A Discrete Model for a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation: Numerical Efficiency and Energy Dissipation | Chapter 9 | Research Updates in Mathematics and Computer Science Vol. 5

The investigation of mathematical models with fractional derivatives has become a fruitful area of research in recent decades. For the first time, a new dissipationpreserving scheme is proposed and analyzed to solve a Caputo–Riesz time-spacefractional multidimensional nonlinear wave equation with generalized potential. Classically, Caputo-like fractional operators have found potential applications in studying systems with memory effects. In the present work, we consider a fractional extension of the nonlinear Klein–Gordon equation with damping, which involves Caputo temporal derivatives and Riesz spatial derivatives. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hypercube. We introduce an energy-type function and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employed a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carried out a number of numerical simulations to verify the validity of our theoretical results.

Author(s) Details:

Jorge E. Macías-Díaz,
Department of Mathematics and Didactics of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia and Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20100, Mexico.

Tassos Bountis,
Department of Mathematics, University of Patras, 26500 Patras, Greece and Center for Integrable Systems, P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia.

Please see the link here: https://stm.bookpi.org/RUMCS-V5/article/view/14345

Numerical Analysis of the Effect of Dynamic Viscosity Associated with a Continuously Moving Heated Horizontal Plate | Chapter 11 | Novel Perspectives of Engineering Research Vol. 6

 The impact of changing dynamic viscosity in combination with viscous dissipation on convective heat transfer in a fluid-filled rectangular cavity is studied using numerical simulations. At the top of the chamber, a heated horizontal plate is kept at temperature Tw and moves at a constant velocity Uw. Additionally, cold isothermal walls limit it on the left vertical and lower horizontal sides, whereas an adiabatic vertical wall limits it on the right. In the cavity, a quenching media was injected. The governing equations were transformed to non-dimensional form and solved using the C++ computer language's finite difference scheme numerical approach. The experiment was carried out on a variety of fluids, including oil with a Prandtl number of 10, air with a Prandtl number of 0.7, and liquid metal with a Prandtl number of 0.01, for a range of viscosities, parameters in the range 5*10-1 to 9*10-1, heat capacity in the range 1Cp10, fixed Eckert numbers, Ec = 1.0, and mixed convection parameter, Gr/Re2 = 1.0, Profiles are used to show the numerical results that have been generated. When the specific heat capacity and Prandtl number are greater than unity and the viscous dissipation is fixed, the results show that dynamic viscosity has a significant impact on velocity and temperature profiles. Furthermore, for Prandtl values greater than unity, raising the dynamic viscosity reduces the fluid's maximum velocity significantly. The findings will aid in the design of heat exchanger equipment and serve as a benchmark for industries such as wire drawing, continuous rolling, and glass fibre manufacture.

Author(S) Details

M. A. Waheed
Department of Mechanical Engineering, College of Engineering, University of Agriculture, P. M. B. 2240, Abeokuta, Nigeria.

E. O. Sangotayo
Department of Mechanical Engineering, Ladoke Akintola University of Technology, P. M. B. 4000, Ogbomoso, Nigeria.

View Book:- https://stm.bookpi.org/NPER-V6/article/view/5505