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This paper investigates quenching solutions of an one-dimensional, two-sided Riemann–Liouville fractional order convection–diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in conjunction with standard and shifted Grünwald formulas. The advective term is handled utilizing a straightforward Euler formula, resulting in a semi-discretized system of nonlinear ordinary differential equations. The conservativeness of the proposed scheme is rigorously proved and validated through simulation experiments. The study is further advanced to a fully discretized, semi-adaptive finite difference method. Detailed analysis is implemented for the monotonicity, positivity and stability of the scheme. Investigations are carried out to assess the potential impacts of the fractional order on quenching location, quenching time, and critical length. The computational results are thoroughly discussed and analyzed, providing a more comprehensive understanding of the quenching phenomena modeled through two-sided fractional order convection-diffusion problems. 

This is for a highlight of the application-oriented numerical computation and optimization-a celebration of 60 years of the IJCM publication. On a usual rainy day in early May 1964, the first issue of the International Journal of Computer Mathematics (IJCM) was published in London, Great Britain. Apparently, mathematicians around the world felt and were excited by the morningtide and shock waves from digital computations, after the success of modern computer hardware configuration and software structures. The twilight of the digital new age inspired the pioneers, and this also led to the birth of the IJCM, even before the invention of the word numerical analysis. The first volume of the IJCM consisted of 4 issues, 314 pages of 16 peer-reviewed research papers, and 4 book reviews. 

Quenching has been an extremely important natural phenomenon observed in many biomedical and multiphysical procedures, such as a rapid cancer cell progression or internal combustion process. The latter has been playing a crucial rule in optimizations of modern solid fuel rocket engine designs. Mathematically, quenching means the blowup of temporal derivatives of the solution function q while the function itself remains to be bounded throughout the underlying procedure. This paper studies a semi-adaptive numerical method for simulating solutions of a singular partial differential equation that models a significant number of quenching data streams. Numerical convergence will be investigated as well as verifying that features of the solution is preserved in the approximation. Orders of the convergence will also be validated through experimental procedures. Milne’s device will be used. Highly accurate data models will be presented to illustrate theoretical predictions. 

Nonlinear Kawarada equations have been used to model solid fuel combustion processes in the oil industry. An effective way to approximate solutions of such equations is to take advantage of the finite difference configurations. Traditionally, the nonlinear term of the equation is linearized while the numerical stability of a difference scheme is investigated. This leaves certain ambiguity and uncertainty in the analysis. Based on nonuniform grids generated through a quenching-seeking moving mesh method in space and adaptation in time, this paper introduces a completely new stability analysis of the approximation without freezing the nonlinearity involved. Pointwise orders of convergence are investigated numerically. Simulation experiments are carried out to accompany the mathematical analysis to strengthen our conclusions. 

This paper studies an extended application of the Glowinski-Le Tallec splitting for approximating solutions of linear and nonlinear partial differential equations. It is shown that the three-level, six-component operator decomposition, originally designed for Lagrangian optimizations, provides a stable second-order operator splitting approximation for the solutions of evolutional partial differential equations. It is also found that the Glowinski-Le Tallec formula not only provides an effective enhancement to conventional two-level, four-component ADI and LOD methods, but also introduces a flexible way for constructing multi-parameter operator splitting strategies in respective spaces where broad spectrums of mathematical models may exist for important natural phenomena and applications. The extended operator splitting is utilized for solving a singular and nonlinear Kawarada problem satisfactorily. Multiple simulation results are presented. 

A preservative scheme is presented and analyzed for the solution of a quenching type convective-diffusion problem modeled through one-sided Riemann-Liouville space-fractional derivatives. Properly weighted Grünwald formulas are employed for the discretization of the fractional derivative. A forward difference approximation is considered in the approximation of the convective term of the nonlinear equation. Temporal steps are optimized via an asymptotic arc-length monitoring mechanism till the quenching point. Under suitable constraints on spatial-temporal discretization steps, the monotonicity, positivity preservations of the numerical solution and numerical stability of the 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 initial-boundary value problem. Effects of the convective function to quenching are discussed. Numerical estimates of the order of convergence are obtained. Computational results obtained are carefully compared with those acquired from conventional integer order quenching convection-diffusion problems for validating anticipated accuracy. The experiments have demonstrated expected accuracy and feasibility of the new method. 

Abstract: Quenching has been an extremely important natural phenomenon observed in many biomedical and multiphysical procedures, such as a rapid cancer cell progression or internal combustion process. The latter has been playing a crucial rule in optimizations of modern solid fuel rocket engine designs. Mathematically, quenching means the blow-up of temporal derivatives of the solution function u while the function itself remains to be bounded throughout the underlying procedure. This paper studies a semi-adaptive numerical method for simulating solutions of a singular partial differential equation that models a significant number of quenching data streams. Numerical convergence will be investigated as well as verifying that features of the solution is preserved in the approximation. Orders of the convergence will also be validated through experimental procedures. Milne's device will be used for data estimate and initial building-up of our deep neural network (DNN) software. Highly accurate data models will be presented to illustrate theoretical predictions. 

The aims of this paper are to investigate and propose a numerical approximation for a quenching type diffusion problem associated with a two-sided Riemann-Liouville space- fractional derivative. The approach adopts weighted Grünwald formulas for suitable spatial discretization. An implicit Crank-Nicolson scheme combined with adaptive technology is then implemented for a temporal integration. Monotonicity, positivity preservation and linearized stability are proved under suitable constraints on spatial and temporal discretization parameters. Two specially designed simulation experiments are presented for illustrating and outreaching properties of the numerical method constructed. Connections between the two-sided fractional differential operator and critical values including quenching time, critical length and quenching location are investigated.