Cole-Hopf Transformation Research Articles

Multistep Methods for the Numerical Simulation of Two-Dimensional Burgers’ Equation

In this paper, a numerical technique is proposed to solve a two-dimensional coupled Burgers’ equation. The two-dimensional Cole–Hopf transformation is applied to convert the nonlinear coupled Burgers’ equation into a two-dimensional linear diffusion equation with Neumann boundary conditions. The diffusion equation with Neumann boundary conditions is semi-discretized using MOL in both x and y directions. This process yielded the system of ordinary differential equations in the time variable. Multistep methods namely backward differentiation formulas of order one, two and three are employed to solve the ode system. Efficiency and accuracy of the proposed methods are verified through numerical experiments. The proposed schemes are simple, accurate, efficient and easy to implement.

How flat is flat in random interface growth?

Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on a large deviation rate function for the rescaled initial data, which arises naturally from the Hopf-Cole transformation. This allows us, in particular, to distinguish the domains of attraction of curved, flat, and Brownian initial data and to identify the boundary between the curved and flat domains of attraction, which turns out to correspond to square root initial data. The distribution of the asymptotic one-point fluctuations is characterized by means of a variational formula written in terms of certain limiting processes (arising as subsequential limits of the spatial fluctuations of the KPZ equation with narrow wedge initial data, as shown in Probab. Theory Related Fields 166 (2016), pp. 67–185) which are widely believed to coincide with the Airy 2 _2 process. In order to identify these distributions for general initial data, we extend earlier results on continuum statistics of the Airy 2 _2 process to probabilities involving the process on the entire line. In particular, this allows us to write an explicit Fredholm determinant formula for the case of square root initial data.

An exploration of quintic Hermite splines to solve Burgers\u2019 equation

An innovative scheme of collocation having quintic Hermite splines as base functions has been followed to solve Burgers’ equation. The scheme relies on approximation of Burgers’ equation directly in non-linear form without using Hopf–Cole transformation (Hopf in Commun Pure Appl Math 3:201–216, 1950; Cole in Q Appl Math 9:225–236, 1951). The significance of the numerical technique is demonstrated by comparing the numerical results to the exact solution and published results (Asaithambi in Appl Math Comput 216:2700–2708, 2010; Mittal and Jain in Appl Math Comput 218:7839–7855, 2012). Five problems with different initial conditions have been examined to validate the efficiency and accuracy of the scheme. Euclidean and supremum norms have been reckoned to scrutinize the stability of the numerical scheme. Results have been demonstrated in plane and surface plots to indicate the effectiveness of the scheme.

Positive solution to extremal Pucci's equations with singular and gradient nonlinearity

In this paper, we establish the existence of a positive solution to \begin{document}$ \begin{equation*} \label{deff} \left\{ \begin{aligned}{} -\mathcal{M}^{+}_{\lambda,\Lambda}(D^{2}u)+H(x,Du)& = \frac{k(x)f(u)}{u^{\alpha}} \;\text{in}\; \Omega, u&>0 \;\text{in}\; \Omega, u& = 0 \;\text{on} \;\partial\Omega, \end{aligned} \right. \end{equation*} $\end{document} under certain conditions on \begin{document}$ k,f $\end{document} and \begin{document}$ H, $\end{document} using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.

Abundant Lump-Type Solutions and Interaction Solutions for a Nonlinear (3+1) Dimensional Model

We explore dynamical features of lump solutions as diversion and propagation in the space. Through the Hirota bilinear method and the Cole-Hopf transformation, lump-type solutions and their interaction solutions with one- or two-stripe solutions have been generated for a generalized (3+1) shallow water-like (SWL) equation, via symbolic computations associated with three different ansatzes. The analyticity and localization of the resulting solutions in the (x,y,z, and t) space have been analyzed. Three-dimensional plots and contour plots are made for some special cases of the solutions to illustrate physical motions and peak dynamics of lump soliton waves in higher dimensions. The study of lump-type solutions moderates the visuality of optics media and oceanography waves.

Green’s Function of the Linearized Logarithmic Keller–Segel–Fisher/KPP System

We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibrium state, and obtain a detailed, pointwise description of the Green’s function. The result provides a complete solution picture for the linear problem. It also helps to shed light on small solutions of the nonlinear system.

The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

This second paper adopts a more rigorous, in-depth approach to modelling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers’ and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve, [6; 7; 8; 9], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most – if not all – of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander’s seminal work, [6; 7; 8; 9]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures both within this paper and the next.

A Geometric Perspective on Regularized Optimal Transport

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the Schrodinger bridge problem and the Yasue problem. We also propose an analogue of the Hopf–Cole transformation in the geometric setting.

Generalized Cole–Hopf Transformation

On the basis of generalization of the Cole–Hopf transformation for parabolic equations with a source, we obtain some new representations of solutions and coefficients of nonlinear parabolic equations of mathematical physics which in fact are differential-algebraic identities. These representations can be used in studying the multidimensional direct and inverse problems.

Stochastic Burgers' equation on the real line: regularity and moment estimates

ABSTRACTIn this project, we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman–Kac representation which solves a stochastic partial differential equation such that its Hopf–Cole transformation solves Burgers' equation. Finally, we obtain Hölder regularity and moment estimates for the solution to Burgers' equation.

Resolving controls for exact and approximate controllability of viscous Burgers’ equation: The Green’s function approach

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported [Formula: see text] functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers’ Equation**Supported by the National Natural Science Foundation of China under Grant No. 51576079

In this paper, we present a Cole-Hopf transformation based lattice Boltzmann (LB) model for solving one-dimensional Burgers’ equation, and compared to available LB models, the effect of nonlinear convection term can be eliminated. Through Chapman-Enskog analysis, it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model. Some numerical tests are also performed to validate the present LB model, and the numerical results show that, similar to previous LB models, the present model also has a second-order convergence rate in space, but it is more accurate than the previous ones.

A Reliable Study of New Nonlinear Equation: Two-Mode Kuramoto–Sivashinsky

In this letter, we use the sense made by Korsunsky (Phys Lett A 185:174–176, 1994 ) to establish a new nonlinear equation called the two-mode Kuramoto–Sivashinsky (TMKS). A finite series in terms of the Tanh function is proposed to be a candidate solution to this new model. Also, we study possible solutions of TMKS by means of the modified simplified bilinear method where a new Cole-Hopf transformation is considered. The new model describes the propagation of two different wave modes simultaneously.

Boundary layers and stabilization of the singular Keller-Segel system

The original Keller-Segel system proposed in [ 23 ] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter e now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as e →0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the effective viscous flux technique to establish the uniform-in- e estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L 1 -estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg–de Vries–Burgers equation

In this paper, we establish a new nonlinear equation which is called the two-mode Korteweg–de Vries–Burgers equation (TMKdV–BE). The new equation describes the propagation of two different wave modes simultaneously. First, we introduce a new Cole–Hopf transformation needed for the modified simplified bilinear method to find necessary conditions on the solution of TMKdV–BE. Also, a finite series in terms of tanh–coth function is presented as an alternative method to study the solution of the equation. Finally, the reliability of the obtained results is discussed in the last section.

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equationΔu+f(u)=0 under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., Δu+up=0 and Δu+eu=0.

A FDM for Burgers’ Equation on Unbounded Domains Using High-Order Artificial Boundary Conditions

Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale’s theory of scale relativity relying on a fractal space-time, we derive a generalized Schrodinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale’s theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrodinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrodinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrodinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrodinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Two-mode Sharma-Tasso-Olver equation and two-mode fourth-order Burgers equation: Multiple kink solutions

We develop two new equations which describe propagation of two different wave modes simultaneously. The first equation is a two-mode Sharma-Tasso-Olver equation, and the second is a two-mode fourth-order Burgers equation. We show that multiple kink solutions for each model exist only for specific values of the nonlinearity and dispersion parameters included in the models. We will use the Cole-Hopf transformation combined with the simplified Hirota’s method to conduct this analysis.

An asymptotic method based on a Hopf–Cole transformation for a kinetic BGK equation in the hyperbolic limit

We present an effective asymptotic method for approximating the density of particles for kinetic equations with a Bhatnagar–Gross–Krook (BGK) relaxation operator in the large scale hyperbolic limit. The density of particles is transformed via a Hopf–Cole transformation, where the phase function is expanded as a power series with respect to the Knudsen number. The expansion terms can be determined by solving a sequence of equations. In particular, it has been proved in [3] that the leading order term is the viscosity solution of an effective Hamilton–Jacobi equation, and we show that the higher order terms can be formally determined by solving a sequence of transport equations. Both the effective Hamilton–Jacobi equation and the transport equations are independent of the Knudsen number, and are formulated in the physical space, where the effective Hamiltonian is obtained as the solution of a nonlinear equation that is given as an integral in the velocity variable, and the coefficients of the transport equations are given as integrals in the velocity variable. With appropriate Gauss quadrature rules for evaluating these integrals effectively, the effective Hamilton–Jacobi equation and the transport equations can be solved efficiently to obtain the expansion terms for approximating the density function. In this work, the zeroth, first and second order terms in the expansion are used to obtain second order accuracy with respect to the Knudsen number. The proposed method balances efficiency and accuracy, and has the potential to deal with kinetic equations with more general BGK models. Numerical experiments verify the effectiveness of the proposed method.