Integral Transform Solutions for Diffusion in Heterogeneous Media

2008 ◽  
Author(s):  
Carolina P. Naveira ◽  
Olivier Fudym ◽  
Renato M. Cotta ◽  
Helcio R. B. Orlande
Keyword(s):  
Exact Solution ◽  
Eigenvalue Problem ◽  
Heterogeneous Media ◽  
Space Variable ◽  
Integral Transform ◽  
Heat Diffusion ◽  
Heat Transfer Analysis ◽  
Layered System ◽  
Varying Coefficients ◽  
Classical Integral

The Generalized Integral Transform Technique is employed in the hybrid numerical-analytical solution of heat diffusion problems in heterogeneous media. The GITT is utilized to handle the associated eigenvalue problem with aribitrarily space variable coefficients, defining an eigenfunction expansion in terms of a Sturm-Liouville problem of known solution. The formal solution is first applied in solving an example of space variable thermophysical properties found in heat transfer analysis of functionally graded materials (FGM), validated by the exact solution obtained through classical integral transforms in the specific situation of exponentially varying coefficients. Then, it is challenged in handling a double-layered system with abrupt variation of properties, and critically compared against the exact solution obtained by the classical integral transform method with the adequate discontinuous multi-region eigenvalue problem. The convergence behavior of the proposed expansions is then critically inspected and numerical results are presented to demonstrate the applicability of the general approach.

Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Renato M. Cotta ◽  
Carolina P. Naveira-Cotta ◽  
Diego C. Knupp
Keyword(s):  
Eigenvalue Problem ◽  
Eigenfunction Expansion ◽  
Space Variable ◽  
Eigenvalue Problems ◽  
Integral Transform ◽  
Diffusion Problem ◽  
Eigenfunction Expansions ◽  
Diffusion Operator ◽  
Convection Diffusion ◽  
Diffusion Problems

The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers' equation.

Generalized Mellin Convolutions and their Asymptotic Expansions

1984 ◽  
Vol 36 (5) ◽  
pp. 924-960 ◽  
Author(s):  
R. Wong ◽  
J. P. Mcclure
Keyword(s):  
Asymptotic Expansions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Integral Transforms ◽  
Generalized Function ◽  
Positive Parameter ◽  
Ordinary Sense ◽  
Integrable Functions ◽  
Classical Integral ◽  
Image Position

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form1.1where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

Green's function of the Lord–Shulman thermo-poroelasticity theory

2020 ◽  
Vol 221 (3) ◽  
pp. 1765-1776 ◽  
Author(s):  
Jia Wei ◽  
Li-Yun Fu ◽  
Zhi-Wei Wang ◽  
Jing Ba ◽  
José M Carcione
Keyword(s):  
Plane Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Heterogeneous Media ◽  
Numerical Algorithms ◽  
Heat Diffusion ◽  
P Wave ◽  
Wave Analysis ◽  
S Wave ◽  
S Waves

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

Symplectic Analysis of Wrinkles in Elastic Layers With Graded Stiffnesses

2018 ◽  
Vol 86 (1) ◽  
Author(s):  
Jianjun Sui ◽  
Junbo Chen ◽  
Xiaoxiao Zhang ◽  
Guohua Nie ◽  
Teng Zhang
Keyword(s):  
Finite Element ◽  
Hamiltonian System ◽  
Eigenvalue Problem ◽  
Time Variable ◽  
Time Varying ◽  
Thickness Direction ◽  
Varying Coefficients ◽  
Elastic Layers ◽  
Critical Strains ◽  
Graded Structures

Wrinkles in layered neo-Hookean structures were recently formulated as a Hamiltonian system by taking the thickness direction as a pseudo-time variable. This enabled an efficient and accurate numerical method to solve the eigenvalue problem for onset wrinkles. Here, we show that wrinkles in graded elastic layers can also be described as a time-varying Hamiltonian system. The connection between wrinkles and the Hamiltonian system is established through an energy method. Within the Hamiltonian framework, the eigenvalue problem of predicting wrinkles is defined by a series of ordinary differential equations with varying coefficients. By modifying the boundary conditions at the top surface, the eigenvalue problem can be efficiently and accurately solved with numerical solvers of boundary value problems. We demonstrated the accuracy of the symplectic analysis by comparing the theoretically predicted displacement eigenfunctions, critical strains, and wavelengths of wrinkles in two typical graded structures with finite element simulations.

scholarly journals The Exact Solution of the Eigenvalue Problem for the Parametric Down Conversion Process in the Kerr Medium

2003 ◽  
Vol 53 (11) ◽  
pp. 1015-1020 ◽  
Author(s):  
Goce Chadzitaskos ◽  
Maciej Horowski ◽  
Anatol Odzijewicz ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Exact Solution ◽  
Eigenvalue Problem ◽  
Conversion Process ◽  
Kerr Medium ◽  
Parametric Down Conversion ◽  
Down Conversion

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

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