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In this paper, the numerical solution of widely used vibration equation with very large membrane is considered. A collocation method with Haar wavelet (HW) is applied for this purpose. The algorithm based on proposed method is very simple and easy to implement. Highly accurate approximate solutions are obtained with a reasonable number of collocation points. The method is tested upon several different exacts solutions of the vibration equation including polynomials and infinite series solutions. The numerical results confirm the accuracy, efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Highlights • A wavelet deconvolution technique for impact force reconstruction is developed by using the Haar wavelet. • The ill-posed nature of deconvolution is mitigated by controlling not only scale but also shift components. • The optimal scales and shifts for reconstruction are determined based on the compromise between the solution and residual norms. • The present technique can be deployed by considering the impulse response function explicitly or implicitly. • The validity of the present technique is verified by numerical experiment on the dynamic crushing of thin-walled columns. Abstract Deconvolution technique has become a useful solution for indirect measurement of impact force, however, a simple deconvolution technique often encounters difficulty and leads to an unsatisfactory reconstruction due to unavoidable error and the ill-posed nature of the inverse problem. This paper is concerned with a deconvolution technique using wavelet approach for reconstructing impact force. Based on the ability of wavelet transform, impact force reconstruction was formulated and implemented through its expansion coefficients of different scaled and shifted versions of wavelets. The reconstruction of impact force thus becomes the reconstruction of its expansion coefficients which are then used to reproduce the profile of impact force. Considerations of appropriate scales and shifts of wavelets in reconstruction can be regarded as a kind of regularization in order to improve the ill-posed problem. The numerical verification on reconstruction of impact force acting on a load-cell during the impact buckling of thin-walled column showed that the present technique is indeed capable to mitigate the ill-posed nature when the scale level of wavelets is controlled properly. The result also revealed that with an appropriate choice of the shifting parameter at a certain scale level, the noise in reconstructed force has been suppressed further. As a consequence, it has been verified that the present wavelet deconvolution technique can successfully reconstruct the impact force with less noise and higher accuracy in comparison with the conventional methods. [ABSTRACT FROM AUTHOR]

A new collocation method based on Haar wavelet is presented for numerical solution of three-dimensional elliptic partial differential equations with Dirichlet boundary conditions. An important advantage of the method is that it can be applied to both linear as well as nonlinear problems. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations. The numerical experiments confirm the accuracy and diverse applicability of the method. [ABSTRACT FROM AUTHOR]

Non-integer order differentiation is changing application of traditional differentiation because it can achieve a continuous interpolation of the integer order differentiation. However, implementation of the non-integer order differentiation is much more complex than that of integer order differentiation. For this purpose, a Haar wavelet-based implementation method of non-integer order differentiation is proposed. The basic idea of the proposed method is to use the operational matrix to compute the non- integer order differentiation of a signal through expanding the signal by the Haar wavelets and constructing Haar wavelet operational matrix of the non-integer order differentiation. The effectiveness of the proposed method was verified by comparison of theoretical results and those obtained by another non-integer order differential filtering method. Finally, non-integer order differentiation was applied to enhance signal. [ABSTRACT FROM AUTHOR]

We consider infinite-horizon optimal control problems. The main idea is to convert the problem into an equivalent finite-horizon nonlinear optimal control problem. The resulting problem is then solved by means of a direct method using Haar wavelets. A local property of Haar wavelets is applied to simplify the calculation process. The accuracy of the present method is demonstrated by two illustrative examples. [ABSTRACT FROM AUTHOR]

Orthogonal frequency-division multiplexing (OFDM) has lately gained a great deal of attention and is considered as a strong candidate for many next-generation wireless communication systems. However, OFDM is very sensitive to nonlinear effects due to the high peak-to-average power ratio (PAPR) owned by the transmitted signals and does not show robustness to spectral null channels. This paper proposes a novel BPSK OFDM system based on Haar wavelet transformation. The Haar wavelet transformation operates decomposition over the data symbol sequence after binary-to-complex mapping shows that half of the data symbols are zeros and the rest are either $${\sqrt{2}}$$ or $${-\sqrt{2}}$$. Then, we have the PAPR reduced by $${10\log_{10} 2\approx 3}$$ dB at most, compared with the conventional OFDM system. We also propose a novel decoding algorithm for the proposed OFDM system to show robustness to spectral null channels, and derive the bit error rate (BER) performance in theory from unbalanced QPSK modulation. Finally, we compare BER performance of our proposed OFDM with the conventional OFDM over different channels to show the excellent performance of our proposed OFDM system. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we present a computational method for solving eigenvalue problems of high-order ordinary differential equations which based on the use of Haar wavelets. The variable and their derivatives in the governing equations are represented by Haar function and their integral. The first transform the spectral coefficients into the nodal variable values. The second, solve the obtained system of algebraic equation. The efficiency of the method is demonstrated by four numerical examples. [Copyright &y& Elsevier]

This study addresses the identification of nonlinear systems. It is assumed that the function form in the nonlinear system is known, leaving some unknown parameters to be estimated. Since Haar wavelets can form a complete orthogonal basis for the appropriate function space, they are used to expand all signals. In doing so, the state equation can be transformed into a set of algebraic equations in unknown parameters. The technique of Kronecker product is utilized to simplify the expressions of the associated algebraic equations. Together with the least square method, the unknown system parameters are estimated. The proposed method is applied to the identification of an experimental two-well chaotic system known as the Moon beam. The identified model is validated by comparing the chaotic characteristics, such as the largest Lyapunov exponent and the correlation dimension, of the experimental data with that of the numerical results. The simple least square approach is also performed for comparison. The results indicate that the proposed method can reliably identify the characteristics of the nonlinear chaotic system. [ABSTRACT FROM AUTHOR]

Abstract: A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm. [Copyright &y& Elsevier]

Abstract: The Haar wavelet based discretization method for solving differential equations is developed. Nonlinear Burgers equation is considered as a test problem. Both, strong and weak formulations based approaches are discussed. The discretization scheme proposed is based on the weak formulation. An attempt is made to combine the advantages of the FEM and Haar wavelets. The obtained numerical results have been validated against a closed form analytical solution as well as FEM results. Good agreement with the exact solution has been observed. [Copyright &y& Elsevier]

Abstract: In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples. [Copyright &y& Elsevier]

This study addresses the identification of linear time varying systems. The identification is based on the expansion of all time functions in the state equations by Haar wavelets. The unknown time function can thus be identified in terms of Haar wavelets. A Haar wavelet is a set of complete, orthogonal basis and is easy to use computations. Several good properties of Haar wavelets are utilized in the algorithm. Both numerical and experimental results verify the analysis. [ABSTRACT FROM AUTHOR]

We present a computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system. The method is simple and computationally advantageous. The approximate optimal trajectory and optimal control are calculated using the Haar wavelet integral, product, and delay operational matrices. An illustrative example is included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

In this paper we present an implementation of the Haar wavelet to the optimal control of linear singularly perturbed systems. The approximated composite control and the slow and fast trajectories with respect to a quadratic cost function are calculated by solving only the linear algebraic equations. The results are illustrated with a simple example. [ABSTRACT FROM AUTHOR]

This letter presents a modeling method of lumped elements by using the Haar-wavelet MRTD (multiresolution time-domain) technique. With the MRTD technique, the conventional FDTD method is partly used to describe the localized characteristics of the lumped elements, which improves the analysis accuracy of the circuits with lumped elements. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 190–192, 2000. [ABSTRACT FROM AUTHOR]