scholarly journals Bayesian Recovery of the Initial Condition for the Heat Equation

2013 ◽  
Vol 42 (7) ◽  
pp. 1294-1313 ◽  
Author(s):  
B. T. Knapik ◽  
A. W. van der Vaart ◽  
J. H. van Zanten
Keyword(s):  
Heat Equation ◽  
Initial Condition

scholarly journals Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chenguang Zhou ◽  
Yongkui Zou ◽  
Shimin Chai ◽  
Fengshan Zhang
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Polynomial Functions ◽  
Weak Galerkin ◽  
Convergence Estimates

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

ASYMPTOTIC BEHAVIOR FOR TWO REGULARIZATIONS OF THE CAUCHY PROBLEM FOR THE BACKWARD HEAT EQUATION

1998 ◽  
Vol 08 (01) ◽  
pp. 187-202 ◽  
Author(s):  
K. A. AMES ◽  
L. E. PAYNE
Keyword(s):  
Asymptotic Behavior ◽  
Heat Equation ◽  
Singularly Perturbed ◽  
Infinite Cylinder ◽  
Initial Condition ◽  
Damped Wave Equation ◽  
Spatial Decay ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Negative Damping

One method of regularizing the initial value problem for the backward heat equation involves replacing the equation by a singularly perturbed hyperbolic equation which is equivalent to a damped wave equation with negative damping. Another regularization of this problem is obtained by perturbing the initial condition rather than the differential equation. For both of these problems, we investigate the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus establish spatial decay results of the Saint-Venant type.

scholarly journals On the Operator ⨁BkRelated to Bessel Heat Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Wanchak Satsanit
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Positive Constant ◽  
Nonnegative Integer ◽  
Unknown Function ◽  
Generalized Function ◽  
Initial Condition

We study the equation(∂/∂t)u(x,t)=c2⊕Bku(x,t)with the initial conditionu(x,0)=f(x)forx∈Rn+.The operator⊕Bkis the operator iterated k-times and is defined by⊕Bk=((∑i=1pBxi)4-(∑j=p+1p+qBxi)4)k, wherep+q=nis the dimension of theRn+,Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2αi+1,αi>-1/2,i=1,2,3,…,n, andkis a nonnegative integer,u(x,t)is an unknown function for(x,t)=(x1,x2,…,xn,t)∈Rn+×(0,∞),f(x)is a given generalized function, andcis a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

scholarly journals Propagação do calor em uma barra homogênea com condições de fronteira e dependendo de um parâmetro real

1981 ◽  
Vol 3 (3) ◽  
pp. 01
Author(s):  
Lilian M. Kieling Reis ◽  
Vanilde Bisognin
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Initial Condition

In this work the permanent temperature, in one homogeneous bar with boundary conditions that depends of a real parameters, was determined. The problem to be resolved was find the analytical solution of the heat equation ut =α2 uxx with the initial condition u (x, 0) = 0, ≤ x ≤ L and the contours conditions u (0, t) = 0 and u (L, t) = sen t, t> 0.

scholarly journals A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

2013 ◽  
Vol 9 (17) ◽  
pp. 11-20
Author(s):  
Carlos Cadavid ◽  
Juan Diego Vélez
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Critical Points ◽  
Morse Function ◽  
Initial Condition ◽  
Morse Functions ◽  
Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

GENERALIZED FEYNMAN–KAC FORMULA WITH STOCHASTIC POTENTIAL

2002 ◽  
Vol 05 (02) ◽  
pp. 243-255 ◽  
Author(s):  
HABIB OUERDIANE ◽  
JOSÉ LUIS SILVA
Keyword(s):  
Heat Equation ◽  
Stochastic Processes ◽  
Function Space ◽  
Stochastic Heat Equation ◽  
Generalized Function ◽  
Initial Condition ◽  
The Stochastic Heat Equation

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space [Formula: see text].

On solutions to the heat equation with the initial condition in the Orlicz—Slobodetskii space

2014 ◽  
Vol 144 (4) ◽  
pp. 787-807 ◽  
Author(s):  
Agnieszka Kałamajska ◽  
Mirosłav Krbec
Keyword(s):  
Sobolev Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Orlicz Function ◽  
Initial Condition ◽  
Lipschitz Boundary

We study the boundary-value problem ũt = Δxũ(x,t), ũ(x, 0) = u(x), where x ∈ Ω, t ∈ (0,T), Ω ⊆ ℝn−1 is a bounded Lipschitz boundary domain, u belongs to a certain Orlicz–Slobodetskii space YR,R(Ω). Under certain assumptions on the Orlicz function R, we prove that the solution u belongs to the Orlicz–Sobolev space W1,R(Ω × (0,T)).

Sign in / Sign up

Export Citation Format

Share Document