Search results for: Laplace’s equation

Authors: Khaled Moaddy

Abstract:

In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: standard finite difference schemes, non-standard schemes, Laplace equation, Dirichlet boundary conditions

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Authors: M. Jamil Amir, Rabia Iqbal, M. Yaseen

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In this paper, we solve some differential equations analytically by using differential transform method. For this purpose, we consider four models of Laplace equation with two Dirichlet and two Neumann boundary conditions and K(2,2) equation and obtain the corresponding exact solutions. The obtained results show the simplicity of the method and massive reduction in calculations when one compares it with other iterative methods, available in literature. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

Keywords: differential transform method, laplace equation, Dirichlet boundary conditions, Neumann boundary conditions

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Authors: Abbas Ebrahimi, Mohammad Zandsalimy

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The main purpose of this study is to investigate the feasibility of using FPGA (Field Programmable Gate Arrays) chips as alternatives for the conventional CPUs to accelerate the numerical solution of the Laplace equation. FPGA is an integrated circuit that contains an array of logic blocks, and its architecture can be reprogrammed and reconfigured after manufacturing. Complex circuits for various applications can be designed and implemented using FPGA hardware. The reconfigurable hardware used in this paper is an SoC (System on a Chip) FPGA type that integrates both microprocessor and FPGA architectures into a single device. In the present study the Laplace equation is implemented and solved numerically on both reconfigurable hardware and CPU. The precision of results and speedups of the calculations are compared together. The computational process on FPGA, is up to 20 times faster than a conventional CPU, with the same data precision. An analytical solution is used to validate the results.

Keywords: accelerating numerical solutions, CFD, FPGA, hardware definition language, numerical solutions, reconfigurable hardware

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Authors: Jorge Gonzalez Camus, Valentin Keyantuo, Mahamadi Warma

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In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula.

Keywords: discrete fractional Laplacian, explicit representation of solutions, fractional heat and wave equations, fundamental

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Authors: Stephen Kirkup

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This paper discusses the implementation of the boundary element method (BEM) on an Excel spreadsheet and how it can be used in teaching vector calculus and simulation. There are two separate spreadheets, within which Laplace equation is solved by the BEM in two dimensions (LIBEM2) and axisymmetric three dimensions (LBEMA). The main algorithms are implemented in the associated programming language within Excel, Visual Basic for Applications (VBA). The BEM only requires a boundary mesh and hence it is a relatively accessible method. The BEM in the open spreadsheet environment is demonstrated as being useful as an aid to teaching and learning. The application of the BEM implemented on a spreadsheet for educational purposes in introductory vector calculus and simulation is explored. The development of assignment work is discussed, and sample results from student work are given. The spreadsheets were found to be useful tools in developing the students’ understanding of vector calculus and in simulating heat conduction.

Keywords: boundary element method, Laplace’s equation, vector calculus, simulation, education

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Authors: Kajingulu Malandala, Ranganai Edmore

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The asymmetric Laplace distribution (ADL) is commonly used as the likelihood function of the Bayesian quantile regression, and it offers different families of likelihood method for quantile regression. Notwithstanding their popularity and practicality, ADL is not smooth and thus making it difficult to maximize its likelihood. Furthermore, Bayesian inference is time consuming and the selection of likelihood may mislead the inference, as the Bayes theorem does not automatically establish the posterior inference. Furthermore, ADL does not account for greater skewness and Kurtosis. This paper develops a new aspect of quantile regression approach for count data based on inverse of the cumulative density function of the Poisson, binomial and Delaporte distributions using the integrated nested Laplace Approximations. Our result validates the benefit of using the integrated nested Laplace Approximations and support the approach for count data.

Keywords: quantile regression, Delaporte distribution, count data, integrated nested Laplace approximation

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: J. Ranjbarn, A. Alibeigloo

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In this paper, we present an analytical method for analysis of nano-scale spherical shell subjected to thermo-mechanical shocks based on nonlocal elasticity theory. Thermo-mechanical properties of nano shpere is assumed to be temperature dependent. Governing partial differential equation of motion is solved analytically by using Laplace transform for time domain and power series for spacial domain. The results in Laplace domain is transferred to time domain by employing the fast inverse Laplace transform (FLIT) method. Accuracy of present approach is assessed by comparing the the numerical results with the results of published work in literature. Furtheremore, the effects of non-local parameter and wall thickness on the dynamic characteristics of the nano-sphere are studied.

Keywords: nano-scale spherical shell, nonlocal elasticity theory, thermomechanical shock, dynamic response

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Authors: Jayanta Pokharel, Gokarna Aryal, Netra Kanaal, Chris Tsokos

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Investment in stocks and shares aims to seek potential gains while weighing the risk of future needs, such as retirement, children's education etc. Analysis of the behavior of the stock market returns and making prediction is important for investors to mitigate risk on investment. Historically, the normal variance models have been used to describe the behavior of stock market returns. However, the returns of the financial assets are actually skewed with higher kurtosis, heavier tails, and a higher center than the normal distribution. The Laplace distribution and its family are natural candidates for modeling stock returns. The Variance-Gamma (VG) distribution is the most sought-after distributions for modeling asset returns and has been extensively discussed in financial literatures. In this paper, it explore the other Laplace family, such as Asymmetric Laplace, Skewed Laplace, Kumaraswamy Laplace (KS) together with Variance-Gamma to model the weekly returns of the S&P 500 Index and it's eleven business sector indices. The method of maximum likelihood is employed to estimate the parameters of the distributions and our empirical inquiry shows that the Kumaraswamy Laplace distribution performs much better for stock returns modeling among the choice of distributions used in this study and in practice, KS can be used as a strong alternative to VG distribution.

Keywords: stock returns, variance-gamma, kumaraswamy laplace, maximum likelihood

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Authors: Noufe H. Aljahdaly

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The Laplace homotopy perturbation method (LHPM) is an approximate method that help to compute the approximate solution for partial differential equations. The method has been used for solving several problems in science. It requires the initial condition, so it solves the initial value problem. In physics, when some important terms are taken in account, we may obtain non-integrable partial differential equations that do not have analytical integrals. This type of PDEs do not have exact solution, therefore, we need to compute the solution without initial condition. In this work, we improved the LHPM to be able to solve non-integrable problem, especially the damped PDEs, which are the PDEs that include a damping term which makes the PDEs non-integrable. We improved the LHPM by setting a perturbation parameter and an embedding parameter as the damping parameter and using the initial condition for damped PDE as the initial condition for non-damped PDE.

Keywords: non-integrable PDEs, modified Kawahara equation;, laplace homotopy perturbation method, damping term

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Authors: M. Abir Hossain, M. Shahjada Tarafder

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This paper deals with the analysis of potential flow past two-dimensional body by discretizing the body into panels where the Laplace equation was applied to each panel. The Laplace equation was solved at each panel by applying the boundary conditions. The boundary condition was applied at each panel to mathematically formulate the problem and then convert the problem into a computer-solvable problem. Kutta condition was applied at both the leading and trailing edges to see whether the condition is satisfied or not. Another approach that is applied for the analysis is Vortex Lattice Method (VLM). A vortex ring is considered at each control point. Using the Biot-Savart Law the strength at each control point is calculated and hence the pressure differentials are measured. For the comparison of the analytic result with the experimental result, different NACA section hydrofoil is used. The analytic result of NACA 0012 and NACA 0015 are compared with the experimental result of Abbott and Doenhoff and found significant conformity with the achieved result.

Keywords: Kutta condition, Law of Biot-Savart, pressure differentials, potential flow, vortex lattice method

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Authors: Hamdy M. Youssef, Mowffaq Oreijah, Hunaydi S. Alsharif

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In this paper, a three-dimensional model of the generalized thermoelasticity with one relaxation time and variable thermal conductivity has been constructed. The resulting non-dimensional governing equations together with the Laplace and double Fourier transforms techniques have been applied to a three-dimensional half-space subjected to thermal loading with rectangular pulse and traction free in the directions of the principle co-ordinates. The inverses of double Fourier transforms, and Laplace transforms have been obtained numerically. Numerical results for the temperature increment, the invariant stress, the invariant strain, and the displacement are represented graphically. The variability of the thermal conductivity has significant effects on the thermal and the mechanical waves.

Keywords: thermoelasticity, thermal conductivity, Laplace transforms, Fourier transforms

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Authors: Sani Isa, Ali Musa

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Burgers’ fluid with a fractional derivatives model in an annulus was analyzed. Combining appropriately the basic equations, with the fractionalized fractional Burger’s fluid model allow us to determine the velocity field, temperature and shear stress. The governing partial differential equation was solved using the combine Laplace transformation method and Riemann sum approximation to give velocity field, temperature and shear stress on the fluid flow. The influence of various parameters like fractional parameters, relaxation time and retardation time, are drawn. The results obtained are simulated using Mathcad software and presented graphically. From the graphical results, we observed that the relaxation time and time helps the flow pattern, on the other hand, other material constants resist the fluid flow while fractional parameters effect on fluid flow is opposite to each other.

Keywords: sani isa, Ali musaburger’s fluid, Laplace transform, fractional derivatives, annulus

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Authors: Yinwei Lin, Cha'o-Kuang Chen

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In this article, the Laplace Adomian transform method (LADM) and double decomposition method (DDM) are used to solve the annular hyperbolic profile fins with variable thermal conductivity. As the thermal conductivity parameter ε is relatively large, the numerical solution using DDM become incorrect. Moreover, when the terms of DDM are more than seven, the numerical solution using DDM is very complicated. However, the present method can be easily calculated as terms are over seven and has more precisely numerical solutions. As the thermal conductivity parameter ε is relatively large, LADM also has better accuracy than DDM.

Keywords: fins, thermal conductivity, Laplace transform, Adomian, nonlinear

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Authors: Farrukh Javed, Krzysztof Podgórski

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We propose a new model that accounts for the asymmetric response of volatility to positive ('good news') and negative ('bad news') shocks in economic time series the so-called leverage effect. In the past, asymmetric powers of errors in the conditionally heteroskedastic models have been used to capture this effect. Our model is using the gamma difference representation of the generalized Laplace distributions that efficiently models the asymmetry. It has one additional natural parameter, the shape, that is used instead of power in the asymmetric power models to capture the strength of a long-lasting effect of shocks. Some fundamental properties of the model are provided including the formula for covariances and an explicit form for the conditional distribution of 'bad' and 'good' news processes given the past the property that is important for the statistical fitting of the model. Relevant features of volatility models are illustrated using S&P 500 historical data.

Keywords: heavy tails, volatility clustering, generalized asymmetric laplace distribution, leverage effect, conditional heteroskedasticity, asymmetric power volatility, GARCH models

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Authors: Geeta Partap, Nitika Chugh

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The present paper deals with the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler-Bernoulli beam resonators, due to Exponential time varying load. Both the axial ends of the beam are assumed to be at simply supported conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection.The analytical results have been numerically analyzed with the help of MATLAB software for magnesium like material. The graphical representations and interpretations have been discussed for Deflection of beam under Simply Supported boundary condition and for distinct considered values of time and space as well. The obtained results are easy to implement for engineering analysis and designs of resonators (sensors), modulators, actuators.

Keywords: microstretch, deflection, exponential load, Laplace transforms, residue theorem, simply supported

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Authors: Sen-Yung Lee, Li-Kuo Chou, Chao-Kuang Chen

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In this study, the problem of temperature transient response of a spiral fin, with its end insulated, is analyzed with base end subjected to a variation of fluid temperature. The hybrid method of Laplace transforms/Adomian decomposed method-Padé, is applied to the temperature transient response of the fin, the result of the temperature distribution and the heat flux at the base of the spiral fin are obtained, show a good agreement in the physical phenomenon.

Keywords: Laplace transforms, Adomian decomposed method- Padé, transient response, heat transfer

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Authors: Ammar Edress Mohamed, Mustafa Aziz, David Wright

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This paper analyses and calculates the head fields of asymmetrical 2D magnetic recording heads when the soft-underlayer is present using the appropriate Green's function to derive the surface potential/field by utilising the surface potential for asymmetrical head without underlayer. The results follow closely the corners, while the gap region shows a linear behaviour for d/g < 0.5 compared with the calculated fields from finite-element.

Keywords: magnetic recording, finite elements, asymmetrical magnetic heads, superposition, Laplace's equation

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Authors: Riki Dutta, Sagardeep Talukdar, Gautam Kumar Saharia, Sudipta Nandy

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Fokas-Lenells equation (FLE) is one of the integrable nonlinear equations use to describe the propagation of ultrashort optical pulses in an optical medium. A 2x2 Lax pair has been introduced for the FLE and from that solving the Riccati equation yields infinitely many conserved quantities. Thereafter for a new field function (S) of the Landau-Lifshitz (LL) system, a gauge equivalence of the FLE with the generalised LL equation has been derived. We hope our findings are useful for the application purpose of FLE in optics and other branches of physics.

Keywords: conserved quantities, fokas-lenells equation, landau-lifshitz equation, lax pair

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Authors: Jian-Jun Shu

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It is common knowledge that many physical problems (such as non-linear shallow-water waves and wave motion in plasmas) can be described by the Korteweg-de Vries (KdV) equation, which possesses certain special solutions, known as solitary waves or solitons. As a marriage of the KdV equation and the classical Burgers (KdVB) equation, the Korteweg-de Vries-Burgers (KdVB) equation is a mathematical model of waves on shallow water surfaces in the presence of viscous dissipation. Asymptotic analysis is a method of describing limiting behavior and is a key tool for exploring the differential equations which arise in the mathematical modeling of real-world phenomena. By using variable transformations, the asymptotic expansion of the KdVB equation is presented in this paper. The asymptotic expansion may provide a good gauge on the validation of the corresponding numerical scheme.

Keywords: asymptotic expansion, differential equation, Korteweg-de Vries-Burgers (KdVB) equation, soliton

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Authors: Muhammad Abdullah, Asma Rashid Butt, Nauman Raza

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Biomagnetic fluids like blood play key role in different applications of medical science and bioengineering. In this paper, the magnetohydrodynamic flow of a viscous fluid with magnetic particles through a cylindrical tube is investigated. The fluid is electrically charged in the presence of a uniform external magnetic field. The movement in the fluid is produced due to the cylindrical tube. Initially, the fluid and tube are at rest and at time t=0⁺, the tube starts to move along its axis. To obtain the mathematical model of flow with fractional derivatives fractional calculus approach is used. The solution of the flow model is obtained by using Laplace transformation. The Simon's numerical algorithm is employed to obtain inverse Laplace transform. The hybrid technique, we are employing has less computational effort as compared to other methods. The numerical calculations have been performed with Mathcad software. As the special cases of our problem, the solution of flow model with ordinary derivatives and flow without magnetic particles has been procured. Finally, the impact of non-integer fractional parameter alpha, Hartmann number Ha, and Reynolds number Re on flow and magnetic particles velocity is analyzed and depicted by graphs.

Keywords: viscous fluid, magnetic particles, fractional calculus, laplace transformation

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Authors: Trilok Mathur, Shivi Agarwal

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This paper deals with the solutions of fuzzy-fractional-order Riccati equations under Caputo-type fuzzy fractional derivatives. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhura difference and strongly generalized fuzzy differentiability. The Laplace-Adomian-Pade method is used for solving fractional Riccati-type initial value differential equations of fractional order. Moreover, we also displayed some examples to illustrate our methods.

Keywords: Caputo-type fuzzy fractional derivative, Fractional Riccati differential equations, Laplace-Adomian-Pade method, Mittag Leffler function

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Authors: Y. Pala, M. O. Ertas

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In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati equation, analytical solution, proper solution, nonlinear

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Authors: Mergim Gasia, Bojan Milovanovica, Sanjin Gumbarevic

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Better energy performance of the building envelope is one of the most important aspects of energy savings if the goals set by the European Union are to be achieved in the future. Dynamic heat transfer simulations are being used for the calculation of building energy consumption because they give more realistic energy demands compared to the stationary calculations that do not take the building’s thermal mass into account. Software used for these dynamic simulation use methods that are based on the analytical models since numerical models are insufficient for longer periods. The analytical models used in this research fall in the category of the conduction transfer functions (CTFs). Two methods for calculating the CTFs covered by this research are the Laplace method and the State-Space method. The literature review showed that the main disadvantage of these methods is that they are inadequate for heavyweight façade elements and shorter time periods used for the calculation. The algorithms for both the Laplace and State-Space methods are implemented in Mathematica, and the results are compared to the results from EnergyPlus and TRNSYS since these software use similar algorithms for the calculation of the building’s energy demand. This research aims to check the efficiency of the Laplace and the State-Space method for calculating the building’s energy demand for heavyweight building elements and shorter sampling time, and it also gives the means for the improvement of the algorithms used by these methods. As the reference point for the boundary heat flux density, the finite difference method (FDM) is used. Even though the dynamic heat transfer simulations are superior to the calculation based on the stationary boundary conditions, they have their limitations and will give unsatisfactory results if not properly used.

Keywords: Laplace method, state-space method, conduction transfer functions, finite difference method

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Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

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A backward or inverse problem is known to be an ill-posed problem due to its instability that easily emerges with any slight change within the conditions of the problem. Therefore, only a limited number of numerical approaches are available to solve a backward problem. This paper considers the Nagumo equation, an equation that describes impulse propagation in nerve axons, which also models population growth with the Allee effect. A creative operator splitting numerical scheme is constructed to solve the inverse Nagumo equation. Computational simulations are used to verify that this scheme is stable, accurate, and efficient.

Keywords: inverse/backward equation, operator-splitting, Nagumo equation, ill-posed, finite-difference

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Authors: Sofia M. Karadimitriou, Kostas Triantafyllopoulos, Timothy Heaton

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Reduced dimension Dynamic Spatio-Temporal Models (DSTMs) jointly describe the spatial and temporal evolution of a function observed subject to noise. A basic state space model is adopted for the discrete temporal variation, while a continuous autoregressive structure describes the continuous spatial evolution. Application of such a DSTM relies upon the pre-selection of a suitable reduced set of basic functions and this can present a challenge in practice. In this talk, we propose an online estimation method for high dimensional spatio-temporal data based upon DSTM and we attempt to resolve this issue by allowing the basis to adapt to the observed data. Specifically, we present a wavelet decomposition in order to obtain a parsimonious approximation of the spatial continuous process. This parsimony can be achieved by placing a Laplace prior distribution on the wavelet coefficients. The aim of using the Laplace prior, is to filter wavelet coefficients with low contribution, and thus achieve the dimension reduction with significant computation savings. We then propose a Hierarchical Bayesian State Space model, for the estimation of which we offer an appropriate particle filter algorithm. The proposed methodology is illustrated using real environmental data.

Keywords: multidimensional Laplace prior, particle filtering, spatio-temporal modelling, wavelets

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Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: Yuan Yan Tang, Lina Yang, Hong Li

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Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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Authors: Jorge Olivares, Fernando Maass, Pablo Martin

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Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function.

Keywords: Caputo, modified Bessel functions, hypergeometric, linear fractional differential equations, transform Laplace

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