Generalized Cole-Hopf Transformation Research Articles

The generalized Cole–Hopf transformation for a generalized Burgers–Fisher equation with spatiotemporal variable coefficients

In this paper, we investigate a generalized Burgers-Fisher equation with spatiotemporal variable coefficients(gvcBF), which can describe the nonlinear convection–diffusion phenomenon in chemical engineering and biology. It is shown that the gvc Burgers-Fisher equation can be exactly linearized as long as the variable coefficient functions satisfy some constraints conditions. We obtain a Bäcklund transformation between the gvc Burgers-Fisher equation and a linear parabolic equation with variable coefficients by the simplified homogeneous balance method. Furthermore, we derive out a generalized Cole-Hopf transformation between a class of the gvc Burgers-Fisher equation and the linear heat conduction equation with the help of a new unknown function transformation. Especially, we obtain the generalized Cole-Hopf transformation between the cylindrical and spherical Burgers-Fisher equations with algebraical decaying damping term and the heat conduction equation. Finally, we obtain a series of explicit exact solutions of the gvc Burgers-Fisher equation.

The generalized Cole–Hopf transformation to a general variable coefficient Burgers equation with linear damping term

A general variable coefficient (gVC) Burgers equation with linear damping term has been investigated by using simplified homogeneous balance (SHB) method. The generalized Cole–Hopf transformation for the gVC Burgers equation with linear damping term has been derived out if the damping coefficient satisfies a certain constraint condition. By means of the generalized Cole–Hopf transformation, the exact solutions of the gVC Burgers equation with the certain damping term have been obtained successfully. Several concrete examples (included (2+1)-dimensional damping Burgers equation) of gVC Burgers equation with the certain damping term are studied, and their corresponding generalized Cole–Hopf transformations as well as their corresponding solutions are all given, respectively.

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.

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.

Solutions to Painleve III and other nonlinear equations by a generalized Cole–Hopf transformation

In this paper, we introduce a generalized form of Cole‐Hopf transformation and apply it to find new closed‐form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.

ECUACIÓN FRACCIONARIA DE BURGERS NO HOMOGÉNEA

En este artículo se estudian diferentes tipos de soluciones de la ecuación fraccionaria unidimensional no lineal de Burgers con un término no homogéneo asociado a fuerzas externas. Esta ecuación es una generalización de la ecuación de difusión no homogénea en la que se incluye una derivada fraccionaria de Caputo que describe una no linealidad no local. Por medio de la transformación de Cole-Hopf generalizada, la ecuación de Burgers fraccionaria no homogénea se convierte en una ecuación diferencial lineal en derivadas parciales, lo que permite obtener soluciones analíticas. Se exploran los efectos asociados al término no homogéneo y al orden de la derivada fraccionaria.

A generalized Cole–Hopf transformation for nonlinear ODEs

We introduce a ‘hybrid’ Cole–Hopf–Darboux transformation to relate the solutions of nonlinear and linear second-order differential equations and derive classification and sufficient condition for this correspondence. We explore physical applications of this correspondence to nonlinear oscillations, the Duffing equation and a nonlinear form of the Schrödinger equation for the nonrelativistic hydrogen atom. In addition, we show that solutions of some nonlinear second-order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the ‘class of special nonlinear equations’ which correspond to the linear differential equations which define the special functions of mathematical physics.

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (ut + uux − uxx)x + s(t)uyy = 0 is transformed into the Riccati equation \(H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0\) via the ansatz \(u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y\), provided that s(t) = t−3/2. Further, a generalized Cole-Hopf transformations \(u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}\), \(\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y\), r(t) = log t is derived to linearize (ut + uux − uxx)x + t−3/2uyy to the parabolic equation \(U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho\).

Damped parametric oscillator and exactly solvable complex Burgers equations

We obtain exact solutions of a parametric Madelung fluid model with dissipation which is linearazible in the form of Schrödinger equation with time variable coefficients. The corresponding complex Burgers equation is solved by a generalized Cole-Hopf transformation and the dynamics of the pole singularities is described explicitly. In particular, we give exact solutions for variable parametric Madelung fluid and complex Burgers equations related with the Sturm-Liouville problems for the classical Hermite, Laguerre and Legendre type orthogonal polynomials.

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole–Hopf transformation.

Diffusion equation coupled to Burgers' equation

Scalar diffusion in one-dimensional Burgers' flow is considered. When the Prandtl number is unity, the diffusion equation with convective term is reduced to a simple diffusion equation by a generalized Cole-Hopf transformation. An exact solution of an initial value problem is obtained in a closed form. When the Prandtl number is arbitrary, a similar analytical treatment is possible for limited classes of Burgers' flow (expansion wave and single shock). The statistics of scalar field are discussed briefly.