Numerical method for inverse laplace transform with Haar wavelet operational matrix

Wavelets have been applied successfully in signal and image processing. Many attempts have been made in mathematics to use orthogonal wavelet function as numerical computational tool. In this work, an orthogonal wavelet function namely Haar wavelet function is considered. We present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix for integration. We proved the method for the cases of the irrational transfer function using the extension of Riemenn-Liouville fractional integral. The proposed method extends the work of J.L.Wu et al. (2001) to cover the whole of time domain. Moreover, this work gives an alternative way to find the solution for inversion of Laplace transform in a faster way. The use of numerical Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Additionally, few benefits come from its great features such as faster computation and attractiveness. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to analytical solution. Examples on solving differential equation by Laplace transform method are also given

A numerical solution for heat transfer past a stretching sheet with ohmic dissipation and suction or injection problem using Haar wavelet quasilinearization method

This paper represents a numerical analysis for heat transfer of a Jeffrey fluid flow past a stretching sheet with ohmic dissipation and suction/injection. The partial differential equations are reduced into a set of convenient nonlinear ordinary differential equations with the boundary conditions. Haar wavelet quasilinearization method (HWQM) is used to solve ordinary differential equations. The effect of various related parameters on velocity and temperature profiles are computed and analyzed. Then, comparison is made between the numerical results of proposed method with existing numerical solutions found in the literature, and reasonable agreement is noted

CHF enhancement of a large heated surface by a honeycomb porous plate and a gridded metal structure in a saturated pool boiling of nanofluid

The enhancement of the critical heat flux (CHF) in saturated pool boiling of water-based nanofluid (containing TiO2 nanoparticles) by the attachment of a honeycomb porous plate (HPP) and a gridded metal structure (GMS) on a horizontal heated surface have been investigated experimentally. The honeycomb porous plate attached to the heated surface enhances the liquid supply due to capillary action to the heated surface and the release of vapor through the vapor escape channel. The deposition of nanoparticles on the heated surface during the boiling of the nanofluid enhances the spread of liquid along the heated surface due to the capillary action. The preceding papers by the present authors revealed that the CHF could be significantly enhanced by 2.2 times that of water boiling by the attachment of the HPP on the heated surface with the nanoparticle deposition layer. According to the hydrodynamic theory by Lienhard et al. (1973), the installation of a gridded structure on the heated surface could also enhance the CHF because the number of the escaping vapor jets each of which allows the liquid flow to the heated surface near the CHF conditions increases with the increment in the number of grid. The present paper describes the results directed toward the further enhancement of the pool boiling CHF of nanofluid by the installation of the GMS onto the HPP on a large heated surface. The tested surface has a diameter of ϕ50 mm, which is 20 times the capillary length, λC(=σ/g(ρl-ρv). For plain surfaces being larger than 20 times the length λC, the CHF can be regarded as being equivalent to that of an infinite large surface. Based on the Lienhard model, grid size of the GMS is chosen so that the CHF of water boiling is increased most effectively. As a result, for simultaneous existence of three factors (the HPP, the GMS and deposition layer of nanoparticles), the CHF has been enhanced to 3.1 MW/m2, which is the higher than either of the HPP in water, the HPP in water-based nanofluid and the GMS in water. High-speed-movie visualization of water boiling revealed that the attachment of the gridded metal structure shortens the hovering period of the coalesced bubble compared to the plain surface. Shortened period causes the more frequent liquid supply to the heated surface. These results illustrate the potential for increasing the safety margin in the IVR (In-Vessel Retention) systems as a heat removal technology

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded

Measurement of a heated surface temperature using a high-speed infrared camera during critical heat flux enhancement by a honeycomb porous plate in a saturated pool boiling of a nanofluid

This article presents an experimental study to investigate the critical heat flux (CHF)enhancement mechanism using honeycomb porous plate (HPP). The CHF enhanced significantly with combination of the HPP and nanofluid, up to 3.2MW/m2 at maximum compared to a plain surface, 1.0MW/m2. The mechanism by which the CHF is improved in this system was elucidated by measuring the temperature of the heated surface using an indium tin oxide (ITO) heater and a high-speed infrared camera. The pool boiling experiment of water and nanofluid is performed under saturated temperature and atmospheric pressure conditions. The CHF values obtained using ITO heater is in good agreement with a conventional CHF pool boiling experiment with HPP attachment. High-speed infrared camera is analyzed to understand the behavior of local temperature at various locations over time. It is observed at the burnout condition, the highest average temperature is occurred at the intersection of HPP wall. Moreover, the reversible dry spots were initiated in the cell part of the HPP, and small dry spots coalesced into a growth of large irreversible dry out that leads to burnout. Further CHF enhancement could be realized if the initiation of the dryout region could be suppressed

Numerical solution for the chemotaxis model by finite difference method

The finite difference method for discretization space fractional chemotaxis model is introduced in this study. The space fractional chemotaxis system is obtained from the classical advection-diffusion equations of the chemotaxis system by replacing the spatial derivative with a generalized derivative of fractional order. We compare the numerical solution of finite difference method and exact solution for a test example. The results reveal that the finite difference method is very simple and efficient for solving space fractional chemotaxis system

Study of linear-correlation based solar irradiance measurement device photovoltaic application

Solar irradiance is the most fundamental element for energy generation from the photovoltaic system. Huge number of measurement instruments are produced commercially for better reading of this parameter. Nevertheless, there are still no detail studies that have ever been reported regarding the critical relationship between solar irradiance and other electrical parameters such as voltage or current in the literature. This study is essential to determine the correct parameters to be considered in developing a precise measurement of solar irradiance. In this study, a linear-correlation based solar irradiance measurement device has been designed to investigate this relationship. The obtained solar irradiance values from the proposed device exhibit an excellent agreement with those measured by the commercial solar meters/sensors which highly suggest that this study should become a trigger for better improvement of the solar measurement system in the future

A study of the hyperbolic heat conduction problem and laplace inversion using generalized haar wavelet operational matrix method / Suazlan bin Mt Aznam

Wavelets have been applied successfully in image and signal processing. Many attempts have been made in mathematics to use wavelet function as numerical computational tool. In this study, an orthogonal wavelet function namely Haar wavelet function is considered. We used the operational matrix based on Haar basis to solve hyperbolic heat conduction equation problem and Laplace inversion. It is remarkably known by many that one of the difficulties encountered in numerical method for non-Fourier heat conduction problem is the numerical oscillation within the vicinity of jump discontinuities at the wave front. We propose a new method of solving non-Fourier heat conduction equation problem which is also a hyperbolic partial differential equation. Our new method for solving partial differential equation of hyperbolic heat conduction equation is a hybrid of finite difference method and pseudo spectral method, where the former for time discretization and the latter for spatial discretization. The time discretization is performed prior to spatial discretization. In this sense, partial differential equation is reduced to ordinary differential equation and solved implicitly with Haar wavelet basis. For pseudo spectral method, Haar wavelet expansion has been used considering its advantage of the absence of the Gibbs phenomenon at the jump discontinuities. Furthermore, definition of Haar wavelet basis in this work allows a pleasant way in computing inverse of Haar wavelet matrix. We also derived generalized Haar wavelet operational matrix in the interval of [0,X). The propose method have been applied into one physical problem namely, thin surface layers. It is found that the proposed numerical ii results could suppress and eliminate the numerical oscillation in the jump vicinity at a certain value of discretization. We also present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix. We prove the method for the case of the transfer function using the extension of Riemann-Liouville fractional integral. The proposed method extends the work of Wu et al. to cover the whole of time domain as we used the generalized Haar wavelet operational matrix. Moreover, this method gives an alternative numerical way to find the solution for inversion of Laplace transform in a simple way. The use of numerical generalized Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Examples in finding Laplace inversion for rational, irrational and exponential transfer function are illustrated.Furthermore, examples on solving differential equation by Laplace transform method are also included. Both of the proposed numerical methods are stable, convergent and easily coded. Numerical results also demonstrate good performance of the method in term of accuracy and competitiveness compared to other numerical methods. Additionally, few benefits come from its great features such as faster computation and attractiveness