Numerical Simulation And Calculation Of Quenching Phenomenon Based On One Dimensional Spatial Fractional Differential Equations

Quenching phenomena can be observed in many scientific and industrial fields,such as cell multiplications,liquid and solid fuel combustion,pipeline decay preventions and infectious disease outbreaks.The content of this article consists of the following two parts.Firstly,this article simulates the quenching phenomenon based on one-dimensional two-sided fractional diffusion reaction equation.The second part simulates the quenching phenomenon of one-dimensional one-sided fractional convection diffusion reaction equation.The specific contents of this article is as follows:In the first part,a numerical method for simulating the quenching phenomenon based on the onedimensional diffusion reaction equation with two-sided Riemann-Liouville space fractional derivatives is proposed and studied.The approach adopts weighted Griinwald formulas for spatial discretization.The time step is optimized through the asymptotic arc length monitoring function.Monotonicity,positivity preservations and linearized stability are proved under suitable constraints on spatial and temporal discretization parameters.Two simulation experiments are conducted to demonstrate the effectiveness of the numerical method.Connections between the two-sided fractional differential operator and critical values including quenching time,critical length and quenching location are investigated.In the second part,a numerical method for simulating the quenching phenomenon based on the one-dimensional convection-diffusion reaction equation with one-sided Riemann-Liouville space fractional derivatives is proposed and studied.Properly weighted Grünwald formulas are employed for the discretization of the fractional derivative.A forward Euler formula is considered in the approximation of the convective term.The time step is optimized through the asymptotic arc length monitoring function.Monotonicity,positivity and stability of the numerical solution generated by this scheme are proved.Three numerical experiments are designed to demonstrate and simulate key characteristics of the semi-adaptive scheme constructed,including critical length,quenching time and quenching location of the fractional quenching phenomena formulated through the one-sided space-fractional convective-diffusion reaction problem.Effects of the convective function to quenching key characteristics are also investigated.Computational results obtained are carefully compared with those acquired from conventional integer order quenching convection-diffusion reaction problems for validating anticipated accuracy.The experimental results show that the method is highly effective and feasible.