An Integral Transform Involving Heun Functions and a Related Eigenvalue Problem

1986 ◽  
Vol 17 (3) ◽  
pp. 688-703 ◽  
Author(s):  
G. Valent
Keyword(s):  
Eigenvalue Problem ◽  
Integral Transform ◽  
Heun Functions

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.

The concentration distribution produced by shear dispersion of solute in Poiseuille flow

1990 ◽  
Vol 210 ◽  
pp. 201-221 ◽  
Author(s):  
A. N. Stokes ◽  
N. G. Barton
Keyword(s):  
Eigenvalue Problem ◽  
Concentration Distribution ◽  
Integral Transform ◽  
Flow Direction ◽  
Asymptotic Methods ◽  
Cross Sectional ◽  
Gaussian Profile ◽  
Shear Dispersion ◽  
Parallel Flows ◽  
Small Times

One of G. I. Taylor's most famous papers concerns the large-time behaviour of a cloud of soluble matter which has been injected into a solvent in laminar flow in a pipe. In the past thirty years, a number of successful attempts have been made to derive differently or extend Taylor's result, which is that the cloud of solute eventually takes a Gaussian profile in the flow direction. The present paper is another examination of this well-worked problem, but this time from the viewpoint of a formal integral transform representation of the solution. This approach leads to a better understanding of the solution; it also enables efficient numerical computations, and leads to extended and new asymptotic expansions.A Laplace transform in time and a Fourier transform in the flow direction leaves a complicated eigenvalue problem to be solved to give the cross-sectional behaviour. This eigenvalue problem is examined in detail, and the transforms are then inverted to give the concentration distribution. Both numerical and asymptotic methods are used. The numerical procedures lead to an accurate description of the concentration distribution, and the method could be generalized to compute dispersion in general parallel flows. The asymptotic procedures use two different classes of eigenvalues to give leading- and trailing-edge approximations for the solute cloud at small times. Meanwhile, at larger times, one eigenvalue branch dominates the solution and Taylor's result is recovered and extended using'the computer to generate extra terms in the approximation. Sixteen terms in the approximation are calculated, and a continued fraction expansion is deduced to enhance the accuracy.

Vector eigenfunction expansion in the integral transform solution of transient natural convection

2019 ◽  
Vol 29 (8) ◽  
pp. 2684-2708 ◽  
Author(s):  
Kleber Marques Lisboa ◽  
Jian Su ◽  
Renato M. Cotta
Keyword(s):  
Natural Convection ◽  
Eigenvalue Problem ◽  
Eigenfunction Expansion ◽  
Stokes Equations ◽  
Integral Transform ◽  
Temperature Fields ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Transient Natural Convection

Purpose The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation. Design/methodology/approach The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme. Findings A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification. Originality/value A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

scholarly journals Convolution algebras arising from Sturm-Liouville transforms and applications

2001 ◽  
Vol 27 (4) ◽  
pp. 221-228
Author(s):  
Jason P. Huffman ◽  
Henry E. Heatherly
Keyword(s):  
Eigenvalue Problem ◽  
Integral Transform ◽  
Operational Calculus ◽  
Liouville Problem ◽  
Generalized Convolution ◽  
Convolution Operation ◽  
Convolution Algebras ◽  
Sturm Liouville ◽  
Linear Integral ◽  
Differential And Integral Equations

A regular Sturm-Liouville eigenvalue problem gives rise to a related linear integral transform. Churchill has shown how such an integral transform yields, under certain circumstances, a generalized convolution operation. In this paper, we study the properties of convolution algebras arising in this fashion from a regular Sturm-Liouville problem. We give applications of these convolution algebras for solving certain differential and integral equations, and we outline an operational calculus for classes of such equations.

Conjugated Convection-Conduction Analysis in Microchannels With Axial Diffusion Effects and a Single Domain Formulation

2013 ◽  
Vol 135 (9) ◽  
Author(s):  
Diego C. Knupp ◽  
Carolina P. Naveira-Cotta ◽  
Renato M. Cotta
Keyword(s):  
Heat Transfer ◽  
Eigenvalue Problem ◽  
Integral Transform ◽  
Convection Heat Transfer ◽  
Solution Procedure ◽  
Variable Coefficients ◽  
Single Domain ◽  
Axial Diffusion ◽  
Diffusion Effects ◽  
Transfer Problems

An extension of a recently proposed single domain formulation of conjugated conduction–convection heat transfer problems is presented, taking into account the axial diffusion effects at both the walls and fluid regions, which are often of relevance in microchannels flows. The single domain formulation simultaneously models the heat transfer phenomena at both the fluid stream and the channel walls, by making use of coefficients represented as space variable functions, with abrupt transitions occurring at the fluid-wall interface. The generalized integral transform technique (GITT) is then employed in the hybrid numerical–analytical solution of the resulting convection–diffusion problem with variable coefficients. With axial diffusion included in the formulation, a nonclassical eigenvalue problem may be preferred in the solution procedure, which is itself handled with the GITT. To allow for critical comparisons against the results obtained by means of this alternative solution path, we have also proposed a more direct solution involving a pseudotransient term, but with the aid of a classical Sturm-Liouville eigenvalue problem. The fully converged results confirm the adequacy of this single domain approach in handling conjugated heat transfer problems in microchannels, when axial diffusion effects must be accounted for.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Sign in / Sign up

Export Citation Format

Share Document