Jordan Hristov

This paper presents a numerical method for solving transient one-dimensional heat conduction problems in multilayer slabs with temperature-dependent thermal conductivities. The governing nonlinear partial differential equations are formulated separately in each layer, allowing for distinct material properties. Perfect thermal contact at internal interfaces is enforced through continuity of temperature and heat flux, while general boundary conditions are imposed at the external boundaries, including prescribed temperature, specified heat flux, and convective exchange. A key feature of the proposed approach is to discretize the partial differential equations first in time using the implicit Euler method, thereby reducing the original problem to a sequence of nonlinear two-point boundary value problems with interface (transmission) conditions. A second-order finite difference scheme is employed for spatial discretization, and the resulting system is expressed in global form using a unified indexing strategy. The system is solved at each time step by Newton linearization, yielding a sparse Jacobian matrix that is tridiagonal in the interior and locally extended at the interfaces. Efficient banded solvers lead to O(N) cost per time step, where N is the number of spatial nodes. Numerical experiments confirm the expected accuracy, unconditional stability, and computational complexity of the method.

Progress has been made in developing sigmoids derived from the Mittag-Leffler function. Generally, the Mittag-Leffler functions of one and two parameters, as well as the Atangana-Baleanu formulation, are used to substitute the traditional exponential in the Verhulst model. Sigmoid formulations have been successfully demonstrated using numerous subcases of the Mittag-Leffler function. Maple and Mathematica computing have enabled us to successfully visualize the results and identify emerging computational problems.

Fractional calculus, the study of derivatives and integrals of arbitrary order

Thermal resistance and entropy generation rate have been analytically formulated as approximate series solutions in response to a semi-Gaussian flux pulse applied at the interface of a semi-infinite medium. Effective implementations of the Riemann-Liouville fractional integral of order 1/2 and a series approximation of the Dawson integral function yielded satisfactory, application-focused outcomes, avoiding the necessity for solutions through the entire domain.

A systematic review on logistic dynamics models and related sigmoidal functions has been developed, bringing together local and non-local interpretations. Special efforts were applied to show how non-locality can be implemented in logistic models, applying the Volterra approach and the causality concepts. An analysis of solutions of fractionalized logistic models is provided. Special attention is paid to logistic maps, both standard and fractional.

This note aims for a non-local extension of the Johnson–Mehl–Avrami–Kolmogorov (JMAK) kinetic equation, describing solid phase transformation through the implementation of the time-fractional Caputo derivative and Mittag-Leffler function instead of the exponential Avrami kinetics. These are preliminary results that include tests on some published data and a clarification of the concept.

Approximate analytical solutions to doubly degenerate reaction-diffusion models pertinent to population dynamics and chemical kinetics have been developed. The double integral-balance method applied to preliminary transformed models and by a direct integration approach has provided physically reasonable results. The model equation scaling has revealed the time and length scales, as well as the characteristic velocity of the process and the Fourier number as the controlling dimensionless group.

The double integral-balance approach and Barenblatt?s assumed profile have been used to create approximate solutions to the Zeldovich equation, both linear and degenerate. The evaluation of the controlling dimensionless groups and proper dimensional scaling have been the main focus of the solution developments and analyses.