exact solutions to nonlinear equations and systems of equations of general form in mathematical...
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Exact solutions to nonlinear Exact solutions to nonlinear equations and systems of equations and systems of equations of general formequations of general formin mathematical physicsin mathematical physics
Andrei PolyaninAndrei Polyanin11, Alexei Zhurov, Alexei Zhurov1,21,2
11 Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow2 2 Cardiff University, Cardiff, Wales, UKCardiff University, Cardiff, Wales, UK
Generalized Separation of VariablesGeneralized Separation of Variables
General form of exact solutions:General form of exact solutions:
Partial differential equations with quadratic or Partial differential equations with quadratic or power nonlinearities:power nonlinearities:
On substituting expression (1) into the differential equation (2), one arrives at a On substituting expression (1) into the differential equation (2), one arrives at a functional-differential equationfunctional-differential equation
for the for the i i ((xx)) and and ii ( ( yy)). The functionals . The functionals jj ((XX)) and and j j ((Y Y )) depend only on depend only on xx and and
yy, respectively,, respectively,
The formulas are written out for the case of a second-order equation (2).The formulas are written out for the case of a second-order equation (2).
Solution of Functional-Differential Solution of Functional-Differential Equations by DifferentiationEquations by Differentiation
General form of exact solutions:General form of exact solutions:
1. Assume that 1. Assume that kk is not identical zero for some is not identical zero for some kk. . Dividing the equation by Dividing the equation by kk
and differentiating w.r.t. and differentiating w.r.t. yy, we obtain a similar equation but with fewer terms, we obtain a similar equation but with fewer terms
2. We continue the above procedure until a simple separable two-term equation 2. We continue the above procedure until a simple separable two-term equation is obtained:is obtained:
3. The case 3. The case kk 00 should be treated separately (since we divided the should be treated separately (since we divided the
equation by equation by k k at the first stage).at the first stage).
Information on Solution Methods Information on Solution Methods
A.D. Polyanin, V.F. Zaitsev, A.I. Zhurov, Solution methods for A.D. Polyanin, V.F. Zaitsev, A.I. Zhurov, Solution methods for nonlinear equations of mathematical physics and mechanics (in nonlinear equations of mathematical physics and mechanics (in Russian). Moscow: Fizmatlit, 2005.Russian). Moscow: Fizmatlit, 2005.http://eqworld.ipmnet.ru/en/education/edu-pde.htmhttp://eqworld.ipmnet.ru/en/education/edu-pde.htm
Methods for solving mathematical equationsMethods for solving mathematical equationshttp://eqworld.ipmnet.ru/en/methods.htmhttp://eqworld.ipmnet.ru/en/methods.htmhttp://eqworld.ipmnet.ru/ru/methods.htmhttp://eqworld.ipmnet.ru/ru/methods.htm
A.D. Polyanin, Lectures on solution methods for nonlinear partial A.D. Polyanin, Lectures on solution methods for nonlinear partial differential equations of mathematical physics, 2004.differential equations of mathematical physics, 2004.http://eqworld.ipmnet.ru/en/education/edu-pde.htmhttp://eqworld.ipmnet.ru/en/education/edu-pde.htmhttp://eqworld.ipmnet.ru/ru/education/edu-pde.htmhttp://eqworld.ipmnet.ru/ru/education/edu-pde.htm
Exact Solutions to Exact Solutions to Nonlinear Systems ofNonlinear Systems of
EquationsEquations
Generalized separation of variables for nonlinear systemsGeneralized separation of variables for nonlinear systems
We look for nonlinear systems (1), and also their generalizations, that We look for nonlinear systems (1), and also their generalizations, that admit exact solutions in the form:admit exact solutions in the form:
The functions The functions ((ww), ), ((ww), ), ((ww), ), andand ((ww)) are selected so that both equations are selected so that both equations
of system (1) produce the same equation for of system (1) produce the same equation for ((xx,,tt)) ..
Consider systems of nonlinear second-order equations:Consider systems of nonlinear second-order equations:
(1)
Such systems often arise in the theory of mass exchange of reactive media, Such systems often arise in the theory of mass exchange of reactive media, combustion theory, mathematical biology, and biophysics.combustion theory, mathematical biology, and biophysics.
Nonlinear systems. Example 1Nonlinear systems. Example 1
We seek exact solutions in the form:We seek exact solutions in the form:
Let us require that the argument Let us require that the argument bubucwcw is dependent on is dependent on t t only:only:
Consider the nonlinear systemConsider the nonlinear system
(1)
The functions The functions ff((zz), ), gg11((zz) ) andand gg22((zz)) are arbitrary are arbitrary ..
It follows thatIt follows that
Nonlinear systems. Example 1 (continued)Nonlinear systems. Example 1 (continued)
Then Then ((xx, , tt) ) satisfies the linear heat equationsatisfies the linear heat equation
()
For the two equations to coincide, we must require thatFor the two equations to coincide, we must require that
This leads to the following equationsThis leads to the following equations
Nonlinear systems. Example 1 (continued)Nonlinear systems. Example 1 (continued)
Eventually we obtain the following Eventually we obtain the following exact solution:exact solution:
Nonlinear system:Nonlinear system:
(1)
From (*) we find thatFrom (*) we find that
Nonlinear systems. Example 2Nonlinear systems. Example 2
wherewhere
Nonlinear system:Nonlinear system:
It admits exact solutions of the form It admits exact solutions of the form
Nonlinear systems. Example 3Nonlinear systems. Example 3
where where tt and and rr rr((xx, , tt) ) satisfy the equations satisfy the equations
Nonlinear system:Nonlinear system:
Exact solution 1:Exact solution 1:
Exact solution 2:Exact solution 2:
Exact solution 3:Exact solution 3:
Nonlinear systems. Example 4Nonlinear systems. Example 4
where where tt and and rr rr((xx, , tt) ) satisfy the equations satisfy the equations
Nonlinear system:Nonlinear system:
Exact solution:Exact solution:
Nonlinear systems. Example 5Nonlinear systems. Example 5
where where tt and and rr rr((xx, , tt) ) satisfy the equations satisfy the equations
Nonlinear system:Nonlinear system:
Exact solution 1:Exact solution 1:
Exact solution 2:Exact solution 2:
where where LL is an arbitrary linear differential operator in is an arbitrary linear differential operator in xx (of any order with (of any order with respect to the derivatives); the coefficients can depend on respect to the derivatives); the coefficients can depend on xx..
Nonlinear systems. Example 6Nonlinear systems. Example 6
where where tt and and rr rr((xx, , tt) ) satisfy the equations satisfy the equations
Nonlinear system:Nonlinear system:
Exact solution:Exact solution:
where where LL is an arbitrary linear differential operator in is an arbitrary linear differential operator in xx (of any order with (of any order with respect to the derivatives); the coefficients can depend on respect to the derivatives); the coefficients can depend on xx..
Nonlinear systems. Example 7Nonlinear systems. Example 7
where where tt and and rr rr((xx, , tt) ) satisfy the equations satisfy the equations
Nonlinear system:Nonlinear system:
Exact solution:Exact solution:
where where LL is an arbitrary linear differential operator in is an arbitrary linear differential operator in xx (of any order with (of any order with respect to the derivatives); the coefficients can depend on respect to the derivatives); the coefficients can depend on xx..
Nonlinear wave equations. Example 1Nonlinear wave equations. Example 1
Nonlinear equation:Nonlinear equation:
Arises in wave and gas dynamics.Arises in wave and gas dynamics.
Functional separable solutions in implicit form:Functional separable solutions in implicit form:
where where ((ww) ) andand ((ww)) are arbitrary functions. are arbitrary functions.
Nonlinear wave equations. Example 2Nonlinear wave equations. Example 2
Nonlinear Nonlinear nn-dimensional equation:-dimensional equation:
Functional separable solutions in implicit form:Functional separable solutions in implicit form:
where where ((ww), …, ), …, nn((ww), ), ((ww), ), andand ((ww)) are arbitrary functions, and the function are arbitrary functions, and the function
nn((ww) ) satisfies the normalization condition satisfies the normalization condition
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, 2004Chapman & Hall/CRC Press, 2004
Thank youThank you