• Title/Summary/Keyword: The laplace transform method

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COMBINED LAPLACE TRANSFORM WITH ANALYTICAL METHODS FOR SOLVING VOLTERRA INTEGRAL EQUATIONS WITH A CONVOLUTION KERNEL

  • AL-SAAR, FAWZIAH M.;GHADLE, KIRTIWANT P.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.22 no.2
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    • pp.125-136
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    • 2018
  • In this article, a homotopy perturbation transform method (HPTM) and the Laplace transform combined with Taylor expansion method are presented for solving Volterra integral equations with a convolution kernel. The (HPTM) is innovative in Laplace transform algorithm and makes the calculation much simpler while in the Laplace transform and Taylor expansion method we first convert the integral equation to an algebraic equation using Laplace transform then we find its numerical inversion by power series. The numerical solution obtained by the proposed methods indicate that the approaches are easy computationally and its implementation very attractive. The methods are described and numerical examples are given to illustrate its accuracy and stability.

ANALYTICAL SOLUTION OF SINGULAR FOURTH ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS OF VARIABLE COEFFICIENTS BY USING HOMOTOPY PERTURBATION TRANSFORM METHOD

  • Gupta, V.G.;Gupta, Sumit
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.165-177
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    • 2013
  • In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

A Boundary Integral Equation Formulation for an Unsteady Anisotropic-Diffusion Convection Equation of Exponentially Variable Coefficients and Compressible Flow

  • Azis, Mohammad Ivan
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.557-581
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    • 2022
  • The anisotropic-diffusion convection equation with exponentially variable coefficients is discussed in this paper. Numerical solutions are found using a combined Laplace transform and boundary element method. The variable coefficients equation is usually used to model problems of functionally graded media. First the variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation is then Laplace-transformed so that the time variable vanishes. The Laplace-transformed equation is consequently written as a boundary integral equation which involves a time-free fundamental solution. The boundary integral equation is therefore employed to find numerical solutions using a standard boundary element method. Finally the results obtained are inversely transformed numerically using the Stehfest formula to get solutions in the time variable. The combined Laplace transform and boundary element method are easy to implement and accurate for solving unsteady problems of anisotropic exponentially graded media governed by the diffusion convection equation.

A TYPE OF FRACTIONAL KINETIC EQUATIONS ASSOCIATED WITH THE (p, q)-EXTENDED 𝜏-HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Khan, Owais;Khan, Nabiullah;Choi, Junesang;Nisar, Kottakkaran Sooppy
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.2
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    • pp.381-392
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    • 2021
  • During the last several decades, a great variety of fractional kinetic equations involving diverse special functions have been broadly and usefully employed in describing and solving several important problems of physics and astrophysics. In this paper, we aim to find solutions of a type of fractional kinetic equations associated with the (p, q)-extended 𝜏 -hypergeometric function and the (p, q)-extended 𝜏 -confluent hypergeometric function, by mainly using the Laplace transform. It is noted that the main employed techniques for this chosen type of fractional kinetic equations are Laplace transform, Sumudu transform, Laplace and Sumudu transforms, Laplace and Fourier transforms, P𝛘-transform, and an alternative method.

ACCESS TO LAPLACE TRANSFORM OF fg

  • HWAJOON KIM;SOMCHAI LEKCHAROEN
    • Journal of applied mathematics & informatics
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    • v.41 no.1
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    • pp.83-93
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    • 2023
  • We would like to consider Laplace transform of the form of fg, the form of product, and applies it to Burger's equation in general case. This topic has not yet been addressed, and the methodology of this article is done by considerations with respect to several approaches about the transform of the form of f g and the mean value theorem for integrals. This paper has meaning in that the integral transform method is applied to solving nonlinear equations.

Analysis of One-Dimensional Transient Heat Conduction Problems using Hybrid Laplace Transform/finite Element Method (라플라스 변환과 유한요소법의 결합에 의한 1차원 과도 열전도 문제 해석)

  • Song, Byoung-Chul;Jung, Hae-Duk;Lee, Ki-Sik
    • Proceedings of the KIEE Conference
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    • 1997.07a
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    • pp.309-311
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    • 1997
  • In this paper, it is proposed that a algorithm which is applicable to the transient analysis by combined use of the Laplace transform and the finite element method. The proposed method removes the time terms using the Laplace transform and then solves the associated equation with the finite element method. The solution which is solved at frequency domain is transformed into time domain by use of the Laplace inversion. To verify proposed algorithm, heat conduction problem is analysed and found a good agreement with analytic solution.

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Study on the Dynamic Analysis of the Continuous System by Digital Modeling (이산화 기법에 의한 연속계의 동적 응답해석에 관한 연구)

  • 이용관;김인수;홍성욱;췌처린
    • Journal of KSNVE
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    • v.7 no.1
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    • pp.135-142
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    • 1997
  • This paper presents a digital modeling technique of the distributed system. The basic idea of the proposed technique is to discretize a continuous system with respect to the spatial coordinates using bilinear method. The response of the discretized system is analyzed by Laplace transform and z-transform. The computational results in torsional shaft and Timoshenko beam using the proposed technique are compared with the exact solutions and the results of finite element method.

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Transient analysis of cross-ply laminated shells using FSDT: Alternative formulation

  • Sahan, Mehmet Fatih
    • Steel and Composite Structures
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    • v.18 no.4
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    • pp.889-907
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    • 2015
  • This paper aims to present an alternative analytical method for transient vibration analysis of doubly-curved laminated shells subjected to dynamic loads. In the method proposed, the governing differential equations of laminated shell are derived using the dynamic version of the principle of virtual displacements. The governing equations of first order shear deformation laminated shell are obtained by Navier solution procedure. Time-dependent equations are transformed to the Laplace domain and then Laplace parameter dependent equations are solved numerically. The results obtained in the Laplace domain are transformed to the time domain with the help of modified Durbin's numerical inverse Laplace transform method. Verification of the presented method is carried out by comparing the results with those obtained by Newmark method and ANSYS finite element software. Also effects of number of laminates, different material properties and shell geometries are discussed. The numerical results have proved that the presented procedure is a highly accurate and efficient solution method.

THE DOUBLE FUZZY ELZAKI TRANSFORM FOR SOLVING FUZZY PARTIAL DIFFERENTIAL EQUATIONS

  • Kshirsagar, Kishor A.;Nikam, Vasant R.;Gaikwad, Shrikisan B.;Tarate, Shivaji A.
    • Journal of the Chungcheong Mathematical Society
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    • v.35 no.2
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    • pp.177-196
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    • 2022
  • The Elzaki Transform method is fuzzified to fuzzy Elzaki Transform by Rehab Ali Khudair. In this article, we propose a Double fuzzy Elzaki transform (DFET) method to solving fuzzy partial differential equations (FPDEs) and we prove some properties and theorems of DFET, fundamental results of DFET for fuzzy partial derivatives of the nth order, construct the Procedure to find the solution of FPDEs by DFET, provide duality relation of Double Fuzzy Laplace Transform (DFLT) and Double Fuzzy Sumudu Transform(DFST) with proposed Transform. Also we solve the Fuzzy Poisson's equation and fuzzy Telegraph equation to show the DFET method is a powerful mathematical tool for solving FPDEs analytically.

A Numerical Method for One-dimensional Inverse Heat Conduction Problem Using Laplace Transform (라플라스 변환을 이용한 1차원 열전도의 수치해석)

  • Shin, Woon-Chul;Bae, Sin-Chul
    • Journal of the Korean Society of Safety
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    • v.22 no.4
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    • pp.13-19
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    • 2007
  • An numerical method to estimate thermal diffusivity has been developed for one-dimensional unsteady heat conduction problem, when the temperatures are know at two positions in a semi-infinite body. Using the closed form solution which has already derived an explicit solution for the inverse problem for one-dimensional transient heat conduction using Laplace transform technique, we first estimate the surface temperature. The thermal diffusivity can be estimated by using the estimated surface temperature and measured temperatures, which include some uncertainties. The estimated surface heat flux and thermal diffusivity are found to be in good agreement with those of the experimented conditions. This method will be extended to the simultaneous measurement of thermal diffusivity and thermal conductivity.