Analytical solutions for the multi-term time–space fractional advection–diffusion equations with mixed boundary conditions

2013 ◽  
Vol 14 (2) ◽  
pp. 1026-1033 ◽  
Author(s):  
Xiao-Li Ding ◽  
Yao-Lin Jiang
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Diffusion Equations ◽  
Advection Diffusion ◽  
Time Space

scholarly journals A Data-Driven Based Spatiotemporal Model Reduction for Microwave Heating Process with the Mixed Boundary Conditions

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 827
Author(s):  
Jiaqi Zhong ◽  
Shan Liang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Model Reduction ◽  
Microwave Heating ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Data Driven ◽  
Heating Process ◽  
Spatiotemporal Model ◽  
Time Space

In this paper, a data-driven based spatiotemporal model reduction approach is proposed for predicting the temperature distribution and developing the computation speeds in the microwave heating process. Due to the mixed boundary conditions, it is difficult for the traditional spectral method to directly obtain the analytical eigenfunctions. Motivated by the time/space separation theory, we first propose a general framework of spatiotemporal model reduction, which can effectively develop the computation speeds in the numerical analysis of multi-physical fields. Subsequently, the empirical eigenfunctions are generated by applying the Karhunen–Loève theory to decompose the snapshots. Then, the partial differential Equation (PDE) model is discretized into a class of recursive equations and transformed as the reduced-order ordinary differential Equation (ODE) model. Finally, the effectiveness and superiority of the proposed approach is demonstrated by a comparison study with a traditional method on the microwave heating Debye medium.

Existence theorems for the quantum drift–diffusion equations with mixed boundary conditions

2016 ◽  
Vol 18 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Existence Theorems ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Approximate Solutions ◽  
Diffusion Equations ◽  
Drift Diffusion ◽  
High Regularity ◽  
Recent Theorem

In this paper, we construct a weak solution to the unipolar quantum drift–diffusion equations coupled with initial and mixed boundary conditions in up to four space dimensions. We only assume that the boundary of the domain is Lipschitz and the interface between the Dirichlet boundary and the Neumann boundary is a Lipschitzian hypersurface, and thus the lack of high regularity for solutions of such problems is an issue. The convergence of a sequence of approximate solutions is established via an application of a recent theorem by the author on the logarithmic upper bound for solutions of elliptic equations with the same type of boundary conditions [Logarithmic up bounds for solutions of elliptic partial differential equations, Proc. Amer. Math. Soc. 139 (2011) 3485–3490].

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