scholarly journals Variational approach to fractal reaction-diffusion equations with fractal derivatives

2021 ◽  
pp. 42-42
Author(s):  
Yue Wu
Keyword(s):  
Variational Principle ◽  
Diffusion Process ◽  
Variational Formulation ◽  
Variational Approach ◽  
Solution Process ◽  
Fractal Dimensions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Fractal Space

A fractal modification of the reaction-diffusion process is proposed with fractal derivatives, and a fractal variational principle is established in a fractal space. The concentration of the substrate can be determined according to the minimal value of the variational formulation. The solution process is illustrated step by step for ease applications in engineering, and the effect of fractal dimensions on solution morphology is elucidated graphically.

scholarly journals A variational approach to a porous catalyst

2021 ◽  
pp. 44-44
Author(s):  
Zhi-Qiang Sun
Keyword(s):  
Approximate Solution ◽  
Variational Principle ◽  
Diffusion Process ◽  
Variational Approach ◽  
Fractal Model ◽  
Porous Electrodes ◽  
Porous Catalyst ◽  
Convection Diffusion ◽  
Fractal Derivative ◽  
Fractal Space

The convection-diffusion process in porous electrodes depends greatly upon the porous structure. A fractal model for porous catalyst in a thin-zone bed reactor is established using He?s fractal derivative, and a variational principle is also established in a fractal space, and an approximate solution is obtained. Additionally an ancient Chinese algorithm is adopted to solve an algebraic equation.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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