One-dimensional Heat Equation Research Articles

Inverse Problem with Free Boundary for Heat Equation

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a one-dimensional heat equation with unknown time-dependent leading coefficient in the case where a part of the boundary of the domain is unknown.

Three-level techniques for one-dimensional parabolic equation with nonlinear initial condition

In this paper a boundary value problem for one-dimensional heat equation is considered under the constraint of a nonlocal initial condition. In place of the classical specification of initial data, we impose the nonlocal initial condition. Efficient higher-order algorithms are developed for solving this parabolic partial differential equation with nonstandard initial condition. Several three-level finite difference schemes are presented. These schemes are based on the modification of the centred-time centred-space explicit formula, the three-level (1,3,1) explicit technique, the sixth-order (1,3,3) explicit method and the Dufort–Frankel finite difference scheme. Numerical computations combined with a simple iteration procedure of some examples are given to demonstrate the efficiency and accuracy of the new algorithms.

On maximum norm contractivity of second order damped single step methods

In this paper we consider A(θ)-stable finite difference methods for numerical solutions of dissipative partial differential equations of parabolic type. Combining two rational approximation methods with different orders of accuracy, where the lower order method is applied n 0 times (n 0 fixed) at each time step, we prove the existence of a second order method which is contractive for all time steps. Moreover, we shed light on the conditions on the lower order method which are sufficient (and sometimes necessary) to obtain the optimal order of accuracy. For the one-dimensional heat equation we construct a family of numerical methods which are contractive in the maximum norm for all values of the discretization parameters. We also present numerical examples to illustrate our results.

Asymptotic mass distribution speed for the one-dimensional heat equation with constant drift and stationary potential

We study the long-time behavior of the solution u( t, x) of a Cauchy problem for the one-dimensional heat equation with constant drift and random potential in the quenched setting: u t= 1 2 u xx+hu x+ξu . The initial function is compactly supported. For bounded stationary ergodic potential ξ, we show that u is asymptotically ( t→∞) concentrated in a ball of radius o( t) and center v h t which is independent of the realization of the random potential. There is a critical drift value h cr where we observe a change from sublinear ( v h =0) to linear (0< v h ⩽ h) mass propagation.

Electron and phonon thermal waves in semiconductors: the effect of carrier diffusion and recombination on the photoacoustic signal

The electron and phonon temperature distributions are calculated in bipolar semiconductors by taking into account generation and heating of carriers on the surface due to the incident modulated laser beam as well as the finite carrier diffusion and recombination in the solid. We solve the coupled one-dimensional heat and carrier diffusion equations in the linear approximation using appropriate boundary conditions according to the photoacoustic experimental conditions. In addition, special emphasis is laid on the dependence of the electron system heat flux on the electron and phonon temperature. We also consider the influence of both temperatures on the recombination of the electron–hole pairs. The dependence of the amplitude of the electron and phonon temperatures on the modulation frequency are analyzed for different values of characteristic parameters of the problem: electron–phonon energy exchange and carrier thermal generation rate.

Invariant Solutions and Conservation Laws of the Black-Scholes Equation

As the Black-Scholes equation can be transformed into the one-dimensional linear heat equation via two sets of transformations, an optimal system of one-dimensional subalgebras for the one-dimensional heat equation is exploited to obtain two classes of optimal systems of one-dimensional subalgebras for the well-known Black-Scholes equation of the mathematics of finance. Two methods for the derivation of the two classes of optimal systems of group-invariant solutions for this model are available. We present the simpler approach

Saulyev'S Techniques For Solving A Parabolic Equation With A Non Linear Boundary Specification

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \\mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\\hskip1pty, t) = \\vint u(x, y, t)\\,\\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \\mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.

Brownian Trajectory Is a Regular Lateral Boundary for the Heat Equation

The one-dimensional heat equation in the domain x>wt, t>0, is considered. Here wt is a trajectory of Brownian motion. For almost any trajectory, it is proved that if the boundary data are continuous, then the solution is continuous in the closure of the domain. The proof is based on Davis's law of square root for Brownian motion or on its weaker version, which is obtained by using the theory of stochastic partial differential equations.

A Method of Solution for the One-Dimensional Heat Equation Subject to Nonlocal Conditions

The problem of solving the one-dimensional heat equation ∂φ/∂t - ∂2φ/∂x2 = f(x, t) subject to given initial and nonlocal conditions is considered. It is solved in the Laplace transform domain by taking the Laplace transform of the unknown function φ with respect to time t. The physical solution is recovered with the help of a numerical technique for inverting the Laplace transform.

Grid approximation of a singularly perturbed one-dimensional heat equation on an unbounded domain excluding a rectangle with sides noncollinear to the axes

We consider discrete approximations of the first initial boundary value problem for a singularly perturbed heat equation on a domain outside the 'inclusion', i.e. on the domain = R ×[ t 0 , T ] excluding a rectangle whose sides are noncollinear to the x-, t -axes. The solution of the problem has boundary moving layers, regular by their behaviour, in a neighbourhood of the outside part of the boundary and parabolic transition layers in a neighbourhood of characteristics tangent to the inclusion. In addition, the solution of the problem has a discontinuity-type singularity occurring at the instant the inclusion appears. Using the fitted operator method (in a neighbourhood of the layer appearance, where the singularity develops) and the refining mesh technique (in a neighbourhood of the layers), we construct a special difference scheme convergent ε-uniformly almost everywhere in the grid domain except for the set of the solution discontinuity. When constructing the scheme we use the new coordinates in which the location of boundary layers becomes stationary.

Education

This issue's Education section exhibits two quite different applications of mathematical modeling---different in the types of real-life problems that are examined and in the types of mathematical models that are used to study them---that both make nice instructional modules. The first paper uses partial differential equations to model the behavior of a practical device that is important to our health: a "pulsed amperometric ion sensor." This device is the possible successor to related devices that are used extensively in clinical laboratories, and in hospitals at patients' bedsides, to measure blood electrolytes. "How Do Pulsed Amperometric Ion Sensors Work? A Simple PDE Model," by Eric Bakker and A. J. Meir, is a collaboration between a chemist and a mathematician at Auburn University. Proceeding from a description of the operation of the device, and the relevant physics and chemistry, the authors pose a number of questions that a mathematical model can help to answer. The answers to these questions help provide quantitative information on optimal operating conditions and parameters for the device. The mathematical model constructed to answer these questions is the one-dimensional heat equation posed on a semi-infinite domain. After deriving the model, the paper presents solutions that exhibit a number of important results. The paper is accessible to advanced undergraduates and graduate students taking a first course on partial differential equations. The other paper in this issue involves a very different set of practical pursuits---bicycle riding and suntans! Indeed the paper by Geert Jan Olsder is titled "Bicycle Routing for Maximum Suntan," a topic that surely is of particular interest in the author's home country (The Netherlands), which is not noted for an abundance of sunshine. In the paper, the author presents a sequence of increasingly difficult but accessible optimal control problems related to maximizing exposure to the sun while riding a bicycle. The problems are chosen to have tractable solutions that are easy to interpret. The solutions also illustrate both the fundamental ideas and the subtleties associated with control theory. The paper makes a nice module that can be used to accompany the teaching of basic principles in control theory. I feel compelled to add that in my part of the world (Colorado), where sunshine is overly abundant as well as very intense, faculty may need to ask students to develop a corresponding strategy to achieve bicycle routing for minimal suntan!

Wavelet and error estimation of surface heat flux

We consider the determination of the surface heat flux of a body from a measured temperature history at a fixed location inside the body. Mathematically, it is formulated as a problem for the one-dimensional heat equation in a quarter plane with data given along the line x=1, and the gradient of the solution is wanted for 0< x<1. This problem is severely ill-posed and there are few theoretic results. The results available in the literatures are mainly devoted to the case of determining surface temperature. The aim of this paper is to indicate a possibility of a new approach for solving this problem based on an application of wavelet basis decomposition of measured data. The error estimate with Hölder type is given and the convergence of regularization approximation solution at zero is also obtained which is a difficult problem left over by some authors.

Modelization of resistive heating of carbon nanotubes during field emission

In this paper we simulate the resistive heating of carbon nanotubes (CNT's) during field emission (FE) using the one-dimensional heat equation including radiation and conductive losses. The simulations are in relatively good agreement with our recent experiments in which we measured the heating of individual CNT's to as high as 2000 K during FE for currents ${I}_{\mathrm{FE}}$ of $\ensuremath{\approx}2 \ensuremath{\mu}\mathrm{A}.$ For low temperatures where radiation is negligible the simulations reduce to an analytic solution that provides a universal guide to estimating current-induced heating in CNT's. The effects of this heating are included in the calculations of ${I}_{\mathrm{FE}}$ as a function of voltage.

ON THE ASYMPTOTIC SPATIAL BEHAVIOR IN LINEAR THERMOELASTICITY OF MATERIALS WITH VOIDS

The comparison principle involving solutions of the one-dimensional heat equation is used to investigate the spatial decay of solutions to initial-boundary value problems for the linear theory of thermoelasticity with voids. A spatial decay estimate is established for an appropriate volumetric measure of the solution in an unbounded body. This proves that at large distances from the support of the external given data the decay of the mechanical and thermal effects is controlled only by the thermal characteristic coefficients of the material.

PROPER ORTHOGONAL DECOMPOSITION BASED OPTIMAL CONTROL DESIGN OF HEAT EQUATION WITH DISCRETE ACTUATORS USING NEURAL NETWORKS

A new method is presented for the optimal control design of one-dimensional heat equation with the actuators being concentrated at discrete points in the spatial domain. This systematic methodology incorporates the advanced concept of proper orthogonal decomposition for the model reduction of distributed parameter systems. After deigning a set of problem oriented basis functions an analogous optimal control problem in the lumped domain is formulated. The optimal control problem is then solved in the time domain, in a state feedback sense, following the philosophy of adaptive critic neural networks. The control solution is then mapped back to the spatial domain using the same basis functions. Numerical simulation results are presented for a linear and a nonlinear one-dimensional heat equation problem.

Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density

We consider the 1−d heat equation with rapidly oscillating periodic density in a bounded interval with Dirichlet boundary conditions. When the period tends to zero and the density weakly converges to its average we prove that the boundary controls converge to a control of the limit, constant coefficient heat equation when the density is C2.The proof is based on a control strategy in three steps in which: we first control the low frequencies of the system, we then let the system to evolve freeely and, finally, we control to zero the whole solution. We use the theory of real exponentials to analyze the low frequencies and Carleman inequalities to control the whole solution.The result is in constrast with the divergent behavior of the null controls for the wave equation with rapidly oscillating coefficients.

Sharpening the estimate of the stability constant in the maximum-norm of the Crank–Nicolson scheme for the one-dimensional heat equation

This paper is concerned with the stability constant C ∞ in the maximum-norm of the Crank–Nicolson scheme applied to the one-dimensional heat equation. A well known result due to S.J. Serdyukova is that C ∞<23. In the present paper, by using a sharp resolvent estimate for the discrete Laplacian together with the Cauchy formula, it is shown that 3⩽ C ∞<4.325. This bound also holds when the heat equation is considered on a bounded interval along with Dirichlet or Neumann boundary conditions.

On the monotonicity conservation in numerical solutions of the heat equation

It is important to choose such numerical methods in practice that mirror the characteristic properties of the described process beyond the stability and convergence. The investigated qualitative property in this paper is the conservation of the monotonicity in space of the initial heat distribution. We prove some statements about the monotonicity conservation and total monotonicity of one-step vector-iterations. Then, applying these results, we consider the numerical solutions of the one-dimensional heat equation. Our main theorem formulates the necessary and sufficient condition of the uniform monotonicity conservation. The sharpness of the conditions is demonstrated by numerical examples.

Singular invariant equation for the (1+1) Fokker–Planck equation

Semi-invariants for the linear parabolic equations with two independent variables (time variable t and space variable x) and one dependent variable u are derived under the transformation of the independent variables, by using the infinitesimal method. We also obtain the joint invariant equation for the above-mentioned equation under equivalence transformation. In fact, we prove a necessary and sufficient condition for a (1 + 1) parabolic equation to be reducible via a local equivalence transformation to the one-dimensional classical heat equation. This result provides practical criteria for reduction. Finally, examples of (1 + 1) Fokker–Planck equations from applications are given to verify the results obtained.

On the convergence of finite difference schemes for the heat equation with concentrated capacity

We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is $c\left( x\right) =1+K\delta \left( x-\xi \right). K = \const > 0$ represents the magnitude of the heat capacity concentrated at the point $x=\xi$ . An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained.