Ebraheem O. Alzahrani

A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

We consider the ergodic (or additive eigenvalue) problem for the Neumann type boundary value problem for Hamilton-Jacobi equations and the corresponding discounted problems. When denoting by u the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family {u}λ>0 to a solution of the ergodic problem, as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting byproduct, we introduce Mather measures associated with Hamilton-Jacobi equations with the Neumann type boundary condtitions. These results are variants of the main results in the paper “Convergence of the solutions of the discounted equations” by A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, where they study the same convergence problem on smooth compact manifolds without boundary. Acknowledgments. The third author is grateful to Dr. Andrea Davini for sending him the preprint [6] at a timely occasion. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. 2-130-1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support. The work of the third author was supported in part by KAKENHI #21340032, #21224001, #23340028 and #23244015, JSPS.

Quiescence as an explanation of Gompertzian tumor growth revisited

Gompertzs empirical equation remains the most popular one in describing cancer cell population growth in a wide spectrum of bio-medical situations due to its good fit to data and simplicity. Many efforts were documented in the literature aimed at understanding the mechanisms that may support Gompertzs elegant model equation. One of the most convincing efforts was carried out by Gyllenberg and Webb. They divide the cancer cell population into the proliferative cells and the quiescent cells. In their two dimensional model, the dead cells are assumed to be removed from the tumor instantly. In this paper, we modify their model by keeping track of the dead cells remaining in the tumor. We perform mathematical and computational studies on this three dimensional model and compare the model dynamics to that of the model of Gyllenberg and Webb. Our mathematical findings suggest that if an avascular tumor grows according to our three-compartment model, then as the death rate of quiescent cells decreases to zero, the percentage of proliferative cells also approaches to zero. Moreover, a slow dying quiescent population will increase the size of the tumor. On the other hand, while the tumor size does not depend on the dead cell removal rate, its early and intermediate growth stages are very sensitive to it.

Correction: New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

The affiliation for the fifth author is incorrect. Abdulrahim A. Alzahrani is not affiliated with #3, but with: Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia.

Generalized magneto-thermoviscoelasticity in a perfectly conducting thermodiffusive medium with a spherical cavity

In this work, the effects of viscosity and diffusion on thermoelastic interactions in an infinite medium with a spherical cavity are studied. The formulation is applied to the generalized thermoelasticity based on the theory of generalized thermoelastic diffusion with one relaxation time. The surface of the spherical cavity is taken to be traction free and subjected to both heating and external constant magnetic field. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain obtained numerically using a method based on Fourier expansion techniques. The temperature, displacement, stress, concentration as well as the chemical potential are obtained and represented graphically. Comparisons are made within the theory in the presence and absence of viscosity and diffusion.

Dynamical analysis of cigarette smoking model with a saturated incidence rate

In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.

GLOBAL EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR A GENERALIZED QUASILINEAR PARABOLIC EQUATION WITH APPLICATION TO A GLIOBLASTOMA GROWTH MODEL

This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, Lp-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.