scholarly journals On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 921
Author(s):  
Alexander Kazakov ◽  
Lev Spevak ◽  
Olga Nefedova ◽  
Anna Lempert
Keyword(s):  
Source Term ◽  
High Efficiency ◽  
Numerical Study ◽  
Nonlinear Heat Equation ◽  
Two Dimensional ◽  
Dual Reciprocity Method ◽  
Reciprocity Method ◽  
Nonlinear Parabolic ◽  
Solution Existence And Uniqueness ◽  
Dimensional Boundary

The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

scholarly journals Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Jamal H. Al-Smail ◽  
Salim A. Messaoudi ◽  
Ala A. Talahmeh
Keyword(s):  
Parabolic Equation ◽  
Operator Theory ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Nonlinear Operator ◽  
Numerical Study ◽  
Well Posedness ◽  
Two Dimensional ◽  
Nonlinear Parabolic ◽  
Decay Of Solutions

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t), where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

A Numerical Study of the Diffusion Performance of a Terraced Classroom

2011 ◽  
Vol 97 (5) ◽  
pp. 890-899
Author(s):  
Zhaorong Zhang ◽  
C. M. Mak ◽  
Jianliang Li
Keyword(s):  
Boundary Element Method ◽  
Porous Material ◽  
Sound Wave ◽  
Sound Pressure ◽  
Numerical Study ◽  
Quadratic Residue ◽  
Pressure Level ◽  
Two Dimensional ◽  
Rear Wall ◽  
Dimensional Boundary

The acoustic diffusion performance of a terraced classroom was investigated using a two-dimensional boundary element method. Quadratic residue diffuser (QRD) and porous material were employed on the ceiling and rear wall, respectively, to improve the diffusion performance. The diffusion gains of various models were calculated to compare the diffusion performance. It is found that terraces in a rectangular classroom raise the lowest sound pressure level and provide slight diffusion improvement. The QRD ceiling enhances the diffusion by scattering the sound wave to be more evenly distributed, but at some frequencies the diffusion improvement is minor or even negative. The absorption rear wall provides useful diffusion gain mainly at higher frequencies by absorbing parts of the reflected sound. When the parameters of the QRD ceiling and porous material change, the diffusion improvement first increases and then begins to decrease. In a terraced classroom with both treatments, the diffusion at lower frequencies is similar to that with the QRD ceiling, while at higher frequencies it resembles that with the absorption rear wall. The results clearly indicate that the combination of the two treatments produces the most desirable diffusion performance of the tested models.

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