This paper develops a hybrid post-processed Haar wavelet collocation method (HP-HWCM) for initial value problems of fractional differential equations (FDEs). Unlike the conventional Haar wavelet collocation method (HWCM), which directly approximates the fractional derivative, the proposed approach approximates the lowest-order integer derivative that strictly exceeds the fractional order, thereby substantially reducing the dominant error source. For the resulting nonlinear algebraic systems, a damped Newton method with backtracking line search (DNM-BLS) is employed to ensure robust and efficient convergence. Two complementary post-processing techniques are seamlessly integrated: polynomial spectral post-processing (PSPP) for FDEs with exact solutions of polynomial form, elevating the experimental order of convergence (EOC) from the theoretical ceiling of 2 inherent to the standard HWCM to a range of 2.5–2.8. For FDEs whose exact solutions contain a specific fractional-power term of the form t^δ ( δ≥ n+1 , n=[α ]+1 ), we introduce a three-grid Richardson extrapolation that systematically restores high-order convergence, yielding robust EOC values between 3 and 4. A rigorous error analysis confirms the theoretical convergence rates of the proposed framework. Numerical experiments on both linear and nonlinear problems for collocation points ranging from m=8 to m=512 demonstrate that the HP-HWCM consistently reduces errors by 2–6 orders of magnitude compared to the classical HWCM. Remarkably, by tuning the relaxation parameter, the PSPP technique can further elevate the numerical accuracy to an accuracy of 10^-19 – 10^-15 . Furthermore, comprehensive comparisons with various state-of-the-art schemes reveal that the proposed framework attains superior accuracy while requiring substantially fewer degrees of freedom.
Stiff ordinary differential equations frequently arise in chemical kinetics because reaction time scales can span many orders of magnitude. This work presents a systematic methodological and numerical study of high-order implicit–explicit Runge–Kutta schemes on the Akzo Nobel, Robertson, HIRES, and 20-species air-pollution benchmarks. For each problem, a reaction-based partition is specified and then assessed along a reference trajectory through the spectral radii of the full, implicit, and explicit Jacobians. Classical fixed-step RK4 and adaptive ode45 are used to demonstrate the explicit stability restriction, while comparisons with the corresponding fourth- and fifth-order DIRK schemes, which share the same implicit tableaux, isolate the effect of the IMEX partition. In the reported tests, the IMEX schemes attain accuracy comparable to their matched DIRK counterparts at lower computational cost. Linear invariants are preserved to machine precision, whereas nonnegativity is not guaranteed and can be lost on coarse grids. These results clarify both the practical benefits and the limitations of high-order IMEX integration for stiff chemical kinetics.
The Lengyel–Epstein mathematical model for the CIMA chemical reaction is studied. The reaction is envisaged to occur in a rectangularly-shaped batch spatial reactor. In addition, flow terms have been added to the original model, as an approximation to exchange of reagents with those in a well-mixed reservoir. These novel flow terms are responsible for the creation of parameter zones in which three different equilibrium states may exist. We demonstrate that large-amplitude limit-cycle oscillations can occur, enclosing three equilibria. Turing bifurcations are studied in detail, and it is shown that two-dimensional spatial patterns of chemical concentration can occur, both as time-dependent transient configurations and also as stationary spatial patterns.
The initial rate method is commonly used to study the kinetics of reactions proceeding by a general modifier mechanism. Its application assumes that the total enzyme concentration (E0) is small compared to the total substrate concentration (S0). Under in vivo conditions, however, the enzyme concentration is usually comparable to the substrate concentration. In this study, therefore, the maximum instantaneous reaction rate has been proposed as an alternative to the initial rate; the values of these rates are practically identical under the condition S0 >> E0. Accurate calculation of the maximum instantaneous rate requires solving a system of differential equations, which complicates the practical analysis of reaction kinetics. In this study, an approximate method for calculating this rate has been proposed, which consists in solving a seventh-degree algebraic equation. This method’s effectiveness has been demonstrated in examples for both low and high enzyme concentrations.
This article presents a parameter-uniform computational approach for singularly perturbed convection–diffusion problems arising in catalytic surface reactions. We compute the evolution of reactant concentration at different time levels in reactive flow systems. The interplay between convection and diffusion causes sharp variations in reactant concentration, leading to the formation of twin boundary layers. These layers are accurately resolved by combining uniform time discretization via the implicit Euler method with exponential B-spline collocation on a layer-adapted spatial mesh. The stability and convergence of the scheme are rigorously analysed, and numerical experiments demonstrate that it accurately captures reactant distributions even for very small perturbation parameters. The numerical results are compared with existing work.
