Dimensional Heat Equation Research Articles

An alternative explicit expression of the kernel of the one dimensional heat equation with Dirichlet conditions

This paper is devoted to the study of the one dimensional non homogeneous heat equation coupled to Dirichlet Boundary Conditions. We obtain the explicit expression of the solution of the linear equation by means of a direct integral in an unbounded domain. The main novelty of this expression relies in the fact that the solution is not given as a series of infinity terms. On our expression the solution is given as a sum of two integrals with a finite number of terms on the kernel. The main novelty is that, on the contrary to the classical method, where the solutions are derived by a direct application of the separation of variables method, on the basis of the spectral theory and the Fourier Series expansion, the solution is obtained by means of the application of the Laplace Transform with respect to the time variable. As a consequence, for any t ≥ 0 fixed, we must solve an Ordinary Differential Equation on the spatial variable, coupled to Dirichlet Boundary conditions. The solution of such a problem is given by the construction of the related Green’s function.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

The Edwards–Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.

Third Order Parallel Splitting Method for Nonhomogeneous Heat Equation with Integral Boundary Conditions

A third order parallel algorithm is proposed in this article to solve one dimensional non-homogenous heat equation with integral boundary conditions. For this purpose, we approximate the space derivative by third order finite difference approximation. This parallel splitting technique is combined with Simpson’s 1/3 rule to tackle the nonlocal part of this problem. The algorithm developed here is tested on two model problems. We conclude that our method provides better accuracy due to the availability of real arithmetic.

On Strongly Anisotropic Type I Blowup

We consider the energy super critical 4 dimensional semilinear heat equation ∂tu = ∆u + |u| p−1 u, x ∈ R 4 , p > 5. Let Φ(r) be a three dimensional radial self similar solution for the three supercrit-ical probmem as exhibited and studied in [7]. We show the finite codimensional transversal stability of the corresponding blow up solution by exhibiting a man-ifold of finite energy blow up solutions of the four dimensional problem with cylindrical symmetry which blows up as u(t, x) ∼ 1 (T − t) 1 p−1 U (t, Y), Y = x √ T − t with the profile U given to leading order by U (t, Y) ∼ 1 (1 + b(t)z 2) 1 p−1 Φ r 1 + b(t)z 2 , Y = (r, z), b(t) = c |log(T − t)| corresponding to a constant profile Φ(r) in the z direction reconnected to zero along the moving free boundary |z(t)| ∼ 1 √ b ∼ |log(T − t)|. Our analysis revis-its the stability analysis of the self similar ODE blow up [1, 29, 30] and combines it with the study of the Type I self similar blow up [7]. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.

Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

Exponential Convergence of Online Enrichment in Localized Reduced Basis Methods

Online enrichment is the extension of a reduced solution space based on the solution of the reduced model. Procedures for online enrichment were published for many localized model order reduction techniques. We show that residual based online enrichment on overlapping domains converges exponentially. Furthermore, we present an optimal enrichment strategy which couples the global reduced space with a local fine space. Numerical experiments on the two dimensional stationary heat equation with high contrast and channels confirm and illustrate the results.

Anisotropic $$(2+1)$$d growth and Gaussian limits of q-Whittaker processes

We consider a discrete model for anisotropic $$(2+1)$$ -dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space–time limit of covariances to those of the $$(2+1)$$ -dimensional additive stochastic heat equation (or Edwards–Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.

Real-Time Prediction of Temperature Elevation During Robotic Bone Drilling Using the Torque Signal.

Bone drilling is a surgical procedure commonly required in many surgical fields, particularly orthopedics, dentistry and head and neck surgeries. While the long-term effects of thermal bone necrosis are unknown, the thermal damage to nerves in spinal or otolaryngological surgeries might lead to partial paralysis. Previous models to predict the temperature elevation have been suggested, but were not validated or have the disadvantages of computation time and complexity which does not allow real time predictions. Within this study, an analytical temperature prediction model is proposed which uses the torque signal of the drilling process to model the heat production of the drill bit. A simple Green's disk source function is used to solve the three dimensional heat equation along the drilling axis. Additionally, an extensive experimental study was carried out to validate the model. A custom CNC-setup with a load cell and a thermal camera was used to measure the axial drilling torque and force as well as temperature elevations. Bones with different sets of bone volume fraction were drilled with two drill bits ([Formula: see text]1.8 mm and [Formula: see text]2.5 mm) and repeated eight times. The model was calibrated with 5 of 40 measurements and successfully validated with the rest of the data ([Formula: see text]C). It was also found that the temperature elevation can be predicted using only the torque signal of the drilling process. In the future, the model could be used to monitor and control the drilling process of surgeries close to vulnerable structures.

Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.

Intensive wave power and steel quenching 3-D model for cylindrical sample. Time direct and reverse formulations and solutions

In this paper we develop mathematical models for three dimensional hyperbolic heat equations (wave equation or telegraph equation) with inner source power and construct their analytical solutions for the determination of the initial heat flux for cylindrical sample. As additional conditions the temperature and heat flux at the end time are given. In some cases we give expression of wave energy. In some cases we give expression of wave energy. Some solutions of time inverse problems are obtained in the form of first kind Fredholm integral equation, but others has been obtained in closed analytical form as series. We viewed both direct and inverse problems at the time. For the time inverse problem we use inversion in the time argument.

Hollow system with fin. Transient Green function method combination for two hollow cylinders

In this paper we develop mathematical model for three dimensional heat equation for the system with hollow wall and fin and construct its analytical solution for two hollow cylindrical sample. The method of solution is based on Green function method for one hollow cylinder. On the conjugation conditions between both hollow cylinders we construct solution for system wall with fin. As result we come to integral equation on the surface between both hollow cylinders. Solution is obtained in the form of second kind Fredholm integral equation. The generalizing of Green function method allows us to use Green function method for regular non-canonical domains.

Application of the functional calculus to solving of infinite dimensional heat equation

In this paper we study infinite dimensional heat equation associated with the Gross Laplacian. Using the functional calculus method, we obtain the solution of appropriate Cauchy problem in the space of polynomial ultradifferentiable functions. The semigroup approach is considered as well.

Symmetry methods for option pricing

Abstract We obtain a solution of the Black–Scholes equation with a non-smooth boundary condition using symmetry methods. The Black–Scholes equation along with its boundary condition are first transformed into the one dimensional heat equation and an initial condition respectively. We then find an appropriate general symmetry generator of the heat equation using symmetries and the fundamental solution of the heat equation. The symmetry generator is chosen such that the boundary condition is left invariant; the symmetry can be used to solve the heat equation and hence the Black–Scholes equation.

Asymptotic Behavior of the Linearized Semigroup at Space-Periodic Stationary Solution of the Compressible Navier–Stokes Equation

The asymptotic behavior of the linearized semigroup at spatially periodic stationary solution of the compressible Navier–Stokes equation in a periodic layer of $$\mathbb {R}^n$$ $$(n=2,3)$$ is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of an $$n-1$$ dimensional linear heat equation as time goes to infinity and the other one decays exponentially.

Hermite spectral method for solving inverse heat source problems in multiple dimensions

In this paper, we in multiple dimensions. We present a stability estimate for determining the source term in the multiple dimensional heat equation in an unbounded domain, and the regularization parameter is chosen by a discrepancy principle. Error estimate between the exact solution and its regularization solution is given. Numerical experiments for the one-dimension and two-dimension cases show the effectiveness of the method.

Stochastic Heat Equation Limit of a (2 + 1)d Growth Model

We determine a $q\to 1$ limit of the two-dimensional $q$-Whittaker driven particle system on the torus studied previously in [Corwin-Toninelli, arXiv:1509.01605]. This has an interpretation as a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called anisotropic Kardar-Parisi-Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the $(2+1)$-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in $(2+1)$-dimension is irrelevant.

Blowup profile for a complex valued semilinear heat equation

This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We classify blow-up solutions and derive their blow-up profiles under some assumptions. In particular, we discuss the possibility of a nonsimultaneous blow-up.

On weak convergence of stochastic heat equation with colored noise

In this work we are going to show weak convergence of probability measures. The measure corresponding to the solution of the following one dimensional nonlinear stochastic heat equation ∂∂tut(x)=κ2∂2∂x2ut(x)+σ(ut(x))ηα with colored noise ηα will converge to the measure corresponding to the solution of the same equation but with white noise η, as α↑1. Function σ is taken to be Lipschitz and the Gaussian noise ηα is assumed to be colored in space and its covariance is given by E[ηα(t,x)ηα(s,y)]=δ(t−s)fα(x−y) where fα is the Riesz kernel fα(x)∝1/|x|α. We will work with the classical notion of weak convergence of measures, that is convergence of probability measures on a space of continuous function with compact domain and sup–norm topology. We will also state a result about continuity of measures in α, for α∈(0,1).

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion. One dimensional hyperbolic non-homogeneous heat, wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem. The method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations. The system is then solved by the optimal four stage three order strong stability-preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method has been tested on seven examples. The stability of the method is also discussed and found to be unconditionally stable.