Dimensional Heat Equation Research Articles

The Geometry of Vector Fields and two Dimensional Heat Equation

The Geometry of Vector Fields and two Dimensional Heat Equation

KPZ equation limit of sticky Brownian motion

We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the (1+1) dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [66,7] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an N-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order (log⁡N)2.

Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

AbstractWe consider the Dirichlet problem for the energy-critical heat equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+u^5 &{} \text {in} \quad \Omega \times \mathbb {R}^+,\\ u=0 &{}\text {on} \quad \partial \Omega \times \mathbb {R}^+,\\ u(x,0)=u_0(x) &{} \text {in} \quad \Omega , \end{array}\right. } \end{aligned}$$ u t = Δ u + u 5 in Ω × R + , u = 0 on ∂ Ω × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^3$$ R 3 . Let $$H_\gamma (x,y)$$ H γ ( x , y ) be the regular part of the Green function of $$-\Delta -\gamma $$ - Δ - γ in $$\Omega $$ Ω , where $$\gamma \in (0,\lambda _1)$$ γ ∈ ( 0 , λ 1 ) and $$\lambda _1$$ λ 1 is the first Dirichlet eigenvalue of $$-\Delta $$ - Δ . Then, given a point $$q\in \Omega $$ q ∈ Ω such that $$3\gamma (q)<\lambda _1$$ 3 γ ( q ) < λ 1 , where $$\begin{aligned} \gamma (q) :=\sup \{ \gamma>0: H_\gamma (q,q)>0 \}, \end{aligned}$$ γ ( q ) : = sup { γ > 0 : H γ ( q , q ) > 0 } , we prove the existence of a non-radial global positive and smooth solution u(x, t) which blows up in infinite time with spike in q. The solution has the asymptotic profile $$\begin{aligned} u(x,t)\sim 3^{\frac{1}{4}} \left( \frac{\mu (t)}{\mu (t)^2+\vert x-\xi (t)\vert ^2}\right) ^{\frac{1}{2}} \quad \text {as}\quad t \rightarrow \infty , \end{aligned}$$ u ( x , t ) ∼ 3 1 4 μ ( t ) μ ( t ) 2 + | x - ξ ( t ) | 2 1 2 as t → ∞ , where $$\begin{aligned} -\ln (\mu (t))= 2\gamma (q) t(1+o(1)),\quad \xi (t)=q+O(\mu (t)) \quad \text {as}\quad t \rightarrow \infty . \end{aligned}$$ - ln ( μ ( t ) ) = 2 γ ( q ) t ( 1 + o ( 1 ) ) , ξ ( t ) = q + O ( μ ( t ) ) as t → ∞ .

On a family of nonzero solutions to the heat equation ut = △u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ ℝn+1

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data (i.e. vanishing on the hyperplane [Formula: see text]) can be easily obtained as a consequence of the examples on [Formula: see text] Finally, given an arbitrary hyperplane [Formula: see text], we can construct a family of nonzero solutions which vanish on [Formula: see text]

Extended symmetry analysis of (1+2)‐dimensional fine Kolmogorov backward equation

AbstractWithin the class of (1+2)‐dimensional ultraparabolic linear equations, we distinguish a fine Kolmogorov backward equation with a quadratic diffusivity. Modulo the point equivalence, it is a unique equation within the class whose essential Lie invariance algebra is five‐dimensional and nonsolvable. Using the direct method, we compute the point symmetry pseudogroup of this equation and analyze its structure. In particular, we single out its essential subgroup and classify its discrete elements. We exhaustively classify all subalgebras of the corresponding essential Lie invariance algebra up to inner automorphisms and up to the action of the essential point‐symmetry group. This allowed us to classify Lie reductions and Lie invariant solutions of the equation under consideration. We also discuss the generation of its solutions using point and linear generalized symmetries and carry out its peculiar generalized reductions. As a result, we construct wide families of its solutions parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)‐dimensional linear heat equation or one or two arbitrary solutions of (1+1)‐dimensional linear heat equations with inverse square potentials.

Quantitative Rapid and Finite Time Stabilization of the Heat Equation

The finite time stabilizability of the one dimensional heat equation is proved by Coron-Nguyên [J.-M. Coron and H.-M. Nguyen, Arch. Ration. Mech. Anal. 225 (2017) 993–1023], while the same question for multidimensional spaces remained open. Inspired by Coron-Trélat [J.-M. Coron and E. Trélat, SIAM J. Control Optim. 43 (2004) 549–569] we introduce a new method to stabilize multidimensional heat equations quantitatively in finite time and call it Frequency Lyapunov method. This method naturally combines spectral inequality [G. Lebeau and L. Robbiano, Comm. Partial Diff. Equ. 20 (1995) 335–356] and constructive feedback stabilization. As application this approach also yields a constructive proof for null controllability, which gives sharing optimal cost CeC/T with explicit controls and works perfectly for related nonlinear models such as Navier–Stokes equations [S. Xiang, Ann. Inst. H. Poincaré C Anal. Non Lineaire 40 (2023) 1487–1511.].

Numerical Solution for the Heat Equation Using Crank-Nicolson Difference Method

Abstract: In this paper, the Crank-Nicolson difference method is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using Crank-Nicolson difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of Crank-Nicolson difference method difference method of heat equation.

Numerical Solution for the Heat Equation Using Forward Time Centered Space Difference Method

Abstract: In this paper, the forward time centered space is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using forward time centered space difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of explicit forward time centered space difference method of heat equation.

Entropy analysis of Cu-Al2O3 based hybrid and Cu based mono nanofluid flows through porous medium: A comparative study

The motive of this research article, is to present a comparative study of entropy generation for a system of MHD mixed convection non-Darcy porous medium flow of mono and hybrid nanofluids over a melting surface. Darcy–Forchheimer model for the flow through porous medium is employed to formulate the problem under consideration. The dimensional momentum and heat equations governing the flow are transformed into a non-dimensional ODEs with the help of similarity variables. The similarity solution of the dimensionless differential equation is computed with the MATLAB’s bvp4c package. The influence of various embedded parameters on the flow fields and entropy generation are examined. The study reveals that the entropy generation in Cu−Al2O3 water based hybrid nanofluid is smaller compared to that of Cu-water based mono nanofluid. The findings of the study have bearing in the fields involving oil flow filtration, crude oil production, electronic cooling equipment etc.

A Terminal Condition in Linear Bond-pricing Under Symmetry Invariance

In this paper, we examine a general bond-pricing model with respect to its solutions that satisfy a given terminal condition. Firstly, we obtain reversible transformations that change the model to a classical and well known partial differential equation, the one dimensional heat equation. We further show that the terminal condition is transformed into a nonsmooth initial condition. The important result that emerges is that the Lie symmetries are adopted to solve the equation subject to its unique configuration of initial conditions.

Fully discrete Schwarz waveform relaxation analysis for the heat equation on a finite spatial domain

Schwarz waveform relaxation methods provide space-time parallelism for the solution of time dependent partial differential equations. The algorithms are differentiated by the choice of the transmission conditions enforced at the introduced space-time boundaries. Early results considered the theoretical analysis of these algorithms in the continuous and semi-discrete (in space) settings for various families of linear partial differential equations. Later, fully discrete results were obtained under the simplifying assumption of an infinite spatial domain. In this paper, we provide a first analysis of a fully discrete classical Schwarz Waveform algorithm for the one-dimensional heat equation on an arbitrary but finite number of bounded subdomains. The θ-method is chosen as the time integrator. Convergence results are given in both the infinity norm and two norm, with an explicit contraction given in the case of a uniform partitioning. The results are compared to the numerics and to the earlier theoretical results.

Point and generalized symmetries of the heat equation revisited

We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected components. That is, the heat equation admits a single independent discrete symmetry, which can be chosen to be alternating the sign of the dependent variable. We introduce the notion of pseudo-discrete elements of a Lie group and show that alternating the sign of the space variable, which was for a long time misinterpreted as a discrete symmetry of the heat equation, is in fact a pseudo-discrete element of its essential point symmetry group. The classification of subalgebras of the essential Lie invariance algebra of the heat equation is enhanced and the description of generalized symmetries of this equation is refined as well. We also consider the Burgers equation because of its relation to the heat equation and prove that it admits no discrete point symmetries. The developed approach to point-symmetry groups whose elements have components that are linear fractional in some variables can directly be extended to many other linear and nonlinear differential equations.

Extended symmetry analysis of remarkable (1+2)-dimensional Fokker–Planck equation

AbstractWe carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

The Geometry of Vector Fields and Two Dimensional Heat Equation

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications in the theory of control systems, in dynamic systems, in geometry and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.

FRACTIONAL ORDER DISTRIBUTION ON HEAT FLUX FOR CRYSTALLINE CONCRETE MATERIAL

The present work devoted to analyse the impact of fractional order one dimensional heat equation on heat transfer in concrete substances, during conduction process. Throughout the evolution the thermal properties of substances are considered as constant parameters. Investigations were done through statistical phenomena with linear representation of curve.

Determination of time-dependent coefficient in time fractional heat equation

The aim of this work is to determine the time-dependent heat coefficient in a type of inverse problem for one dimensional time-fractional heat equations defined by the Caputo operator for (0<α<1) by applying a method based on the finite difference scheme and Tikhonov’s regularization. First, a stable implicit finite difference scheme is applied to find a numerical solution to the (forward) direct problem. While the inverse problem was reformulated as a nonlinear least-square minimization problem with a simple physical bound on the unknown coefficient and solved efficiently by MATLAB routine lsqnonlin from the optimization toolbox. But the latter problem will remain ill-posed because the presence of any error in the input data will lead to a large error in the output data. So, Tikhonov’s technique is applied to obtain stable results. In the end, two numerical test examples show that the proposed method is stable, accurate, and works well with different noise levels

Use Different Mathematical Methods to Solve Three Dimensional Conduction Heat Equation in Cartesian Coordinate

In this paper three-dimensional heat conduction equation in cartesian coordinate has been solved in two different methods one of which depends on the separation of variables and the other depends on the integral transform .The results are got and plotted by using Matlab. And the results obtained showed the difference between the two methods that were used in the solution . That difference is evident in the illustrations . According to the results it was concluded that the integral transform method is the best because it has fewer steps to reached to the solution

Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as \(\varepsilon \downarrow 0\). The tool we used here is the Malliavin-Stein’s method. We also give a functional version of this result.

Determination of thermophysical characteristics in a nonlinear inverse heat transfer problem

This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas’ method, and the nonlinear difference problem is solved by the Newton’s method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0,tmax), the quadratic convergence of Newton’s method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton’s method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (tmax,2tmax), the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2tmax,3tmax), the density of the soil is determined. Also, based on the measured initial data on the segment (3tmax,4tmax), the heat transfer coefficient is determined.Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed.The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown.

AN EFFICIENT NUMERICAL TECHNIQUE FOR SOLVING HEAT EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

A third order parallel algorithm is proposed 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 develop here is tested on two model problems. We conclude that our method provides better accuracy due to availability of real arithmetic.