Group method solution for solving nonlinear heat diffusion problems

The linear transformation group approach is developed to simulate heat diffusion problems in a media with the thermal conductivity and the heat capacity are nonlinear and obeyed a striking power law relation, subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equation with the boundary conditions reduces to an ordinary differential equation with appropriate corresponding conditions. The Runge–Kutta shooting method is used to solve the nonlinear ordinary differential equation. Different parametric studies are worked out and plotted to study the effect of heat transfer coefficient, density and radiation number on the surface temperature.     

Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux

The problem of heat and mass transfer in an unsteady free-convection flow over a continuous moving vertical sheet in an ambient fluid is investigated for constant heat flux using the group theoretical method. The nonlinear coupled partial differential equation governing the flow and the boundary conditions are transformed to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved numerically using the shooting method. The effect of Prandlt number on the velocity and temperature of the boundary-layer is plotted in curves. A comparison with previous work is presented.     

Analytic transient solutions of a cylindrical heat equation with a heat source

The method of separation of variables is applied in order to investigate the analytical solutions of a certain two-dimensional cylindrical heat equation. In the analysis presented here, the partial differential equation is directly transformed into ordinary differential equations. The closed-form transient temperature distributions and heat transfer rates are generalized for a linear combination of the products of Fourier-Bessel series of the exponential type. Relevant connections with some other closely-related recent works are also indicated.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source, where the flow is generated due to a linear stretching of the sheet and influenced by uniform magnetic field applied vertically. Here an analysis has been carried out to study the effect of magnetic field on the visco-elastic liquid flow and heat transfer over a stretching sheet with non-uniform heat source. The non-linear boundary layer equation for momentum is converted into ordinary differential equation by means of similarity transformation and is solved exactly. Heat transfer differential equation is also solved analytically. The effect of magnetic field on velocity, skin friction and temperature profiles are presented graphically and discussed.     

Similarity analysis of a nonlinear fin equation

A nonlinear fin equation in which the thermal conductivity is an arbitrary function of the temperature and the heat transfer coefficient is an arbitrary function of a spatial variable is considered. Scaling, translational and spiral group symmetries of the equations are determined. Classification of the functions for which these symmetries exist is performed. In general, no useful symmetries exist for arbitrary thermal conductivity and heat transfer coefficients. However, for some restricted forms of the functions, useful symmetries exist. A similarity transformation is used to reduce the partial differential equation to an ordinary differential equation as an example.     

Diffusion approximation for nonlinear evolutionary equations with large interaction and fast boundary fluctuation

Approximations are derived for both nonlinear heat equations and singularly perturbed nonlinear wave equations with highly oscillating random force on boundary and strong interaction. By a diffusion approximation method, if the interaction is large and the singular perturbation is small enough, the approximation of the nonlinear wave equation is an one dimensional stochastic ordinary differential equation with white noise from the boundary which is exactly the same as that of the nonlinear heat equation.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

Transmutations and irregular singular points

We consider transumtations for a class of problems in partial differential equations where the underlying equation, involving two assignable parameters, is an associated ordinary differential equation with an irregular singular point. An integral formula for the solution of this associated problem, valid for negative values of a timelike variable t, permits relating the solution of the problems in partial differential equations to be bounded or slow groth solutions of generalized heat problems. Applications of the formulas are made to Cauchy and boundary type problems.     

A diagonal splitting method for solving semidiscretized parabolic partial differential equations

In this work, a diagonal splitting idea is presented for solving linear systems of ordinary differential equations. The resulting methods are specially efficient for solving systems which have arisen from semidiscretization of parabolic partial differential equations (PDEs). Unconditional stability of methods for heat equation and advection–diffusion equation is shown in maximum norm. Generalization of the methods in higher dimensions is discussed. Some illustrative examples are presented to show efficiency of the new methods.     

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

Numerical studies of a nonlinear heat equation with square root reaction term

Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions

Refined integral heat balance is developed for Stefan problem with time-dependent temperature applied to exchange surface. The method is applied to phase change in the half-plane and ordinary differential equation is obtained for the solid/liquid interface. The results are compared to those obtained by heat balance integral, perturbation and numerical methods.     

多约束二阶非线性常微分方程拐点的数值求法

提出了一种求解多约束二阶非线性常微分方程拐点的数值解法,并以具有平行反应的化学放热系统为例,给出了一个具体算例.     

Internal Dynamics Stability of a Heat Conduction System Reduced via Orthogonal Collocation

This paper deals with the crucial problem of choice of the collocation points in the reduction of a one-dimensional heat transfer problem into an ordinary differential equation system. It is shown that, for its parabolic differential equation, the use of such an approach can either lead to systems with or without a NMP behaviour, which is not characteristic of the original behaviour of the initial system. A simple rule of choice, restrained to this class of systems, is given to avoid this phenomenon.     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.     

On the question of representation of ultrafilters in a product of measurable spaces

We consider nonlinear ordinary differential equations up to the sixth order that are associated with the heat equation. Each of them is subjected to the Painlevé analysis. For the fourth- and sixth-order equations we obtain a criterion for having the Painlevé property; for the fifth-order equation we formulate necessary conditions for passing the Painlevé test. We also present a fifth-order equation analogous to the Chazy-3 equation.     

Conditional invariance and exact solutions of the nonlinear equation

The conditional invariance of the nonlinear heat equation is studied. Conditionalinvariance operators are applied for reducing the original equation to ordinary differential equations, and also for finding its exact solutions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1370–1376, October, 1990.     

Crystallization waves in thin amorphous layers on heat conducting substrates

A crystallization process in thin films is considered, where, driven by the release of the latent heat of fusion, the transformation from an amorphous state to the crystalline state takes place in a progressing wave of invariant shape. The crystallization rate is determined by a rate equation. The influence of the heat loss due to heat conduction into the substrate is taken into account. The resulting system of an ordinary differential equation and an integro-differential equation is solved numerically using a collocation method. The propagation speed of the wave in dependence on a non-dimensional heat loss parameter is determined. It turns out that the existence of a self-sustaining crystallization wave requires the heat loss parameter to be smaller than a certain critical value. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.