Functional Separation Of Variables Research Articles

Exact solutions of steady Euler equations in two dimension by functional separation

Exact solutions of two dimensional Euler equations for incompressible fluid flows are found by the functional separation of variables. More specifically, two independent variables are considered in the Cartesian coordinates, polar coordinates and also general orthogonal curvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several new solutions are obtained and other classical solutions are rediscovered.

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

Integrable mathfrak{sl} (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for mathfrak{sl} (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

New exact solutions of nonlinear wave type PDEs with delay

We consider nonlinear wave type PDEs with delay of the form utt+H1(u)ut=[G(u)ux]x+H2(u)ux+F(u,w),where u=u(x,t) is the unknown function, w=u(x,t−τ), and τ is the delay time. The source function F(u,w) depends on one or several arbitrary functions of one argument. Using the modified method of functional constraints, we obtain a number of new exact solutions with generalized and functional separation of variables, as well as traveling-wave solutions. Most solutions are expressed in terms of elementary functions and contain free parameters. These solutions can be used to formulate test problems intended to evaluate the accuracy of numerical methods for solving nonlinear delay PDEs.

Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

Functional separable solutions of two classes of nonlinear mathematical physics equations

The study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions; the specific expressions of these functions are determined in the subsequent analysis of the arising functional differential equations. The effectiveness of the method is illustrated by examples of nonlinear reaction-diffusion equations and Klein-Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained.

Functional Separable Solutions of Two Classes of Nonlinear Mathematical Physics Equations

This study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions (specific expressions for these functions are determined by analyzing the arising functional differential equations). The effectiveness of the method is illustrated by examples of nonlinear reaction–diffusion equations and Klein–Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained.

Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion–convection equations with weak source

In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion–convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

Transformations, properties, and exact solutions of unsteady axisymmetric boundary layer equations for non-Newtonian fluids

Unsteady axisymmetric boundary layer equations for power-law non-Newtonian fluids are analyzed. A number of new exact solutions containing arbitrary functions and free parameters are constructed using generalized or functional separation of variables. The solutions are obtained using a Crocco-type transformation reducing the order of the equations examined and simpler point transformations. Along with the exact solutions to axisymmetric boundary layer equations, some new exact solutions to planar boundary layer equations for non-Newtonian fluids are constructed. Several properties have been discovered that allow the exact solutions of the unsteady axisymmetric boundary layer equations to be generalized by including additional arbitrary functions therein. All results refer to an arbitrarily shaped streamlined solid of revolution.

One-dimensional reductions and functional separable solutions to unsteady plane and axisymmetric boundary-layer equations for non-Newtonian fluids

The paper deals with equations describing the unsteady axisymmetric boundary layer of power-law non-Newtonian fluids on a body of revolution. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a single third-order PDE of the formwtz+wzwxz−wxwzz=κrn+1(x)wzzn−1wzzz+F(t,x),where n is a rheological parameter of the fluid; the function r(x), determining the shape of the body, is assumed arbitrary. For this non-linear PDE, we describe one-dimensional reductions as well as a number of new generalized and functional separable solutions, which depend on two to five arbitrary functions. The solutions are obtained with the direct method of functional separation of variables by using particular solutions to an auxiliary ODE and systems of first-order PDEs. Many of the solutions are expressed in terms of elementary functions, which, together with significant arbitrariness, makes them especially useful for solving certain model problems and testing numerical and approximate analytical methods in fluid dynamics. Apart from power-law fluids, the paper looks at three-parameter polynomial and generalized Sisko models of non-Newtonian fluids. The unsteady plane boundary-layer equations for the general non-Newtonian fluid model are shown to admit a reduction to an ODE. The paper presents new exact solutions to the plane boundary-layer equations for power-law fluids as well as some other rheologically complex fluids.

Exact solutions of unsteady boundary layer equations for power-law non-Newtonian fluids

A number of new exact solutions (with the generalized and functional separation of variables) of unsteady equations of a planar and asymmetric boundary layer of power-law non-Newtonian fluids are described. To find the solutions, the Crocco transformation reducing the order of the equations considered and simpler point transformations are used. Two theorems allowing one to generalize exact solutions of the unsteady axisymmetric boundary layer equations including additional arbitrary functions into them are proven.

Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations

The paper deals with equations describing the unsteady axisymmetric laminar boundary layer on an extensive body of revolution as well as axisymmetric jet flows. Such equations are shown to reduce to a single nonlinear third-order PDE with variable coefficientswtz+wzwxz−wxwzz=νzwzzz+F(t,x),where w is a modified stream function. We describe a number of new generalized and functional separable solutions to this equation, which depend on two to four arbitrary functions of a single argument (a few solutions depend on an arbitrary function of two arguments). We use three methods to construct the exact solutions: (i) direct method for symmetry reductions, (ii) direct method of functional separation of variables (a special form of solutions with six undetermined functions is preset and particular solutions to an auxiliary ODE are used), and (iii) method of generalized separation of variables. Most of the solutions obtained are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for testing numerical and approximate analytical methods for nonlinear hydrodynamic-type PDEs and solving certain model problems. The direct method of functional separation of variables used in this paper can also be effective for constructing exact solutions to other nonlinear PDEs.

Unsteady axisymmetric boundary-layer equations: Transformations, properties, exact solutions, order reduction and solution method

The paper deals with equations describing the unsteady axisymmetric boundary layer on a body of revolution. The shape of the body is assumed to be arbitrary. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a plane boundary-layer equation with a streamwise-coordinate-dependent viscosity of the formwtz+wzwxz−wxwzz=νr2(x)wzzz+F(t,x).We describe a number of new generalized and functional separable solutions to this non-linear equation, which depend on two to five arbitrary functions. The solutions are obtained with a new method (direct method of functional separation of variables) based on using particular solutions to an auxiliary ODE. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Two theorems are stated that enable one to generalize exact solutions of unsteady axisymmetric boundary-layer equations by including additional arbitrary functions. Furthermore, we specify a von Mises-type transformation that reduces the unsteady axisymmetric boundary-layer equation to a non-linear second-order PDE. We also present several new exact solutions to the plane boundary-layer equation and solve a boundary layer problem for a non-uniformly heated flat plate in a unidirectional fluid flow with temperature dependent viscosity.The method proposed in this paper can also be effective for constructing exact solutions to many other non-linear PDEs.

Classification and Approximate Functional Separable Solutions to the Generalized Diffusion Equations with Perturbation

In this paper, the generalized diffusion equation with perturbation ut = A(u;ux)uII+eB(u;ux) is studied in terms of the approximate functional variable separation approach. A complete classification of these perturbed equations which admit approximate functional separable solutions is presented. Some approximate solutions to the resulting perturbed equations are obtained by examples.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = (A(u)ux)x + εB(u, ux). A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approximate solutions to the resulting perturbed equations are constructed via examples.

Approximate Functional Variable Separation for the Quasi-Linear Diffusion Equations withWeak Source

As an extension to the functional variable separation approach, the approximate functional variable separation approach is proposed, and it is applied to study the quasi-linear diffusion equations with weak source. A complete classification of these perturbed equations which admit approximate functional separable solutions is obtained. As a result, the corresponding approximate functional separable solutions to the resulting perturbed equations are derived via examples.

Approximate Derivative-Dependent Functional Variable Separation for the Generalized Diffusion Equations with Perturbation

As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.

Approximate functional variable separation for the porous medium equation with perturbed nonlinear source

The approximate functional variable separation for the porous medium equation with perturbed nonlinear source is studied. Complete classification of the perturbed equation which admits approximate functional separable solutions is obtained. The main solving procedure for the approximate functional variable separation approach is shown by way of examples, and the corresponding approximate functional separable solutions to the resulting equations are then constructed.

The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex "oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations.

Solutions and Conditional Lie–Bäcklund Symmetries of Quasi-linear Diffusion-Reaction Equations

New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Bäcklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.