Lecture 3 Differential Constraints Method Andrei D. Polyanin.

Lecture 3Lecture 3

Differential Constraints MethodDifferential Constraints Method

Andrei D. PolyaninAndrei D. Polyanin

Preliminary Remarks. A Simple ExamplePreliminary Remarks. A Simple ExampleAdditive separable solutions in the case of Additive separable solutions in the case of two independent variables are sought in two independent variables are sought in the formthe form

Further, differentiating (2) with respect to Further, differentiating (2) with respect to xx yieldsyields

At the initial stage, the functions At the initial stage, the functions ((xx)) and and ((yy)) are assumed arbitrary and are to be are assumed arbitrary and are to be determined in the subsequent analysis.determined in the subsequent analysis.

Differentiating (1) with respect to Differentiating (1) with respect to yy yields yields

Conversely, relation (2) implies a Conversely, relation (2) implies a representation of the solution in therepresentation of the solution in theform (1).form (1).

Conversely, from (3) we obtain a Conversely, from (3) we obtain a representation of the solution in the representation of the solution in the form (1).form (1).

Thus, the problem of finding exact solutions Thus, the problem of finding exact solutions of the form (1) for a specific partial of the form (1) for a specific partial differential equation may be replaced by andifferential equation may be replaced by anequivalent problem of finding exact equivalent problem of finding exact solutions of the given equation solutions of the given equation supplemented with condition (2) or (3). supplemented with condition (2) or (3). Such supplementary conditions in the form Such supplementary conditions in the form of one or several differential equations will of one or several differential equations will be called be called differential constraintsdifferential constraints..

Simple ExampleSimple ExampleConsider the boundary layer equation for Consider the boundary layer equation for stream functionstream function

Let us seek a solution of equation (1) Let us seek a solution of equation (1) satisfying the linear first-order satisfying the linear first-order differential differential constraintconstraint

The unknown function The unknown function ((yy)) must satisfy must satisfy the the condition of compatibility condition of compatibility of equations of equations (1) and (2).(1) and (2).

First stage.First stage. Successively differentiating (2) Successively differentiating (2) with respect to different variables, we with respect to different variables, we calculate the derivativescalculate the derivatives

It is the compatibility condition for Eqs. (1) It is the compatibility condition for Eqs. (1) and (2).and (2).

Differentiating (1) with respect to Differentiating (1) with respect to xx yields yields

Substituting the derivatives from (2) and (3) Substituting the derivatives from (2) and (3) into (4), we obtaininto (4), we obtain

Simple Example (continued)Simple Example (continued)Second stage.Second stage. In order to construct an In order to construct an exact solution, we integrate equation (2) to exact solution, we integrate equation (2) to obtainobtain

Third stage.Third stage. The function The function ((yy)) is found is found by substituting (6) into (1) and taking into by substituting (6) into (1) and taking into account condition (5). As a result, we arrive account condition (5). As a result, we arrive at the ordinary differential equationat the ordinary differential equation

Finally, we obtain an exact solution of the Finally, we obtain an exact solution of the form (6), with the functions form (6), with the functions and and described by equations (5) and (7).described by equations (5) and (7).

ReminderReminder

Boundary layer equation for stream Boundary layer equation for stream functionfunction

Differential constraintDifferential constraint

General Scheme for the Differential Constraints MethodGeneral Scheme for the Differential Constraints Method

Original equation: ( , , , , F x y w wx w w w wy xx xy yy, , , , ...) = 0

Find conditions for the equations = 0 and = 0F Gcompatbility

Solve the equations for the determining functions

Find an invariant manifold: ( , , , , , , , , ...) = 0g x y w w w w w wx y xx xy yy

Insert resulting solution (with arbitrariness) into original equation

Obtain an exact solution of the original equation

Introduce a supplementary equation

Perform compatibility analysis of the two equations

Obtain equations for the determining functions

Insert the solution into the differential constraint

Solve the equation = 0 for g w

Determine the unknown functions and constants

Differential constraint: ( , , , , , , , , ...) = 0G x y w w w w w wx y xx xy yy

Differential Constraints MethodDifferential Constraints MethodConsider a general second-order evolution Consider a general second-order evolution equation solved for the highest-order equation solved for the highest-order derivative:derivative:

Let us supplement this equation with a first-Let us supplement this equation with a first-order differential constraintorder differential constraint

The condition of compatibility of these The condition of compatibility of these

equations is equations is wwxxtxxtwwtxxtxx. Differentiating . Differentiating

(1) and (2), we find that(1) and (2), we find that

where where DDtt and and DDxx are the total differentiation are the total differentiation

operators with respect to operators with respect to tt and and xx::

The partial derivatives The partial derivatives wwtt, , wwxxxx, , wwxtxt, and , and wwtttt

here should be expressed in terms of here should be expressed in terms of xx, , tt, , ww, and , and wwxx by means of relations (1) and (2) by means of relations (1) and (2)

and those obtained by differentiation of (1) and those obtained by differentiation of (1) and (2).and (2).

Differential Constraints MethodDifferential Constraints MethodExampleExample (Galaktionov, 1994). Consider a (Galaktionov, 1994). Consider a nonlinear heat equation with a source of nonlinear heat equation with a source of general form:general form:

Consider differential constraints of simple Consider differential constraints of simple form:form:

Equations (4) and (5) are special cases ofEquations (4) and (5) are special cases of(1) and (2) with(1) and (2) with

The functions The functions f f ((ww)), , gg((ww)), and , and ((ww)) are are unknown in advance and are to be unknown in advance and are to be determined in the subsequent analysis.determined in the subsequent analysis.

Find partial derivatives and the total Find partial derivatives and the total differentiation operators:differentiation operators:

Differential Constraints Method. Example (continued)Differential Constraints Method. Example (continued)

In order to ensure that this equation holds In order to ensure that this equation holds

true for any true for any wwxx, one should set, one should set

Assuming that Assuming that ff f f ((ww)) is prescribed, we is prescribed, we find the solution of equations (6):find the solution of equations (6):

We substitute We substitute ((ww)) from (7) in equation from (7) in equation (5) and integrate to obtain(5) and integrate to obtain

On differentiating (8) with respect to On differentiating (8) with respect to xxand and tt, we get, we get

We insert the expressions of We insert the expressions of DDtt and and DDxx into into

the compatibility condition the compatibility condition DDtt FF = = DDxx22 GG

and rearrange terms to obtainand rearrange terms to obtain

On substituting these expressions into (4) On substituting these expressions into (4) and taking into account (7), we arrive at a and taking into account (7), we arrive at a linear constant-coefficient equation:linear constant-coefficient equation:

Generalized and Functional Separation of Generalized and Functional Separation of Variables vs. Differential ConstraintsVariables vs. Differential Constraints

Table 1: Second-order differential constraintsTable 1: Second-order differential constraintscorresponding to some classes of exact solutions representable in explicit formcorresponding to some classes of exact solutions representable in explicit form

Type of solutionType of solution Structure of solutionStructure of solution Differential constraintsDifferential constraints

Additive separableAdditive separable

Multiplicative separableMultiplicative separable

Generalized separableGeneralized separable


Functional separableFunctional separable


Table 2: Second-order differential constraintsTable 2: Second-order differential constraintscorresponding to some classes of exact solutions representable in explicit formcorresponding to some classes of exact solutions representable in explicit form

Type of solutionType of solution Structure of solutionStructure of solution Differential constraintsDifferential constraints





Direct Method for Similarity Reductions and Differential Direct Method for Similarity Reductions and Differential Constraints MethodConstraints Method

where where FF((xx,,tt,,uu)) and and zz((xx,,tt)) should be should be selected so as to obtain ultimately aselected so as to obtain ultimately asingle ODE for single ODE for uu((zz))..

Employing the solution structure (1) is Employing the solution structure (1) is equivalent to searching for a solution with equivalent to searching for a solution with the help of a first-order quasilinear the help of a first-order quasilinear differential constraint (Olver, 1994)differential constraint (Olver, 1994)

Indeed, first integrals of the characteristic Indeed, first integrals of the characteristic system of ODEssystem of ODEs

have the formhave the form

Therefore, the general solution of Therefore, the general solution of equation (2) can be written as follows:equation (2) can be written as follows:

A generalized similarity reduction based on A generalized similarity reduction based on a prescribed form of the desired solution a prescribed form of the desired solution (Clarkson, Kruskal, 1989)(Clarkson, Kruskal, 1989)

where where uu((zz)) is an arbitrary function. On is an arbitrary function. On

solving (3) for solving (3) for ww, we obtain a representation , we obtain a representation of the solution in the form (1).of the solution in the form (1).

Nonclassical Method for Similarity Reductions and Nonclassical Method for Similarity Reductions and Differential Constraints MethodDifferential Constraints Method

where where ((xx, , yy, , ww)), , ((xx, , yy, , ww)), and , and ((xx, , yy, , ww)) are unknown functions, are unknown functions,

and the coordinates of the first and the second prolongations and the coordinates of the first and the second prolongations ii and and ijij are defined by are defined by

formulas from the classical method of group analysis.formulas from the classical method of group analysis.

The method for the construction of exact solutions to equation (1) based on using the first-The method for the construction of exact solutions to equation (1) based on using the first-order partial differential equation (2) and the invariance condition (3) corresponds to the order partial differential equation (2) and the invariance condition (3) corresponds to the nonclassical method for similarity reduction (G. W. Bluman, J. D. Cole, 1969).nonclassical method for similarity reduction (G. W. Bluman, J. D. Cole, 1969).

Consider the general second-order equationConsider the general second-order equation

Let us supplement equation (1) with two differential constraintsLet us supplement equation (1) with two differential constraints

ReferenceReference

A. D. Polyanin and V. F. Zaitsev,A. D. Polyanin and V. F. Zaitsev,

Handbook of Nonlinear Partial Handbook of Nonlinear Partial Differential EquationsDifferential Equations,,

Chapman & Hall/CRC Press, 2003Chapman & Hall/CRC Press, 2003

Lecture 3 Differential Constraints Method Andrei D. Polyanin.

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Transcript of Lecture 3 Differential Constraints Method Andrei D. Polyanin.