compute function field solutions for (differential or not) system of equations. - Maple Help

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PDEtools[FunctionFieldSolutions] - compute function field solutions for (differential or not) system of equations.

 Calling Sequence FunctionFieldSolutions(PDESYS, DepVars, options=value)

Parameters

 PDESYS - equation or a set or list of them; it can include PDEs, ODEs, non-differential equations and inequations DepVars - (optional) may be required; function, or a set or list of them; the dependent variables of the problem ivars = ... - (optional) name or set of names; additional independent variables, typically not present in the dependency of the unknowns DepVars mathfunctiondegree = ... - (optional) non-negative integer, or a set of equations where the lhs is one of unknowns (DepVars) and the rhs is a non-negative integer; specifies the upper bound degree for the math functions in its solution numberofsolutions = ... - (optional) non-negative integer or the keyword all; specifies the number of polynomial solutions desired. The default is all having degree as upper bound. orderofderivatives = ... - (optional) non-negative integer, or a set of equations where the lhs is one of unknowns (DepVars) and the rhs is a non-negative integer; specifies the upper bound order for derivatives in its solution polynomialdegree = ... - (optional) non-negative integer, or a set of equations where the lhs is one of unknowns (DepVars) and the rhs is a non-negative integer; specifies the upper bound degree for the degree of polynomials in its solution polynomialdependency = ... - (optional) non-negative integer, a range of them, a name, or a set of names, or a set or list containing any of the previous objects; dependency of the solution simplifier = ... - (optional) simplifier to be used instead of the default simplify/size

Description

 • Given PDESYS, a system of equations for DepVars, possibly including inequations and non-differential equations, FunctionFieldSolutions computes solutions that are polynomial in the mathematical functions of PDESYS and its derivatives, with coefficients that are polynomial in the independent variables. The independent variables are typically the variables DepVars depend on, plus any others optionally specified with ivars = .... FunctionFieldSolutions also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.
 • If DepVars is not given, FunctionFieldSolutions will consider all the differentiated unknown functions in PDESYS as unknowns of the problems. Specifying DepVars however permits not only restricting the unknowns in different ways but also specifying unknowns of the problems which do not appear differentiated in PDESYS.
 • FunctionFieldSolutions uses a refinement of the heuristic used by PolynomialSolutions to compute an upper bound for degree of the solution with respect to the mathematical functions entering PDESYS and their derivatives. The differential order of the derivatives of the mathematical functions of PDESYS entering the solution is equal to the maximal differential order of the equations of PDESYS. The coefficients of these mathematical functions and their derivatives are in turn polynomial in the independent variables, of degree determined by UpperBounds.
 • FunctionFieldSolutions is automatically invoked by Infinitesimals, InvariantSolutions, and SimilaritySolutions when they receive the optional argument typeofsymmetry = functionfield in which case the dependency of the polynomials entering the coefficients of the mathematical functions, as well as their degree, can also be specified directly to those commands using their dependency and degree options.
 • To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$

Consider the following linear system in $F\left(x,y\right),G\left(x,y\right)$ that involves the mathematical function $\mathrm{exp}$

 > $\mathrm{declare}\left(\left(F,G\right)\left(x,y\right)\right)$
 ${F}{}\left({x}{,}{y}\right){}{\mathrm{will now be displayed as}}{}{F}$
 ${G}{}\left({x}{,}{y}\right){}{\mathrm{will now be displayed as}}{}{G}$ (1)
 > ${\mathrm{sys}}_{1}:=\left\{-\left(\frac{\partial }{\partial y}F\left(x,y\right)\right)+\frac{\partial }{\partial x}F\left(x,y\right)=-{ⅇ}^{x}+\frac{1}{y}+{ⅇ}^{x}y+\frac{1}{x},-\left(\frac{\partial }{\partial y}G\left(x,y\right)\right)+\frac{\partial }{\partial x}G\left(x,y\right)=-{ⅇ}^{x}-\frac{1}{y}+{ⅇ}^{x}y+\frac{1}{x}\right\}$
 ${{\mathrm{sys}}}_{{1}}{:=}\left\{{-}{{F}}_{{y}}{+}{{F}}_{{x}}{=}{-}{{ⅇ}}^{{x}}{+}\frac{{1}}{{y}}{+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{1}}{{x}}{,}{-}{{G}}_{{y}}{+}{{G}}_{{x}}{=}{-}{{ⅇ}}^{{x}}{-}\frac{{1}}{{y}}{+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{1}}{{x}}\right\}$ (2)

Typically, systems of this type do not admit polynomial solutions and frequently admit function field solutions:

 > $\mathrm{PolynomialSolutions}\left({\mathrm{sys}}_{1}\right)$
 > ${\mathrm{sol}}_{1}:=\mathrm{FunctionFieldSolutions}\left({\mathrm{sys}}_{1}\right)$
 ${{\mathrm{sol}}}_{{1}}{:=}\left\{{F}{=}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{ln}}{}\left({x}\right){-}{\mathrm{ln}}{}\left({y}\right){+}{\left({x}{+}{y}\right)}^{{2}}{}{\mathrm{_C6}}{+}{\mathrm{_C5}}{}{x}{+}{\mathrm{_C5}}{}{y}{+}{\mathrm{_C4}}{,}{G}{=}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{ln}}{}\left({y}\right){+}{\left({x}{+}{y}\right)}^{{2}}{}{\mathrm{_C3}}{+}{\mathrm{_C2}}{}{x}{+}{\mathrm{_C2}}{}{y}{+}{\mathrm{_C1}}\right\}$ (3)

Verify this solution

 > $\mathrm{pdetest}\left({\mathrm{sol}}_{1},{\mathrm{sys}}_{1}\right)$
 $\left\{{0}\right\}$ (4)

Compare with the general solution, that for this example is computable by pdsolve

 > $\mathrm{pdsolve}\left({\mathrm{sys}}_{1}\right)$
 $\left\{{F}{=}{{ⅇ}}^{{x}}{}{y}{-}{\mathrm{ln}}{}\left({-}{y}\right){+}{\mathrm{_F1}}{}\left({x}{+}{y}\right){+}{\mathrm{ln}}{}\left({x}\right){,}{G}{=}{{ⅇ}}^{{x}}{}{y}{-}{\mathrm{ln}}{}\left({-}{y}\right){+}{\mathrm{ln}}{}\left({x}\right){+}{2}{}{\mathrm{ln}}{}\left({y}\right){+}{\mathrm{_F2}}{}\left({x}{+}{y}\right)\right\}$ (5)

The following system is similar to ${\mathrm{sys}}_{1}$ but contains nonlinear terms and its solution is out of reach for pdsolve

 > ${\mathrm{sys}}_{2}:=\left\{-F\left(x,y\right)\left(\frac{\partial }{\partial y}F\left(x,y\right)\right)+\frac{\partial }{\partial x}F\left(x,y\right)=-F\left(x,y\right)\left({ⅇ}^{x}-\frac{1}{y}\right)+{ⅇ}^{x}y+\frac{1}{x},-F\left(x,y\right)\left(\frac{\partial }{\partial y}G\left(x,y\right)\right)+\frac{\partial }{\partial x}G\left(x,y\right)=-F\left(x,y\right)\left({ⅇ}^{x}+\frac{1}{y}\right)+{ⅇ}^{x}y+\frac{1}{x}\right\}$
 ${{\mathrm{sys}}}_{{2}}{:=}\left\{{-}{F}{}{{F}}_{{y}}{+}{{F}}_{{x}}{=}{-}{F}{}\left({{ⅇ}}^{{x}}{-}\frac{{1}}{{y}}\right){+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{1}}{{x}}{,}{-}{F}{}{{G}}_{{y}}{+}{{G}}_{{x}}{=}{-}{F}{}\left({{ⅇ}}^{{x}}{+}\frac{{1}}{{y}}\right){+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{1}}{{x}}\right\}$ (6)
 > ${\mathrm{sol}}_{2}:=\mathrm{FunctionFieldSolutions}\left({\mathrm{sys}}_{2}\right)$
 ${{\mathrm{sol}}}_{{2}}{:=}\left\{{F}{=}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{ln}}{}\left({x}\right){-}{\mathrm{ln}}{}\left({y}\right){+}{\mathrm{_C1}}{,}{G}{=}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{ln}}{}\left({y}\right){+}{\mathrm{_C2}}\right\}$ (7)
 > $\mathrm{pdetest}\left({\mathrm{sol}}_{2},{\mathrm{sys}}_{2}\right)$
 $\left\{{0}\right\}$ (8)

The following nonlinear system is also not solvable by pdsolve and its solution via FunctionFieldSolutions takes time to be computed; in cases like this one, the use of optional arguments to restrict the degrees in which the solution would depend on mathematical functions, can greatly diminish the computational time

 > ${\mathrm{sys}}_{3}:=\left\{-F\left(x,y\right)\left(\frac{\partial }{\partial y}F\left(x,y\right)\right)+\frac{\partial }{\partial x}F\left(x,y\right)=-F\left(x,y\right)\left({ⅇ}^{x}-\frac{1{ⅇ}^{\frac{x}{y}}}{y}\right)+{ⅇ}^{x}y+\frac{{ⅇ}^{\frac{x}{y}}}{x},-F\left(x,y\right)\left(\frac{\partial }{\partial y}G\left(x,y\right)\right)+\frac{\partial }{\partial x}G\left(x,y\right)=-F\left(x,y\right)\left({ⅇ}^{x}+\frac{\mathrm{cos}\left(xy\right)}{y}\right)+{ⅇ}^{x}y+\frac{\mathrm{cos}\left(xy\right)}{x}\right\}$
 ${{\mathrm{sys}}}_{{3}}{:=}\left\{{-}{F}{}{{F}}_{{y}}{+}{{F}}_{{x}}{=}{-}{F}{}\left({{ⅇ}}^{{x}}{-}\frac{{{ⅇ}}^{\frac{{x}}{{y}}}}{{y}}\right){+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{{ⅇ}}^{\frac{{x}}{{y}}}}{{x}}{,}{-}{F}{}{{G}}_{{y}}{+}{{G}}_{{x}}{=}{-}{F}{}\left({{ⅇ}}^{{x}}{+}\frac{{\mathrm{cos}}{}\left({y}{}{x}\right)}{{y}}\right){+}{{ⅇ}}^{{x}}{}{y}{+}\frac{{\mathrm{cos}}{}\left({y}{}{x}\right)}{{x}}\right\}$ (9)
 > ${\mathrm{sol}}_{3}:=\mathrm{FunctionFieldSolutions}\left({\mathrm{sys}}_{3},\mathrm{mathfunctiondegree}=1\right)$
 ${{\mathrm{sol}}}_{{3}}{:=}\left\{{F}{=}{\mathrm{Ei}}{}\left(\frac{{x}}{{y}}\right){+}{\mathrm{_C1}}{}{\mathrm{Si}}{}\left({y}{}{x}\right){-}{\mathrm{_C1}}{}{\mathrm{Ssi}}{}\left({y}{}{x}\right){+}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{_C3}}{,}{G}{=}{\mathrm{Ci}}{}\left({y}{}{x}\right){+}{\mathrm{_C4}}{}{\mathrm{Si}}{}\left({y}{}{x}\right){-}{\mathrm{_C4}}{}{\mathrm{Ssi}}{}\left({y}{}{x}\right){+}{{ⅇ}}^{{x}}{}{y}{+}{\mathrm{_C2}}\right\}$ (10)
 > $\mathrm{pdetest}\left({\mathrm{sol}}_{3},{\mathrm{sys}}_{3}\right)$
 $\left\{{0}\right\}$ (11)