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Analysis of the hysteresis of the enzyme-substrate due to enzyme flow calorimetry is presented based on the mathematical modeling of an immobilized enzyme using kinetic and diffusion parameters. The model is represented by ordinary differential equations containing a nonlinear term representing the substrate inhibition kinetics of the enzymatic reaction. In this paper, analytical expressions of substrate concentration for planar, cylindrical, and spherical particles at steady state condition are derived using a new analytical method. To confirm the validity and accuracy of the proposed method, the results will be compared with those obtained by the well-established homotopy perturbation method and the numerical results obtained by the highly-reputed fourth order Runge-Kutta method. Also, the results of the proposed method will be used to conduct a sensitivity analysis to understand the effect of parameters on concentration profiles.

Recently, conformable calculus has appeared in many abstract uses in mathematics and several practical applications in engineering and science. In addition, many methods and numerical algorithms have been adapted to it. In this paper, we will demonstrate, use, and construct the cubic B-spline algorithm to deal with conformable systems of differential boundary value problems concerning two points and two fractional parameters in both regular and singular types. Here, several linear and nonlinear examples will be presented, and a model for the Lane-Emden will be one of the applications presented. Indeed, we will show the complete construction of the used spline through the conformable derivative along with the convergence theory, and the error orders together with other results that we will present in detail in the form of tables and graphs using Mathematica software. Through the results we obtained, it became clear to us that the spline approach is effective and fast, and it requires little compulsive and mathematical burden in solving the problems presented. At the end of the article, we presented a summary that contains the most important findings, what we calculated, and some future suggestions.