ReducedForm receives two PDE systems, PDESYS_1 and PDESYS_2, in that order, and reduces PDESYS_1 with respect to PDESYS_2. In some sense, ReducedForm is the "differential equation" command equivalent to simplify/siderels. ReducedForm also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.

Generally speaking, that is equivalent to taking the highest derivative of each equation in PDESYS_2, isolating it and using the equation to replace occurrences of this highest derivative in PDESYS_1, and at the end simplifying the resulting system taking into account its integrability conditions. In more technical words, the output is the reduced form of PDESYS_1 modulo the radical differential ideal of PDESYS_2

If RedVars is not specified, ReducedForm will consider all the differentiated unknown functions in PDESYS_2 as (reducing) variables of the problems.

The reduction process performed by ReducedForm consists of the following steps

[-] Departing from PDESYS_1 = [eq1, eq2, ...], a new system PDESYS_tmp = [Y1 - eq1, Y1 - eq2, ..., Yn - eqn] is constructed introducing a set of auxiliary variables {Y1, ..., Yn}, where n is the number of equations in PDESYS_1.

[-] One new system of equations is now built with the equations in PDESYS_tmp and in PDESYS_2 and a differential elimination process is ran ranking the auxiliary variables Yn higher than the dependent (reducing) variables - say F1 ... Fk - of PDESYS_2. So this ranking is of the form [[Y1, ... Yn], [F1, ... Fk]] and hence the elimination process runs using lexicographical ordering when comparing any Yn with any Fk but with the less expensive total degree ordering when comparing the Yn between themselves and the Fk between themselves.

[-] From the output of the differential elimination process, only the equations containing the Yn are preserved, so substituting the auxiliary Yn by zero results in the desired reduction of PDESYS_1 with respect to PDESYS_2, with the maximum number of Fk removed from the problem.

Note that it is possible to reduce one PDE system (PDESYS_1) using another one (PDESYS_2) in cases where the two systems are not consistent with each other, and that generally speaking the reduction obtained is useless in that its solution does not solve any of the two given systems. So you may want to request a check for consistency before proceeding with the reduction - for this purpose add the keyword checkconsistency anywhere in the calling sequence passed to ReducedForm - an example of this is at the end of the Examples section.

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.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{ReducedForm}\,\mathrm{declare}\,\mathrm{diff\_table}\,\mathrm{dsubs}\right)$

$\left[\mathrm{ReducedForm}\,\mathrm{declare}\,\mathrm{diff\_table}\,\mathrm{dsubs}\right]$

$U≔\mathrm{diff\_table}\left(u\left(x\,y\,z\,t\right)\right)\:$

$\mathrm{declare}\left(u\left(x\,y\,z\,t\right)\right)$

$u\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$\mathrm{pde}\left[1\right]≔U\left[x,y,z\right]+U\left[t,y,z\right]+U\left[t,x,z\right]-U\left[t,t\right]$

${\mathrm{pde}}_1≔u_{x,y,z}+u_{t,y,z}+u_{t,x,z}-u_{t,t}$

$\mathrm{pde}\left[2\right]≔U\left[x\right]=0$

${\mathrm{pde}}_2≔u_x=0$

$\mathrm{ReducedForm}\left(\mathrm{pde}\left[1\right]\,\mathrm{pde}\left[2\right]\right)$

$\left[u_{t,y,z}-u_{t,t}\right]\mathbf{where}\left[\right]$

$\mathrm{dsubs}\left(\mathrm{pde}\left[2\right]\,\mathrm{pde}\left[1\right]\right)$

$u_{t,y,z}-u_{t,t}$

$\mathrm{PDEtools}:-\mathrm{casesplit}\left(\left[\mathrm{pde}\left[1\right]\,\mathrm{pde}\left[2\right]\right]\right)$

$\left[u_{t,y,z}=u_{t,t}\,u_x=0\right]\mathbf{where}\left[\right]$

$\mathrm{declare}\left(\left(u\,v\right)\left(x\,y\right)\right)$

$u\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}u$

$v\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}v$

$U,V≔\mathrm{diff\_table}\left(u\left(x\,y\right)\right),\mathrm{diff\_table}\left(v\left(x\,y\right)\right)\:$

$\mathrm{PDESYS\_2}≔\left[V\left[\right]U\left[x,x\right]-V\left[x\right]\,U\left[x,y\right]V\left[y\right]\,{V\left[y,y\right]}^2-1\right]$

$\mathrm{PDESYS\_2}≔\left[u_{x,x}v-v_x\,u_{x,y}{v}_y\,{v_{y,y}}^2-1\right]$

$\mathrm{PDESYS\_1}≔\mathrm{diff}\left(\mathrm{map}\left(\mathrm{`^`}\,\mathrm{PDESYS\_2}\,2\right)\,x\right)-\mathrm{diff}\left(\mathrm{PDESYS\_2}\,y\right)$

$\mathrm{PDESYS\_1}≔\left[-u_{x,x,y}v-u_{x,x}{v}_y+v_{x,y}+2\left(u_{x,x}v-v_x\right)\left(u_{x,x,x}v+u_{x,x}{v}_x-v_{x,x}\right)\,-u_{x,y,y}{v}_y-u_{x,y}{v}_{y,y}+2u_{x,y}{v_y}^2{u}_{x,x,y}+2{u_{x,y}}^2{v}_y{v}_{x,y}\,-2v_{y,y}{v}_{y,y,y}+4\left({v_{y,y}}^2-1\right)v_{y,y}{v}_{x,y,y}\right]$

$\mathrm{ReducedForm}\left(\mathrm{PDESYS\_1}\,\mathrm{PDESYS\_2}\right)$

$\left[0\,0\,0\right]\mathbf{where}\left[\right]$

$\mathrm{ode}\left[1\right]≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{\left(-\mathrm{\alpha }-\mathrm{\beta }y\left(x\right)\right)x^2-2x\mathrm{diff}\left(y\left(x\right)\,x\right)}{x^2}$

${\mathrm{ode}}_1≔y_{x,x}=\frac{\left(-\mathrm{\alpha }-\mathrm{\beta }y\left(x\right)\right)x^2-2x{y}_x}{x^2}$

$\mathrm{ode}\left[2\right]≔\mathrm{diff}\left(y\left(x\right)\,x\right)=-\frac{h\left(y\left(x\right)-\mathrm{\kappa }\right)}{\mathrm{\rho }}$

${\mathrm{ode}}_2≔y_x=-\frac{h\left(y\left(x\right)-\mathrm{\kappa }\right)}{\mathrm{\rho }}$

$\mathrm{PDEtools}:-\mathrm{casesplit}\left(\left[\mathrm{ode}\left[1\right]\,\mathrm{ode}\left[2\right]\right]\right)$

Warning: System is inconsistent

$\mathrm{dsubs}\left(\mathrm{ode}\left[2\right]\,\mathrm{ode}\left[1\right]\right)$

$\frac{h^2\left(y\left(x\right)-\mathrm{\kappa }\right)}{{\mathrm{\rho }}^2}=-\frac{y\left(x\right)\mathrm{\beta }\mathrm{\rho }x+\mathrm{\alpha }\mathrm{\rho }x-2hy\left(x\right)+2h\mathrm{\kappa }}{x\mathrm{\rho }}$

$\mathrm{isolate}\left(\,y\left(x\right)\right)$

$y\left(x\right)=\frac{-\mathrm{\alpha }{\mathrm{\rho }}^{2}x+h^{2}x\mathrm{\kappa }-2h\mathrm{\kappa }\mathrm{\rho }}{\mathrm{\beta }{\mathrm{\rho }}^{2}x+h^{2}x-2h\mathrm{\rho }}$

$\mathrm{odetest}\left(\,\left[\mathrm{ode}\left[1\right]\,\mathrm{ode}\left[2\right]\right]\right)$

$\left[\frac{h\left({\mathrm{\beta }}^{3}h\mathrm{\kappa }{\mathrm{\rho }}^4{x}^4+\mathrm{\alpha }{\mathrm{\beta }}^{2}h{\mathrm{\rho }}^4{x}^4-2{\mathrm{\beta }}^3\mathrm{\kappa }{\mathrm{\rho }}^5{x}^3+2{\mathrm{\beta }}^2{h}^3\mathrm{\kappa }{\mathrm{\rho }}^2{x}^4-2\mathrm{\alpha }{\mathrm{\beta }}^2{\mathrm{\rho }}^5{x}^3+2\mathrm{\alpha }\mathrm{\beta }{h}^3{\mathrm{\rho }}^2{x}^4-8{\mathrm{\beta }}^2{h}^2\mathrm{\kappa }{\mathrm{\rho }}^3{x}^3+\mathrm{\beta }{h}^5\mathrm{\kappa }{x}^4-8\mathrm{\alpha }\mathrm{\beta }{h}^2{\mathrm{\rho }}^3{x}^3+\mathrm{\alpha }{h}^5{x}^4+8{\mathrm{\beta }}^{2}h\mathrm{\kappa }{\mathrm{\rho }}^4{x}^2-6\mathrm{\beta }{h}^4\mathrm{\kappa }\mathrm{\rho }{x}^3+8\mathrm{\alpha }\mathrm{\beta }h{\mathrm{\rho }}^4{x}^2-6\mathrm{\alpha }{h}^4\mathrm{\rho }{x}^3+12\mathrm{\beta }{h}^3\mathrm{\kappa }{\mathrm{\rho }}^2{x}^2+12\mathrm{\alpha }{h}^3{\mathrm{\rho }}^2{x}^2-8\mathrm{\beta }{h}^2\mathrm{\kappa }{\mathrm{\rho }}^{3}x-8\mathrm{\alpha }{h}^2{\mathrm{\rho }}^{3}x-8\mathrm{\beta }h\mathrm{\kappa }{\mathrm{\rho }}^4-8\mathrm{\alpha }h{\mathrm{\rho }}^4\right)}{x{\left(\mathrm{\beta }{\mathrm{\rho }}^{2}x+h^{2}x-2h\mathrm{\rho }\right)}^3}\,-\frac{\mathrm{\rho }h\left({\mathrm{\beta }}^2\mathrm{\kappa }{\mathrm{\rho }}^2{x}^2+\mathrm{\alpha }\mathrm{\beta }{\mathrm{\rho }}^2{x}^2+\mathrm{\beta }{h}^2\mathrm{\kappa }{x}^2+\mathrm{\alpha }{h}^2{x}^2-2\mathrm{\beta }h\mathrm{\kappa }\mathrm{\rho }x-2\mathrm{\alpha }h\mathrm{\rho }x-2\mathrm{\beta }\mathrm{\kappa }{\mathrm{\rho }}^2-2\mathrm{\alpha }{\mathrm{\rho }}^2\right)}{{\left(\mathrm{\beta }{\mathrm{\rho }}^{2}x+h^{2}x-2h\mathrm{\rho }\right)}^2}\right]$

$\mathrm{ReducedForm}\left(\mathrm{ode}\left[1\right]\,\mathrm{ode}\left[2\right]\right)$

$\left[\frac{{\mathrm{\rho }}^{2}y\left(x\right)\mathrm{\beta }x+\mathrm{\alpha }{\mathrm{\rho }}^{2}x+y\left(x\right)h^{2}x-h^{2}x\mathrm{\kappa }-2\mathrm{\rho }y\left(x\right)h+2h\mathrm{\kappa }\mathrm{\rho }}{{\mathrm{\rho }}^{2}x}\right]\mathbf{where}\left[\right]$

$\mathrm{isolate}\left(\mathrm{op}\left(\left[1\,1\right]\,\right)\,y\left(x\right)\right)$

$y\left(x\right)=\frac{-\mathrm{\alpha }{\mathrm{\rho }}^{2}x+h^{2}x\mathrm{\kappa }-2h\mathrm{\kappa }\mathrm{\rho }}{\mathrm{\beta }{\mathrm{\rho }}^{2}x+h^{2}x-2h\mathrm{\rho }}$

$\mathrm{ReducedForm}\left(\mathrm{ode}\left[1\right]\,\mathrm{ode}\left[2\right]\,\mathrm{checkconsistency}\right)$

Error, (in PDEtools:-ReducedForm) the given systems of equations are inconsistent with each other

$\mathrm{with}\left(\mathrm{Physics}\right)$

$\left[\mathrm{`*`}\,\mathrm{`.`}\,\mathrm{Annihilation}\,\mathrm{AntiCommutator}\,\mathrm{Antisymmetrize}\,\mathrm{Assume}\,\mathrm{Bra}\,\mathrm{Bracket}\,\mathrm{Check}\,\mathrm{Christoffel}\,\mathrm{Coefficients}\,\mathrm{Commutator}\,\mathrm{CompactDisplay}\,\mathrm{Coordinates}\,\mathrm{Creation}\,\mathrm{D\_}\,\mathrm{Dagger}\,\mathrm{Decompose}\,\mathrm{Define}\,\mathrm{D\γ}\,\mathrm{DiracConjugate}\,\mathrm{Einstein}\,\mathrm{EnergyMomentum}\,\mathrm{Expand}\,\mathrm{ExteriorDerivative}\,\mathrm{Factor}\,\mathrm{FeynmanDiagrams}\,\mathrm{FeynmanIntegral}\,\mathrm{Fundiff}\,\mathrm{Geodesics}\,\mathrm{GrassmannParity}\,\mathrm{Gtaylor}\,\mathrm{Intc}\,\mathrm{Inverse}\,\mathrm{Ket}\,\mathrm{KillingVectors}\,\mathrm{KroneckerDelta}\,\mathrm{LagrangeEquations}\,\mathrm{LeviCivita}\,\mathrm{Library}\,\mathrm{LieBracket}\,\mathrm{LieDerivative}\,\mathrm{Normal}\,\mathrm{NumericalRelativity}\,\mathrm{Parameters}\,\mathrm{PerformOnAnticommutativeSystem}\,\mathrm{Projector}\,\mathrm{Psigma}\,\mathrm{Redefine}\,\mathrm{Ricci}\,\mathrm{Riemann}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{SortProducts}\,\mathrm{SpaceTimeVector}\,\mathrm{StandardModel}\,\mathrm{Substitute}\,\mathrm{SubstituteTensor}\,\mathrm{SubstituteTensorIndices}\,\mathrm{SumOverRepeatedIndices}\,\mathrm{Symmetrize}\,\mathrm{TensorArray}\,\mathrm{Tetrads}\,\mathrm{ThreePlusOne}\,\mathrm{ToContravariant}\,\mathrm{ToCovariant}\,\mathrm{ToFieldComponents}\,\mathrm{ToSuperfields}\,\mathrm{Trace}\,\mathrm{TransformCoordinates}\,\mathrm{Vectors}\,\mathrm{Weyl}\,\mathrm{`^`}\,\mathrm{dAlembertian}\,\mathrm{d\_}\,\mathrm{diff}\,\mathrm{g\_}\,\mathrm{gamma\_}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\left\{Q\,\mathrm{\theta }\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\,\mathrm{\theta }\right\}\right]$

$\mathrm{PDEtools}:-\mathrm{declare}\left(Q\left(x\,y\,\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)\right)$

$Q\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}Q$

$q≔\mathrm{PDEtools}:-\mathrm{diff\_table}\left(Q\left(x\,y\,\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)\right)\:$

$\mathrm{pde}\left[1\right]≔q\left[x,y,\mathrm{\theta }\left[1\right]\right]+q\left[x,y,\mathrm{\theta }\left[2\right]\right]-q\left[y,\mathrm{\theta }\left[1\right],\mathrm{\theta }\left[2\right]\right]=0$

${\mathrm{pde}}_1≔Q_{x,y,{\mathrm{\theta }}_1}+Q_{x,y,{\mathrm{\theta }}_2}-Q_{y,{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0$

$\mathrm{pde}\left[2\right]≔q\left[\mathrm{\theta }\left[1\right]\right]=0$

${\mathrm{pde}}_2≔Q_{{\mathrm{\theta }}_1}=0$

$\mathrm{ReducedForm}\left(\mathrm{pde}\left[1\right]\,\mathrm{pde}\left[2\right]\right)$

$\left[Q_{x,y,{\mathrm{\theta }}_2}\right]\mathbf{where}\left[\right]$

$\mathrm{pdsolve}\left(\right)$

$\left\{Q=\mathrm{f\_\_9}\left(x\,y\right)\mathrm{\_\λ1}+\mathrm{f\_\_10}\left(x\,y\right){\mathrm{\theta }}_1+\left(\mathrm{f\_\_6}\left(x\right)+\mathrm{f\_\_5}\left(y\right)\right){\mathrm{\theta }}_2+\left(\mathrm{f\_\_8}\left(x\right)+\mathrm{f\_\_7}\left(y\right)\right)\mathrm{\_\λ2}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}$