scholarly journals Generalization of Finite Integral Transforms for Treating Nonlinear Problems in Heat Diffusion. Part II: Application to a nonlinear case.(Dept.P)

2021 ◽  
Vol 15 (1) ◽  
pp. 12-21
Author(s):  
Bishri Abdel-Hamed
Keyword(s):  
Nonlinear Problems ◽  
Integral Transforms ◽  
Heat Diffusion ◽  
Nonlinear Case ◽  
Finite Integral

Nusselt Numbers in Rectangular Ducts With Laminar Viscous Dissipation

1999 ◽  
Vol 121 (4) ◽  
pp. 1083-1087 ◽  
Author(s):  
G. L. Morini ◽  
M. Spiga
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Viscous Dissipation ◽  
Integral Transforms ◽  
Navier Stokes ◽  
Brinkman Number ◽  
Balance Equations ◽  
Nusselt Numbers ◽  
Finite Integral ◽  
Axial Heat

In this paper, the steady temperature distribution and the Nusselt numbers are analytically determined for a Newtonian incompressible fluid in a rectangular duct, in fully developed laminar flow with viscous dissipation, for any combination of heated and adiabatic sides of the duct, in H1 boundary condition, and neglecting the axial heat conduction in the fluid. The Navier-Stokes and the energy balance equations are solved using the technique of the finite integral transforms. For a duct with four uniformly heated sides (4 version), the temperature distribution and the Nusselt numbers are obtained as a function of the aspect ratio and of the Brinkman number and presented in graphs and tables. Finally it is proved that the temperature field in a fully developed T boundary condition can be obtained as a particular case of the H1 problem and that the corresponding Nusselt numbers do not depend on the Brinkman number.

Practical numerical considerations for the Alekseev–Mikhailenko method

1990 ◽  
Vol 27 (8) ◽  
pp. 1023-1030 ◽  
Author(s):  
P. F. Daley ◽  
F. Hron
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Radial Symmetry ◽  
Hyperbolic Partial Differential Equations ◽  
Finite Integral ◽  
Finite Hankel Transform ◽  
Difference Methods ◽  
New Generation

Programs that utilize the Alekseev–Mikhailenko method are becoming viable seismic interpretation aids because of the availability of a new generation of supercomputers. This method is highly numerically accurate, employing a combination of finite integral transforms and finite difference methods, for the solution of hyperbolic partial differential equations, to yield the total seismic wave field.In this paper two questions of a numerical nature are addressed. For coupled P–Sv wave propagation with radial symmetry, Hankel transforms of order 0 and 1 are required to cast the problem in a form suitable for solution by finite difference methods. The inverse series summations would normally require that the two sets of roots of the transcendental equations be employed, corresponding to the zeroes of the Bessel functions of order 0 and 1. This matter is clarified, and it is shown that both inverse series summations may be performed by considering only one set of roots.The second topic involves providing practical means of determining the lower and upper bounds of a truncated series that suitably approximates the infinite inverse series summation of the finite Hankel transform. It is shown that the number of terms in the truncated series generally decreases with increasing duration of the source pulse and that the truncated series may be further reduced if near-vertical-incidence seismic traces are avoided.

Rectangular Plates on Unilateral Edge Supports: Part 1—Theory and Numerical Analysis

1986 ◽  
Vol 53 (1) ◽  
pp. 146-150 ◽  
Author(s):  
J. P. Dempsey ◽  
Hui Li
Keyword(s):  
Singular Integral Equations ◽  
Integral Transforms ◽  
Rectangular Plates ◽  
Centrally Symmetric ◽  
Solution Methods ◽  
Dual Series Equations ◽  
Finite Integral ◽  
Loss Of Contact ◽  
Laterally Loaded ◽  
Numerical Approaches

The corners of a simply supported, laterally loaded rectangular plate must be anchored to prevent them from lifting off the supports. If no such anchors are provided, and the supports are unilateral or capable of exerting forces in one direction only, parts of the plate will bend away from the supports upon loading. The loss of contact when uplift of laterally loaded rectangular plates is not prevented is examined in this paper. Arbitrary centrally symmetric loading is considered. Finite integral transforms convert the coupled dual-series equations that result from the Levy-Nadai approach to two coupled singular integral equations. Different solution methods are applicable for sagged and unsagged supports; these two numerical approaches are discussed in detail.

Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Changpin Li ◽  
Zhiqiang Li
Keyword(s):  
Differential System ◽  
Integral Transforms ◽  
Nonlinear Case ◽  
Algebraic Decay ◽  
Future Studies ◽  
Linearization Theorem ◽  
Fractional Differential System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
The Stability

Abstract In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

scholarly journals Free Transverse Vibration of Rectangular Orthotropic Plates with Two Opposite Edges Rotationally Restrained and Remaining Others Free

2018 ◽  
Vol 9 (1) ◽  
pp. 22 ◽  
Author(s):  
Yuan Zhang ◽  
Sigong Zhang
Keyword(s):  
Boundary Conditions ◽  
Transverse Vibration ◽  
Integral Transforms ◽  
Mode Shapes ◽  
Alternative Formulation ◽  
Orthotropic Plates ◽  
Engineering Structures ◽  
Current Design ◽  
Classical Boundary ◽  
Finite Integral

Many types of engineering structures can be effectively modelled as orthotropic plates with opposite free edges such as bridge decks. The other two edges, however, are usually treated as simply supported or fully clamped in current design practice, although the practical boundary conditions are intermediate between these two limiting cases. Frequent applications of orthotropic plates in structures have generated the need for a better understanding of the dynamic behaviour of orthotropic plates with non-classical boundary conditions. In the present study, the transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained with the remaining others free was studied by applying the method of finite integral transforms. A new alternative formulation was developed for vibration analysis, which provides much easier solutions. Exact series solutions were derived, and the excellent accuracy and efficiency of the method are demonstrated through considerable numerical studies and comparisons with existing results. Some new results have been presented. In addition, the effect of different degrees of rotational restraints on the mode shapes was also demonstrated. The present analytical method is straightforward and systematic, and the derived characteristic equation for eigenvalues can be easily adapted for broad applications.

scholarly journals Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

2016 ◽  
Vol 2016 ◽  
pp. 1-22
Author(s):  
Inês Simões ◽  
António Tadeu ◽  
Nuno Simões
Keyword(s):  
Heat Conduction ◽  
Frequency Domain ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Heat Diffusion ◽  
One Dimensional ◽  
The Time Domain ◽  
Point Line

This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensionalfundamental solutionor 2.5D Green’s functions. These equations are very useful for formulating three-dimensional thermodynamic problems by means of integral transforms methods and/or boundary elements. The image source technique is used to build up different geometries such as half-spaces, corners, rectangular pipes, and parallelepiped boxes. The final expressions are verified here by applying the equations to problems for which the solution is known analytically in the time domain.

New Straightforward Benchmark Solutions for Bending and Free Vibration of Clamped Anisotropic Rectangular Thin Plates

2021 ◽  
pp. 1-24
Author(s):  
Dongqi An ◽  
Zhuofan Ni ◽  
Dian Xu ◽  
Rui Li
Keyword(s):  
Free Vibration ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Integral Transforms ◽  
Thin Plates ◽  
Orthotropic Plates ◽  
Transform Method ◽  
Finite Integral ◽  
Benchmark Solutions ◽  
Similar Complex

Abstract This study presents new straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method. Being different from the previous studies that took pure trigonometric functions as the integral kernels, the exponential functions are adopted, and the unknowns to be determined are constituted after the integral transform, which overcomes the difficulty in solving the governing higher-order partial differential equations with odd derivatives with respect to both the in-plane coordinate variables, thus goes beyond the limit of conventional finite integral transforms that are only applicable to isotropic or orthotropic plates. The present study provides an easy-to-implement approach for similar complex problems, extending the scope of finite integral transforms with applications to plate problems. The validity of the method and accuracy of the new solutions that can serve as benchmarks are well confirmed by satisfactory comparison with the numerical solutions.

On a heat flow problem in a hollow circular cylinder

1968 ◽  
Vol 64 (1) ◽  
pp. 193-202
Author(s):  
Nuretti̇n Y. Ölçer
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Eigenvalue Problems ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Frequency Equation ◽  
Hollow Circular Cylinder ◽  
New Ideas ◽  
Finite Integral

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

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