Identification scheme for fractional Hammerstein models with the delayed Haar wavelet

The parameter identification of a nonlinear Hammerstein-type process is likely to be complex and challenging due to the existence of significant nonlinearity at the input side. In this paper, a new parameter identification strategy for a block-oriented Hammerstein process is proposed using the Haar wavelet operational matrix (HWOM). To determine all the parameters in the Hammerstein model, a special input excitation is utilized to separate the identification problem of the linear subsystem from the complete nonlinear process. During the first test period, a simple step response data is utilized to estimate the linear subsystem dynamics. Then, the overall system response to sinusoidal input is used to estimate nonlinearity in the process. A single-pole fractional order transfer function with time delay is used to model the linear subsystem. In order to reduce the mathematical complexity resulting from the fractional derivatives of signals, a HWOM based algebraic approach is developed. The proposed method is proven to be simple and robust in the presence of measurement noises. The numerical study illustrates the efficiency of the proposed modeling technique through four different nonlinear processes and results are compared with existing methods.

A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

The Lotka‐Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams‐Bashforth‐Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve. To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided.

Constructive Existence of (1,1)-Solutions to Two-Point Value Problems for Fuzzy Linear Multiterm Fractional Differential Equations

In this paper, we consider the following two-point boundary value problems of fuzzy linear fractional differential equations: (Dc1,1αy)(t)⊕b(t)⊗(Dc1,1βy)(t)⊕c(t)⊗y(t)=f(t), t∈(0,1), y(0)=y0 and y(1)=y1, where b,c∈C(I), b(t),c(t)≥0, y,f∈C(I,RF), I=[0,1], y0,y1∈RF and 1<β<α≤2. Our existence result is based on Banach fixed point theorem and the approximate solution of our problem is obtained by applying the Haar wavelet operational matrix.

A Numerical Algorithm to Capture Spin Patterns of Fractional Bloch Nuclear Magnetic Resonance Flow Models

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.

Wavelet Approximation of Functions Belonging to Generalized Lipschitz Class by Extended Haar Wavelet Series and its Application in the Solution of Bessel Differential Equation Using Haar Operational Matrix

In this paper, a class $$Lip_\alpha ^{(s)}[0,1]$$ is introduced. This class of functions is a generalization of known Lipschitz class, i.e., $$Lip_\alpha [0,1], 0<\alpha \le 1$$ of functions. Four new estimators $$E_N^{(1)}(f), E_{J,N}^{(2)}(f), E_N^{(3)}(f)$$ and $$E_N^{(4)}(f)$$ of functions of $$Lip_\alpha [0,1]$$ and $$Lip_\alpha ^{(s)}[0,1]$$ classes have been obtained. These estimators are sharper and best possible in the approximation of functions by wavelet methods. The developed estimators, the solution of Bessel differential equation of order zero by Haar wavelet operational matrix and its comparison with the exact solution are the main significant achievements of this research paper.

Generalized Haar wavelet operational matrix method for solving hyperbolic heat conduction in thin surface layers

It is remarkably known that one of the difficulties encountered in a numerical method for hyperbolic heat conduction equation is the numerical oscillation within the vicinity of jump discontinuities at the wave front. In this paper, a new method is proposed for solving non-Fourier heat conduction problem. It is a combination of finite difference and pseudospectral methods in which the time discretization is performed prior to spatial discretization. In this sense, a partial differential equation is reduced to an ordinary differential equation and solved implicitly with Haar wavelet basis. For the pseudospectral method, Haar wavelet expansion has been using considering its advantage of the absence of the Gibbs phenomenon at the jump continuities. We also derived generalized Haar operational matrix that extend usual domain (0,1] to (0,X]. The proposed method has been applied to one physical problem, namely thin surface layers. It is found that the proposed numerical results could suppress and eliminate the numerical oscillation in the vicinity jump and in good agreement with the analytic solution. In addition, our method is stable, convergent and easily coded. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to other numerical methods.

A novel approach of fractional-order time delay system modeling based on Haar wavelet

In this paper, fractional-order time delay system modeling is presented using Haar wavelet operational matrix of integration. Therefore, it does not require any prior knowledge of transfer function structure or partial information about fractional differentiation order. It allows the estimation of the implicit time delay parameter together with other model parameters by utilizing new delay operational matrix of Haar wavelet based on Riemann-Liouville definition. The proposed technique reduces the complexity of identification by converting the complex fractional calculus equations into simple algebra. The efficacy of the approach is verified on various integer and non-integer (fractional) order systems in simulation. For realistic condition, proposed method is verified in the presence of noise in simulation and also demonstrated on the real-time process control temperature system.

A Wavelet Based Rationalized Approach for the Numerical Solution of Differential and Integral Equations

A wavelet based rationalized method is presented for the numerical solution of differential, integral and integro-differential equations. Rationalized Haar functions are used to estimate the solution. Their fundamental properties are discussed. A rigorous convergence analysis is presented. The operational matrix of the product of two rationalized Haar functions is used to reduce the dynamical system to an algebraic system. A variety of model problems are taken into account so as to test the efficiency of the proposed method. The result so obtained are compared with the available exact solutions. In addition, proposed scheme is compared with some state of the art existing methods. It is found that the Haar wavelet operational matrix is the fastest. Moreover, the results obtained are mathematically simple and the desired accuracy of the solution is obtained using small number of grid points. The main advantages of the wavelet method are its simplicity, fast transformation, possibility of implementation of fast algorithms and low computational cost with high accuracy.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

Haar Wavelet Operational Matrix Method to Solve Initial Value Problems: A Short Survey

In this paper we give a short survey of Haar wavelet operational matrix method which is used to solve initial value problems up to second order, both linear and nonlinear. First we give a short review of Haar wavelet and operational matrices obtained from it. We then describe the method for linear and nonlinear ordinary differential equations both for first and second order. At the end, we provide a stability analysis and it is shown that the proposed method is of second order.

Numerical Solution of Fractional Differential Equations Using Haar Wavelet Operational Matrix Method

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.