Two-Dimensional and Three-Dimensional Thermal Structures in a Medium with Nonlinear Thermal Conductivity

The article considers self-similar solutions of the nonlinear heat equation with a three-dimensional source that evolve in a blow-up setting. The self-similar problem is a boundary-value problem for a nonlinear equation of elliptical type that has a nonunique solution. We investigate the eigenfunction spectrum of the self-similar problem in two- and three-dimensional space. The problem is solved on a grid by Newton’s iteration method. The implementation of Newton’s method requires analysis of a linearized equation and construction of initial approximations. The eigenfunctions are continued in a parameter. Structures of various symmetry are obtained. New types of multidimensional structures are observed: these are multiply connected three-dimensional heat localization regions.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 84–111, 2004.     

Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system

We consider a structural acoustic wave equation with nonlinear acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with boundary conditions on the interface. We prove wellposedness in the Hadamard sense for strong and weak solutions. The main tool used in the proof is the theory of nonlinear semigroups. We present the system of partial differential equations as a suitable Cauchy problem . Though the operator A is not maximally dissipative we are able to show that it is a translate of a maximally dissipative operator. The obtained semigroup solution is shown to satisfy a suitable variational equality, thus giving weak solutions to the system of PDEs. The results obtained (i) dispel the notion that the model does not generate semigroup solutions, (ii) provide treatment of nonlinear models, and (iii) provide existence of a correct state space which is invariant under the flow-thus showing that physical model under consideration is a dynamical system. The latter is obtained by eliminating compatibility conditions which have been assumed in previous work (on the linear case).     

Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in a two-dimensional heat conduction equation using a final temperature measurement. Problems of this type have important applications in several fields of applied science. Under certain assumptions, this problem can be transformed into a one-dimensional problem where the heat source only depends on the variable r  . However, being different from other one-dimensional inverse heat source problems, there exists singularity on the coefficient of our model, which may make the analysis more difficult, regardless of theoretical or numerical. The inverse problem is reduced to an operator equation of the first kind and the corresponding adjoint operator is deuced. For the two dimensional case, i.e., f=f(r,θ)$f=f(r\text{,}\theta )$, theoretical analysis can be done by similar derivation. Based on the landweber regularization framework, an iterative algorithm is proposed to obtain the numerical solution. Some typical numerical examples are presented to show the validity of the inversion method.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:

     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

Blow-up phenomena for <Emphasis Type="Italic">p</Emphasis>-adic semilinear heat equations

The problem of existence of solutions to p-adic semilinear heat equations with particular nonlinear terms has already been studied in the literature but the occurrence of blow-up phenomena has not been considered yet. We initiate the study of finite time blow-up for solutions of this kind of p-adic semilinear equations, proving that this phenomenon always arises under appropriate assumptions in the case when the exponent of nonlinearity times the dimension is strictly less than the order of the operator.     

Inapproximability results for no-wait job shop scheduling

We investigate the approximability of the no-wait job shop scheduling problem under the makespan criterion. We show that this problem is -hard (i) for the case of two machines with at most five operations per job, and (ii) for the case of three machines with at most three operations per job. Hence, these problems do not possess a polynomial time approximation scheme, unless .     

A note on Nasr's and Wong's papers

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form

     

Initial blow-up rates and universal bounds for nonlinear heat equations

We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation

     

A one-dimensional nonlinear heat equation with a singular term

In this paper we are concerned with the Dirichlet problem for the one-dimensional nonlinear heat equation with a singular term:

     

Extended and unscented filtering algorithms using one-step randomly delayed observations

In this paper, the least squares filtering problem is investigated for a class of nonlinear discrete-time stochastic systems using observations with stochastic delays contaminated by additive white noise. The delay is considered to be random and modelled by a binary white noise with values of zero or one; these values indicate that the measurement arrives on time or that it is delayed by one sampling time. Using two different approximations of the first and second-order statistics of a nonlinear transformation of a random vector, we propose two filtering algorithms; the first is based on linear approximations of the system equations and the second on approximations using the scaled unscented transformation. These algorithms generalize the extended and unscented Kalman filters to the case in which the arrival of measurements can be one-step delayed and, hence, the measurement available to estimate the state may not be up-to-date. The accuracy of the different approximations is also analyzed and the performance of the proposed algorithms is compared in a numerical simulation example.     

Numerical solution of the nonlinear heat equation in heterogeneous media 1

The nonlinear multi-dimensional heat transfer problem in discontinuously heterogeneous. but piecewise homogeneous media is treated numerically by using the enthalpy lormulation, certain regularization of the contact conditions between the homogeneous subdomains (like in [1,2]), the fully implicit time discretization and linear finite elements in space with Imear interpolation and numerical integration. A convergence is proved by using a technique that does not check the time derivative of temperature. Phase transitions with a positive latent heat (i.e. Stefan problems) are covered, as well. Besides, the problem need not be of a strongly parabolic type. Some numerical experience with the nonlinear Gauss-Seidel algorithm to solve the created nonlinear algebraic systems is presented, too.     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.     

Group solution of a time dependent chemical convective process

The time dependent progress of a chemical reaction over a flat vertical plate is here considered. The problem is solved using the two parameter group method which reduces the number of independent by one and leads to a set of nonlinear ordinary differential equation. The behavior of the process is numerically investigated for the chemical reaction order and different Schmidt numbers. As the problem shows a singularity at n = 1, the nonlinear system of ordinary differential equation resulting from the transformation of the problem are analytically solved through the perturbation method. The velocity and concentration of chemicals based on the analytical and numerical solutions are presented, compared and discussed.     

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubins Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.     

Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α?5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2<p?10. And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.     

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

Asymptotic expansions for the variance of stopping times in nonlinear renewal theory

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to Wiener processes, we obtain an asymptotic expansion up to vanishing terms in the continuous time case. Applying the same methods to sums of i.i.d. random variables, we also provide an expansion in the discrete time case up to terms of order o(b^1/2) where the leading term is of order O(b), as b → ∞. The possibly unbounded term is the covariance of nonlinear excess and stopping time.