Value Problem For Heat Equation Research Articles

Numerical solution of the mixed boundary value problem for the heat equation in two-dimensional domains of complex shape

A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.

Direct and Inverse Initial Boundary Value Problems for Heat Equation with Non-Classical Boundary Condition

Direct and Inverse Initial Boundary Value Problems for Heat Equation with Non-Classical Boundary Condition

Approximate Solution of Boundary Value Problem for Heat Equation after Represented by Volterra Integral Equation of the First Kind

In this work, we study the regularization method for solving the Boundary Value Problem (BVP) for heat equation. The discretization method applied with two variables on Volterra integral equation in order to covert the problem into a linear operator equation after applied the separation of variables method to solve the partial differential equation. The regularization way used to obtain the estimate solution by using the Lavrentiev regularization method.

A Finite Difference formula of Explicit Type for Pure Initial Value Problem of Heat Equation

In this paper we derive an iterative finite difference formula of explicit type to solve pure initial value problem of heat equation. To explain the development of improved finite difference formula for a given parabolic partial differential equation the simple dimensionless heat equation is used. In unbounded domains implicit methods are not applicable as they generate infinite matrices. Thus it is necessary to find more improved equations of explicit type. We apply finite difference formulas in a specific way and obtain a new formula where the truncation error gets reduced. Also the convergence limit and therefore stability gets improved.

Thin plate approximation in active infrared thermography

In this work, we find and test a new approximated formula (based on the thin plate approximation), for recovering small, unknown damages on the inaccessible surface of a thin conducting (aluminium) plate. We solve this inverse problem from a controlled heat flux and a sequence of temperature maps on the accessible front boundary of our sample. We heat the front boundary by means of a sinusoidal flux. In the meanwhile, we take a sequence of temperature maps of the same side by means of an infrared camera. This procedure is called active infrared thermography. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data. We use domain derivative to linearize the boundary value problem for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply perturbation theory in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data by means of the infrared camera. We use domain derivative to linearize the BVP for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply the perturbation theory in (a is the thickness of our sample) in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging.

Some endpoint estimates for heat equation with application to Navier-Stokes equations

We obtain some estimates in Hardy, BMO and Besov spaces for solutions to the initial value problem of heat equation. Combined with Coifmann-Lions-Meyer-Semmes' result on compensated compactness in Hardy spaces, we also give some endpoint estimates for the second order derivatives of global weak solutions to NavierStokes equations.

Explicit Solution of a Nonlocal Boundary Value Problem for Heat Equation

An initial-boundary value problem for one-dimensional heat equation with a nonlocal boundary condition is studied. This boundary condition ensures both simple and triple eigenvalues of the corresponding one-dimensional spec- tral problem. Applying spectral projectors, we find a series solution of the problem for a special choice of the initial function. Then, using operational calculus approach of Dimovski, we obtain an explicit representation of the so- lution in the general case. The expression obtained contains a non-classical convolution product of the particular solution and an arbitrary initial function. This result is an extension of the classical Duhamel principle, but for the space variable.

A Boundary Element Formulation for the Pricing of Barrier Options

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

Hyers-Ulam-Rassias Stability for the Heat Equation

In this paper we apply the Fourier transform to prove the Hyers-Ulam-Rassias stability for one dimensional heat equation on an infinite rod. Further, the paper investigates the stability of heat equation in with initial condition, in the sense of Hyers-Ulam-Rassias. We have also used Laplace transform to establish the modified Hyers-Ulam-Rassias stability of initial-boundary value problem for heat equation on a finite rod. Some illustrative examples are given.

Boundary value problems for heat equation with certain potentials in Lipschitz cylinders

Abstract The Dirichlet and Neumann problems of heat equation with certain potentials of the form ∂ t - Δ + V = 0 $\partial _t-\Delta +V=0$ in Lipschitz cylinders are studied, where V is a nonnegative potential satisfying certain conditions.

A boundary element approximation to mixed boundary value problem of heat equation via collocation and Galerkin method

This article considers the collocation and Galerkin methods applied to boundary integral equations of transient heat problem with mixed boundary condition. The convergence rate of collocation methods is investigated and distinguished from Galerkin methods especially for negative Dirichlet data. The numerical examples of both methods are presented.

A finite difference method for a non-local boundary value problem for two-dimensional heat equation

A second-order finite difference scheme is given for the numerical solution of a two-dimensional non-local boundary value problem for heat equation. Using a suitable transformation, the solution of this problem is equivalent to the solution of two other problems. The first problem which is a one-dimensional non-local boundary value problem giving the value of μ through using a second-order finite difference scheme. Using this result, the second problem will be changed to a classical two-dimensional problem with Nuemann's boundary condition which will be solved numerically. The stability properties and truncation error of the new method are discussed and the results of numerical experiments are presented.