Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions

In Voller, Swenson and Paola [V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 (2010) 538-549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.     

Temperature distribution in a laminar plane wall jet

In the present note the temperature distribution in a laminar plane wall jet has been studied. It is found that a similarity solution of the energy equation exists. The resulting ordinary differential equation is reduced to a hypergeometric equation by a suitable transformation of the similarity variable and the solution, for arbitrary values of the Prandtl number, is obtained. It is concluded that the heat transfer at the wall at a given section and the product of volume and heat-flux through any cross-section of the boundary layer increase with the increase in the value of Prandtl number respectively.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

任意维数的强阻尼非线性波动方程（Ⅰ）—初边值问题

本文研究任意维数的强阻非线性波动方程ｕｔｔ－ａΔｕｔ－Δｕ＝ｆ（ｕ）具第一类齐边界条件的初值问题，设ｆ∈Ｃ＾１，ｆ＾１（ｕ）上方有界，且当ｎ≥４时存在常数Ａ，Ｂ和ｐ，使｜ｆ＾１（ｕ）｜≤Ａ｜ｕ｜＾ｐ＋Ｂ，其中０＜ｐ≤４／（ｎ－４）（ｎ＞４）：０＜ｐ＜∞（ｎ＝４），得到唯一整体强解，从而改进和推广了已知结果。     

On dynamic effects in an elastic half-space under “thermal impact” : PMM vol. 38, n 6, 1974, pp. 1105–1113

The general uncoupled dynamical problem of thermoelasticity for a half-space under the condition of a thermal impact with a finite rate of change in temperature on its boundary is solved by the method of principal (fundamental) functions within the framework of a generalized theory of heat conduction.An elastic steel half-space is analyzed as an illustration. The problem on thermal stresses originating in an elastic half-space due to thermal impact produced by a jump change in temperature on the boundary was first analyzed in [1]. Since the temperature change on the boundary occurs at a finite rate, it is generally impossible to realize the thermal impact considered in [1] physically. The dynamic effects in an elastic half-space under a thermal impact with finite rate of change in the temperature on the boundary have been studied in [2]. For high rates of change of the heat flux we obtain a generalized wave equation of heat conduction [3] taking into account the finite velocity of heat propagation. Hence, the solution of the ordinary parabolic heat conduction equation used in [1, 2] does not correspond to the true temperature field. The problems of [1, 2] have been examined in [4, 5], respectively, within the framework of a generalized theory of heat conduction.     

Similarity solutions to the power-law generalized Newtonian fluid

The authors of this paper study a second-order nonlinear parabolic equation with a background describing the motion of the power-law generalized Newtonian fluid. The existence of similarity solutions is obtained and the asymptotic behavior of the solution is studied.     

Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux

In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to obtain the analytical solution for a nonlinear ordinary deferential equation that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. Nonlinear ODE which is obtained by similarity solution has been solved through homotopy analysis method (HAM). The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The obtained analytical solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.     

A note on separation of variables solutions of generalized nonlinear diffusion equations

In this letter we show how more cases of the generalized nonlinear diffusion equation that contain separable solutions to those found by Zhang et al. (2002, 2003) via the generalized conditional symmetries method, can be found. Importantly we demonstrate with an example that can be used to describe water flow in unsaturated soil, and which can provide new insights when plant-root uptake is affected by water speed.     

From the classical to the generalized von Kármán and Marguerre–von Kármán equations

In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre–von Kármán shallow shells.

First, we briefly review the “classical” von Kármán and Marguerre–von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations.

In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data.

Following recent joint works, we then reduce these more general equations to a single “cubic” operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^?1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.     

Cauchy Problem for a Generalized Nonlinear Dispersive Equation

The solvability of global smooth solution for the Cauchy problem of a generalized nonlinear dispersive equation is studied by using the continuation method. In addition, the convergences of solution for this problem are also discussed.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

Rational scaled generalized Laguerre function collocation method for solving the Blasius equation

In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate.     

An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM

In this article, the approximate solution of nonlinear heat diffusion and heat transfer equation are developed via homotopy analysis method (HAM). This method is a strong and easy‐to‐use analytic tool for investigating nonlinear problems, which does not need small parameters. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ?, we can obtain reasonable solutions for large modulus. In this study, we compare HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature‐dependent thermal conductivity and the second one is the two‐ and three‐dimensional unsteady diffusion problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A convergent three‐level finite difference scheme for solving a dual‐phase‐lagging heat transport equation in spherical coordinates

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three‐level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60–71, 2004.     

Existence of Solutions to Time-Dependent Nonlinear Diffusion Equations via Convex Optimization

This paper aims at providing new existence results for time-dependent nonlinear diffusion equations by following a variational principle. More specifically, the nonlinear equation is reduced to a convex optimization problem via the Lagrange–Fenchel duality relations. We prove that, in the case when the potential related to the diffusivity function is continuous and has a polynomial growth with respect to the solution, the optimization problem is equivalent with the original diffusion equation. In the situation when the potential is singular, the minimization problem has a solution which can be viewed as a generalized solution to the diffusion equation. In this case, it is proved, however, that the null minimizer in the optimization problem in which the state boundedness is considered in addition is the weak solution to the original diffusion problem. This technique allows one to prove the existence in the cases when standard methods do not apply. The physical interpretation of the second case is intimately related to a flow in which two phases separated by a free boundary evolve in time, and has an immediate application to fluid filtration in porous media.     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

Integrability properties of a generalized reduction of the KdV6 equation

We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz-Ramani-Segur algorithm. In addition we give a Bäcklund transformation to a related equation, giving also an auto-Bäcklund transformation for this last. Our results mirror those for the Korteweg-de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Bäcklund transformation to the second Painlevé equation, this last having an auto-Bäcklund transformation.     

Polynomial decay and control of a 1−d hyperbolic-parabolic coupled system

In this paper we consider a linearized model for fluid-structure interaction in one space dimension. The domain where the system evolves consists in two parts in which the wave and heat equations evolve, respectively, with transmission conditions at the interface. First of all we develop a careful spectral asymptotic analysis on high frequencies for the underlying semigroup. It is shown that the semigroup governed by the system can be split into a parabolic and a hyperbolic projection. The dissipative mechanism of the system in the domain where the heat equation holds produces a slow decay of the hyperbolic component of solutions. According to this analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Next, we discuss the problem of null-controllability of the system when the control acts on the boundary of the domain where the heat equation holds. The key observability inequality of the dual system with observation on the heat component is derived though a new Ingham-type inequality, which in turn, thanks to our spectral analysis, is a consequence of a known observability inequality of the same system but with observation on the wave component.