Solving cylindrical geothermal problems using the Gaver‐Stehfest inverse Laplace transform

Geophysics ◽  
1985 ◽  
Vol 50 (10) ◽  
pp. 1581-1587 ◽  
Author(s):  
Heinrich Villinger
Keyword(s):  
Heat Flow ◽  
Laplace Transform ◽  
Numerical Procedure ◽  
Heat Diffusion ◽  
Inverse Laplace Transform ◽  
Flow Measurements ◽  
Model Calculations ◽  
Heat Diffusion Equation ◽  
Laplace Domain ◽  
Whole Space

The inhomogeneous one‐dimensional heat‐diffusion equation is solved semianalytically in a cylindrically layered whole space. An analytic solution of the problem can be derived in the Laplace domain in a straightforward way assuming continuity of temperature and heat flow at the layer boundaries. The time‐dependent heat sources can be distributed spatially over the layered whole space. Due to the complexity of the solution in the Laplace domain, the inverse Laplace transform is calculated using a numerical procedure (Gaver‐Stehfest algorithm). It is shown that the algorithm is fast, stable, easy to use, and simple to adjust to various models and boundary conditions. The results obtained are accurate to about 0.01 percent. Therefore, it is an excellent tool for geothermal model calculations with cylindrical symmetry and could be useful for the correction of BHT-values or marine heat flow measurements.

scholarly journals Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xiaoli Qiang ◽  
Kamran ◽  
Abid Mahboob ◽  
Yu-Ming Chu
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Fractional Order ◽  
Integrodifferential Equations ◽  
Inverse Laplace Transform ◽  
Laplace Domain ◽  
The Real ◽  
Volterra Integrodifferential Equation ◽  
Engineering Sciences ◽  
Real Domain

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

scholarly journals Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2299-2305 ◽  
Author(s):  
Ilknur Koca ◽  
Abdon Atangana
Keyword(s):  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Heat Diffusion ◽  
Relaxation Kernel ◽  
Extended Version ◽  
Heat Diffusion Equation ◽  
The Laplace Transform ◽  
Non Local

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

Analysis of Two-Dimensional Heat Diffusion Using FEA and the Fundamental Collocation Method

2016 ◽  
Author(s):  
Amir Khalilollahi ◽  
Enayat Mahajerin ◽  
Gary Burgess
Keyword(s):  
Laplace Transform ◽  
Collocation Method ◽  
Initial Conditions ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Two Dimensional ◽  
Heat Diffusion Equation ◽  
The Laplace Transform

Finite Element Analysis (FEA) and the Laplace Transform-Based Fundamental Collocation Method (FCM) are used to solve the heat diffusion equation in two-dimensional regions having arbitrary shapes and subjected to arbitrary initial and mixed type boundary conditions. In the FEA method, the time derivative is replaced with a finite difference approximation. The resulting time dependent global equations are solved incrementally starting with the initial conditions. The FCM approach is applied in the Laplace transform domain to obtain temperatures in the s-domain, T(x,y,s). An inversion technique is used to retrieve the time domain solution, T(x,y,t). To compare applicability and accuracy of these methods, both techniques are applied to transient heat flow problems for which exact solutions are known.

scholarly journals THERMAL ANALYSIS OF DIFFERENT CONFIGURATIONS FOR BUILDING INSULATION SYSTEMS WITH ACTUAL MATERIALS USING REAL SOLAR RADIATION AND TEMPERATURE DATA

2021 ◽  
Vol 20 (1) ◽  
pp. 52
Author(s):  
C. B. P. Sa ◽  
M. F. Curi
Keyword(s):  
Heat Flow ◽  
Air Conditioning ◽  
Temperature Data ◽  
Heat Diffusion ◽  
Launch Vehicles ◽  
Heat Diffusion Equation ◽  
Insulating Material ◽  
Building Insulation ◽  
Insulation Systems ◽  
Flow Through

Thermal insulation is present in several engineering areas, such as in civilconstruction, in industrial ovens, in satellites and launch vehicles and even indomestic applications, such as refrigerators. This work aims to presentmultilayer insulation for civil construction, thus reducing electrical energyexpenditure on air conditioning the environment and increasing thermalcomfort. To determine the heat flow through the wall, it was necessary tosolve the heat diffusion equation in rectangular Cartesian coordinates for eachinsulation layer, and the boundary conditions of each differential equationdepend on the neighboring layers, thus forming a system of equations. TheWolfram Mathematica computational software was used to solve themathematical equations. The heat flow was determined for the four seasonsof the year throughout the day for different configurations of thermalinsulation, varying the insulating material and its thickness. After analyzingthe results, it was possible to observe the great efficiency of the models withthermal insulation compared to traditional walls, making the internal wall'stemperature profile more constant throughout the day.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Quadrature Integration Techniques for Random Hyperbolic PDE Problems

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 160
Author(s):  
Rafael Company ◽  
Vera N. Egorova ◽  
Lucas Jódar
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Quadrature Rule ◽  
Inverse Laplace Transform ◽  
Process Time ◽  
Efficient Computation ◽  
Hyperbolic Pde ◽  
Numerical Convergence ◽  
Laplace Transform Technique ◽  
Integration Techniques

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Modeling electro-osmotic flow and thermal transport of Caputo fractional Burgers fluid through a micro-channel

2021 ◽  
pp. 095440892110259
Author(s):  
Mohammed Abdulhameed ◽  
Garba Tahiru Adamu ◽  
Gulibur Yakubu Dauda
Keyword(s):  
Electric Field ◽  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Micro Channel ◽  
Osmotic Flow ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Burgers Fluid ◽  
Electro Osmotic Flow

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

MACSYMS's inverse Laplace transform

1989 ◽  
Vol 23 (1) ◽  
pp. 33-38
Author(s):  
M. Clarkson
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform
Sign in / Sign up

Export Citation Format

Share Document