Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients

1989 ◽  
Vol 283 (2) ◽  
pp. 211-239 ◽  
Author(s):  
Nicola Garofalo ◽  
Ermanno Lanconelli
Keyword(s):  
Asymptotic Behavior ◽  
Potential Theory ◽  
Fundamental Solutions ◽  
Variable Coefficients

METHOD OF FUNDAMENTAL SOLUTION FOR AN INVERSE HEAT CONDUCTION PROBLEM WITH VARIABLE COEFFICIENTS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341009 ◽  
Author(s):  
MING LI ◽  
XIANG-TUAN XIONG ◽  
YAN LI
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Inverse Heat Conduction Problem ◽  
Method Of Fundamental Solutions ◽  
Inverse Heat Conduction ◽  
Modified Method ◽  
On Line ◽  
Ill Posed

In this paper, we consider an inverse heat conduction problem with variable coefficient a(t). In many practical situations such as an on-line testing, we cannot know the initial condition for example because we have to estimate the problem for the heat process which was already started. Based on the method of fundamental solutions, we give a numerical scheme for solving the reconstruction problem. Since the governing equation contains variable coefficients, modified method of fundamental solutions was used to solve this kind of ill-posed problems. Some numerical examples are given for verifying the efficiency and accuracy of the presented method.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

scholarly journals THE METHOD OF APPROXIMATE PARTICULAR SOLUTIONS FOR SOLVING ELLIPTIC PROBLEMS WITH VARIABLE COEFFICIENTS

2011 ◽  
Vol 08 (03) ◽  
pp. 545-559 ◽  
Author(s):  
C. S. CHEN ◽  
C. M. FAN ◽  
P. H. WEN
Keyword(s):  
Solution Process ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Left Hand ◽  
Particular Solutions ◽  
Particular Solution ◽  
The Right ◽  
Close Form

A new version of the method of approximate particular solutions (MAPSs) using radial basis functions (RBFs) has been proposed for solving a general class of elliptic partial differential equations. In the solution process, the Laplacian is kept on the left-hand side as a main differential operator. The other terms are moved to the right-hand side and treated as part of the forcing term. In this way, the close-form particular solution is easy to obtain using various RBFs. The numerical scheme of the new MAPSs is simple to implement and yet very accurate. Three numerical examples are given and the results are compared to Kansa's method and the method of fundamental solutions.

Bending Vibrations of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

1994 ◽  
Vol 61 (4) ◽  
pp. 949-955 ◽  
Author(s):  
Sen Yung Lee ◽  
Shueei Muh Lin
Keyword(s):  
Differential Equations ◽  
Coupling Effect ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Bending Vibrations ◽  
Rotating Speed ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Elastically Restrained ◽  
Angle Of Rotation

Without considering the Coriolis force, the governing differential equations for the pure bending vibrations of a rotating nonuniform Timoshenko beam are derived. The two coupled differential equations are reduced into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The explicit relation between the flexural displacement and the angle of rotation due to bending is established. The frequency equations of the beam with a general elastically restrained root are derived and expressed in terms of the four normalized fundamental solutions of the associated governing differential equations. Consequently, if the geometric and material properties of the beam are in polynomial forms, then the exact solution for the problem can be obtained. Finally, the limiting cases are examined. The influence of the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and taper ratio on the natural frequencies, and the phenomenon of divergence instability (tension buckling) are investigated.

scholarly journals Fundamental solutions for a class of equations with several singular coefficients

1968 ◽  
Vol 8 (3) ◽  
pp. 575-583 ◽  
Author(s):  
R. J. Weinacht
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Potential Theory ◽  
Fundamental Solutions ◽  
Elliptic Partial Differential Equations ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Symmetric Potential ◽  
Singular Coefficients

This paper is concerned with a class of singular elliptic partial differential equations related to the operator of Weinstein's generalized axially symmetric potential theory (GASPT) [1, 2] which has numerous applications.

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