Perturbation analysis of the heat transfer in porous media with small thermal conductivity

An approximate analytical solution for the one-dimensional problem of heat transfer between an inert gas and a porous semi-infinite medium is presented. Perturbation methods based on Laplace transforms have been applied using the solid thermal conductivity as small parameter. The leading order approximation is the solution of Nusselt (or Schumann) problem. Such solution is corrected by means of an outer approximation. The boundary condition at the origin has been taking into account using an inner approximation for a boundary layer. The gas temperature presents a discontinuous front (due to the incompatibility between initial and boundary conditions) which propagates at constant velocity. The solid temperature at the front has been smoothed out using an internal layer asymptotic approximation. The good accuracy of the resulting asymptotic expansion shows its usefulness in several engineering problems such as heat transfer in porous media, in exhausted chemical reactions, mass transfer in packed beds, or in the analysis of capillary electrochromatography techniques.     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions

The work presents a novel coupling of the Laplace Transform and the compact fourth‐order finite‐difference discretization scheme for the efficient and accurate solution of linear time‐fractional nonhomogeneous diffusion equations subject to both Dirichlet and Neumann boundary conditions. A translational transformation of the dependent variable ensures the Caputo derivative is aligned with the Riemann‐Louiville fractional derivative. The resulting scheme is computationally efficient and shown to be uniquely solvable in all cases, accurate and convergent to in the spatial domain. The convergence rates in the temporal domain are contour dependent but exhibit geometric convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1184–1199, 2016     

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

In this paper, we obtain approximate‐analytical solutions of a cancer chemotherapy effect model involving fractional derivatives with exponential kernel and with general Mittag‐Leffler function. Laplace homotopy perturbation method and the modified homotopy analysis transform method were applied. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on homotopy polynomial. The cancer chemotherapy effect equations are solved numerically and analytically using the aforesaid methods. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new fractional‐order derivatives with exponential decay law and with general Mittag‐Leffler law.     

Modelling non-Fourier heat conduction with periodic thermal oscillation using the finite integral transform

The non-Fourier heat conduction in a finite medium subjected to a periodic heat flux is modelled using the finite integral transform technique and an analytic solution is obtained. An analogy between thermal oscillation and oscillation of mechanical and electrical systems is drawn. A transition criterion from the non-Fourier heat conduction formulation to the Fourier formulation is obtained and a simple analytical expression of the phase and amplitude of thermal oscillation is derived.     

Multicriteria optimization of a composite cylindrical shell subjected to longitudinal thermal stresses and buckling under the action of torque

The multicriteria optimization of the structure and geometry of a multilayer cylindrical shell under the action of external torque and longitudinal thermal stresses is considered. From known monolayer properties of the composite and given values of variable structural and geometric parameters, the thermoelastic properties of the anisotropic layered composite are determined. The criteria to be optimized — the critical external torque and thermal stresses — depend on two variable parameters and temperature. In the space of optimization criteria, the domain of allowable solutions and the Pareto-optimal subdomain are found. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 223–230, March–April, 2009.     

Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem

In this paper, the differential transform is employed to discuss the behaviors of nonlinear heat conduction problem. A hybrid method of differential transform and finite difference approach is proposed to solve the transient responses of a nonlinear heat conduction problem. Different parameters of the equation and boundary conditions are considered to verify the feasibility of the proposed method to such problems. Simulation results are illustrated and discussed in comparison with the linear case. The results show that the hybrid method can achieve good results for such problems.     

Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity,non-uniform heat source and radiation

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a non-Newtonian viscoelastic fluid over a flat sheet with a linear velocity in the presence of thermal radiation and non-uniform heat source. The thermal conductivity is assumed to vary as a linear function of temperature. The basic equations governing the flow and heat transfer are in the form of partial differential equations, the same have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformation. The transformed equations are solved analytically by regular perturbation method. Numerical solution of the problem is also obtained by the efficient shooting method, which agrees well with the analytical solution. The effects of various physical parameters such as viscoelastic parameter, Chandrasekhar number, Prandtl number, variable thermal conductivity parameter, Eckert number, thermal radiation parameter and non-uniform heat source/sink parameters which determine the temperature profiles are shown in several plots and the heat transfer coefficient is tabulated for a range of values of said parameters. Some important findings reported in this work reveals that combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region.     

A problem with directional derivative in the theory of galvanomagnetic effects

We construct an asymptotics of the solution the Laplace equation in a “long” rectangle with the directional derivative given on its “long sides” and Dirichlet data on its “short sides.” By using the asymptotics, we calculate one of the integral characteristics, namely, the magnetoresistance. We obtain new formulas for the low-magnetic field magnetoresistance. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 520–532, April, 1999.     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

General exact solution of the fin problem with the power law temperature‐dependent thermal conductivity

In this paper, the general exact implicit solution of the second‐order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo‐geometric fin parameter is obtained for any value of the mode of heat transfer n and the constant β. Copyright © 2015 John Wiley & Sons, Ltd.     

Exploring the interplay between memory effects and vesicle dynamics: A five-dimensional analysis using rigid sphere models and mapping techniques

The memory effect in vesicle dynamics refers to the persistence of shape changes in lipid vesicles, a type of lipid bilayer membrane that mimics some features of real cells, particularly red blood cells (RBCs). To study this effect, a fractional rigid sphere model in five dimensions has been investigated, which maps the dynamics of a vesicle using the Caputo operator. This model provides new insights into the mechanisms behind the memory effect as well as the emergence of two other motions, namely, tank-treading with underdamped and tank-treading with overdamped vesicle oscillations, in addition to the tank-treading mode where the vesicle rotates about its axis while maintaining a constant shape. Overall, this work represents a significant advance in our understanding of vesicle dynamics and has important implications for understanding the behavior of RBCs.     

Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity

Steady mixed convection micropolar fluid flow towards stagnation point formed on horizontal linearly stretchable melting surface is studied. The vortex viscosity of micropolar fluid along a melting surface is proposed as a constant function of temperature while dynamic viscosity and thermal conductivity are temperature dependent due to the influence of internal heat source on the fluid. Similarity transformations were used to convert the governing equation into non-linear ODE and solved numerically. A parametric study is conducted. An analysis of the results obtained shows that the flow-field is influenced appreciably by heat source, melting, velocity ratio, variable viscosity and thermal conductivity.     

A system of exponential functions with shift and the Kostyuchenko problem

We introduce a system of exponential functions with shift, study its basis properties in L ₂, and examine its connections with the Kostyuchenko problem.     

Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins

In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.     

A Hausdorff‐like moment problem and the inversion of the Laplace transform

We consider the problem of finding u ∈ L ²(I ), I = (0, 1), satisfying ∫_I u (x )x dx = μ _k , where k = 0, 1, 2, …, (α _k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ _kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity

(P)$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+e^u,\hfill & x\in {\mathbf{R}}^N,t>0,\hfill \\ u(x,0)=u_0(x),\hfill & x\in {\mathbf{R}}^N,\hfill \end{array}$

where $u_0$ is a continuous initial function. In this paper, we consider the case where $u_0$ decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for $u_0$ is related to the decay rate of forward self-similar solutions of ${?}_{t}u=\mathrm{\Delta }u+e^u$.