Ordinary and Partial Differential Equations
Maple is the world leader in finding exact solutions to ordinary and partial differential equations. Maple 2021 extends that lead even further with new algorithms and techniques for solving more ODEs and PDEs. For Maple 2021, there are significant improvements in dsolve for the exact solution of 2ndorder linear ODEs using hypergeometric functions. The algorithms implemented are at the frontier of the understanding of this problem, and handle classes of extended equations with apparent singularities as well as the most common linear equations with 4 and 5 singularities.
For ODEs and PDEs, the LieAlgebrasOfVectorFields package in Maple 2021 has a new command MapDE , for analyzing the possible linearization of polynomially nonlinear equations, and determining the corresponding mapping when it exists, allowing in that way for the computation of more approximate and exact solutions.
The new Student[ODEs] package covers the material in a standard first course in ODEs and provides step-by-step tools for solving ODEs as well as interactive visualization. For details, see Student Packages.
Advanced type of Hypergeometric solutions for 2nd order linear ODEs
Linearization of polynomially nonlinear ODE and PDE
When they exist, 2F1, 1F1 and 0F1 hypergeometric solutions for 2ndorder linear ODEs are now computable in a rather general case. An equation with 3 regular singularities and any number of apparent singularities can now be solved in terms of 2F1 equations using the algorithms presented in (6) and (7). Likewise, an equation with 2 or 1 singularities, one of which is irregular, can now be solved in general using 1F1 and 0F1 hypergeometric functions using the algorithms presented in (1), (2) and (3). Common equations with up to 5 regularities (plus any number of apparent singularities) can now be solved using mappings and the tables from references (4) and (5). All this significantly extends the previous capabilities in solving linear equations, and also of higher order and nonlinear ODE and PDE that through a reduction or symmetry process require the solving of intermediate 2ndorder linear ODEs.
The new algorithms are now automatically used by dsolve and also accessible through a new DEtools command, hypergeometricsols.
Examples
Despite the powerful algorithms for hypergeometric solutions of previous Maple releases, none of the following examples could be solved. Now they can. This equation admits 2F1 solutions where the argument is polynomial in x of degree 3
ode__1≔ⅆ2ⅆx2yx=1296a−56x2+−4a3+109x+4a9−49xx−1b+216x−288432x−231+bx−1x2x−1ⅆⅆxyx−9a−56a−16x−232b41+bx−1x2x−1 yx:
dsolveode__1
yx=_C1F12112−a2,−a2+512;12;−bx−1x2+_C2x1−xF12712−a2,1112−a2;32;−bx−1x2
Verify this solution
odetest,ode__1
0
This other equation admits solutions in terms of modified Bessel functions of the first and second kind
ode__2≔ⅆ2ⅆx2yx=8x−482x−92x−15ⅆⅆxyx+4x6−96x5+931x4−4452x3+10539x2−12474x+97204x2x−322x−92x−15 yx:
dsolveode__2
yx=_C1x4x3−36x2+141x−225I1x2+−4x3+24x2−63x+27I2x2x−3+_C2x−4x3+36x2−141x+225K1x2+−4x3+24x2−63x+27K2x2x−3
Moreover, note that each independent solution involves a linear combination of functions. Indeed, while Bessel (or more generally, 0F1) solutions are related to equations with 1 singularity of irregular kind, this example has other regular singularities:
DEtoolssingularitiesode__2
regular=0,3,92,152,irregular=∞
thus requiring the use of a linear combination of pFq functions to construct a solution. A similar situation happens with the next example where, due to the presence of apparent singularities, a linear combination of - this time 1F1 - hypergeometric functions (KummerM and KummerU) is required to solve the problem
ode__3≔ⅆ2ⅆx2yx=16x4+32x2−36x+816x4−16x3+32x2−4xⅆⅆxyx−12x3+13x2+x−9416x4−16x3+32x2−4 x yx:
dsolveode__3
yx=_C112x−122M54,3,x+−20x+5M94,3,x+_C216x−122U54,3,x+20x−5U94,3,x
In addition to computing new solutions out of reach in previous releases, when the linear ODE is of Heun type, so it has 4 regular singularities or one of its confluent cases, and it happens to be one of the special Heun function cases that can be expressible using hypergeometric functions, both kinds of solutions are now computable. This equation is of the Heun triconfluent type
ode__4 ≔ ⅆ2ⅆx2yx = 4 x4−4 x2+10 x−2yx:
The algorithms in dsolve perceive that and solve it accordingly
dsolveode__4
yx=_C1ⅇ−23x3+xHT36234,−152,−613,x6233+_C2ⅇ23x3−xHT36234,152,−613,−x6233
By using the option of indicating the method, a solution in terms of linear combinations of Airy functions (of the 0F1) is now also computable
dsolveode__4,hypergeometricsols
yx=_C12x+1Ai′x2−1+2x2+x−1Aix2−1+_C22x+1Bi′x2−1+2x2+x−1Bix2−1
To express this solution in terms of the more general 0F1 form you can use
convert,hypergeom
yx=_C12x+1−3316Γ232F10;13;−−x2+139π+x2−12313F10;53;−−x2+1396Γ23+2x2+x−1313F10;23;−−x2+1393Γ23−x2−1316Γ23F10;43;−−x2+1392π+_C22x+13323Γ232F10;13;x2−139π+356x−12x+12F10;53;x2−139x−1x+16x2−6Γ23+2x2+x−1356F10;23;−−x2+1393Γ23+323x2−1Γ23F10;43;−−x2+1392π
A similar situation, this time with regards to this equation of the Heun biconfluent class
ode__5≔ⅆ2ⅆx2yx=−x2+3x+43xⅆⅆxyx+x+19x yx:
dsolveode__5
yx=_C1HB13,6,−3,1469,6x6ⅇ−xx+66+_C2HB−13,6,−3,1469,6x6ⅇ−xx+66x13
The solution is expressible in terms of linear combinations of modified Bessel functions of the 1st and 2nd kinds
dsolveode__5,hypergeometricsols
yx=_C1ⅇ−xx+612x2+6x+16I134+x4+xx12+4+x4+xxI434+x4+xx124+xx16+_C2ⅇ−xx+612x2+6x+16K134+x4+xx12−4+x4+xxK434+x4+xx124+xx16
The following example is of the same kind but more general: it belongs to the Heun general class, but again it is one of those special cases where pFq function solutions exist, in this case 2F1 with rational coefficients
ode__6≔ⅆ2ⅆx2yx=−345x2+14x+14x27x2+14x+3ⅆⅆxyx+3x+12b−3b+364x227x2+14x+3 yx:
dsolveode__6
yx=_C1HG−4I2+74I2+7,45927b+4−I2b14−3b28−15141904I2−5983+Ib2+10Ib+51I72+17b256+85b28−484I2+734I2−7256I2b−61904I2−5983+17b,1+b6,251505−6I2115−212301904I2−5983+bI2−32946012+b4I2+734I2−7256I2b−61904I2−5983+17b,b4+1,32,−27x4I2+7x38+b827x+4I2+734−27x+4I2−7−656157113+10604I2456I2+1724I2−724I2+7327x2+14x+314+_C2HG−4I2+74I2+7,1561518b−4−67Ib476+30I172−275b952+15281904I2−5983+Ib2−10Ib+17949I2382−5983b21904+29915b952+4856I2+174I2+734I2−7256I2b−61904I2−5983+17b,1−b12,−503010−6I2115−212301904I2−5983+bI2−329460−6+b4I2+734I2−7256I2b−61904I2−5983+17b,−b4+1,32,−27x4I2+7x38−b827x+4I2+734−27x+4I2−7−656157113+10604I2456I2+1724I2−724I2+7327x2+14x+314
dsolveode__6,hypergeometricsols
yx=_C11+143x+9x2b24−181−x3b12−14x38−b8F1258−b24,−b24+18;1−b12;−256x3x−3227x2+14x+3+_C21+143x+9x2−b24−181−x3−b12−14x38+b8F1258+b24,b24+18;1+b12;−256x3x−3227x2+14x+3
odetest,ode__6
References
(1) M. van Hoeij and Q. Yuan. {em Finding all Bessel type solutions for Linear Differential Equations with Rational Function Coefficients}, ISSAC'2010 Proceedings.
(2) R. Debeerst, M. van Hoeij, W. Koepf, Solving Differential Equations in Terms of Bessel Functions, ISSAC'2008 Proceedings.
(3) M. van Hoeij, J-A. Weil, Solving Second Order Linear Differential Equations with Klein's Theorem. ISSAC'2005 Proceedings.
(4) R. Vidunas, G. Filipuk: A Classification of Covering yielding Heun to Hypergeometric Reductions, Funkcialaj Ekvacioj (2013).
(5) M. van Hoeij, V. Kunwar, Classifying (near)-Belyi maps with Five Exceptional Points. Indagationes Mathematicae (2019).
(6) V. Kunwar and M. van Hoeij, Second Order Differential Equations with Hypergeometric Solutions of Degree Three, ISSAC'2013 Proceedings.
(7) E. Imamoglu and M. van Hoeij, Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases, J. of Symbolic Computation, (2017).
A fingerprint of a linear DE is that linear super-position of its solutions yields other solutions. Encoded geometrically this means that the input DE must have a subgroup of symmetries corresponding to this super-position symmetry (and in particular be an infinite abelian Lie subgroup in the case of linearizable PDE).
IsLinearizable and MapDE leverage the power of the LieAlgebrasOfVectorFields package, introduced in Maple 2020, by algorithmically computing and exploiting symmetry group properties through their associated Lie algebras. Lie symmetry groups, when linearized about the identity symmetry, yield Lie algebras of vector fields tangent to their one parameter group orbits.
This beautiful linearization yields determining systems of linear homogeneous PDE (LHPDE) for the components of the associated Lie algebra of vector fields.
Then algorithmic differential reduction and elimination algorithms are applied to these linear systems, to find both algebraic (e.g. center, lower central series, etc) and geometric (e.g. distribution, invariants, etc) properties of the Lie algebras of vector fields.
The LieAlgebrasOfVectorFields package is a complement to DifferentialGeometry. In particular, the LieAlgebras package in DifferentialGeometry enables the flexible formulation of geometric problems arising in applications, enabling the user to formulate and solve them using exterior differential systems and moving frames while working intrinsically on the relevant manifolds.
The IsLinearizable command illustrating the existence step of MapDE
The IsLinearizable command introduced in Maple 2020 illustrates some basic ideas underlying MapDE. In particular it uses LieAlgebrasOfVectorFields (LAVF) to implement a remarkable algorithm introduced by Lyakhov, Gerdt and Michels (2017).
To illustrate it, consider the nonlinear ODE:
withLieAlgebrasOfVectorFields:
NLODE≔uxⅆ3ⅆx3ux+3ⅆⅆxuxⅆ2ⅆx2ux−3ⅆⅆxux2+uxⅆ2ⅆx2ux+12uxⅆⅆxux+x−8xuxⅆⅆxux+x41+ux2+x2ux2+x2=0:
To check whether this ODE can be linearized you can use IsLinearizable. First, set up the symmetry vector field
X ≔ VectorFieldxix,u,x,etax,u,u, space = x,u
X≔ξx,u∂∂x+ηx,u∂∂u
Then apply IsLinearizable
IsLinearizableNLODE, X
true
Here true means that there exists an invertible change of variables that maps the nonlinear ODE to a linear ODE. IsLinearizable is a good example of how LAVF enables the efficient computation of geometric information, in this case the existence of a linearization map, to determine if more expensive methods can be applied to a problem (in this case to determine the linearization). Here we display the simple procedure, which uses the LAVF commands SymmetryLAVF, SolutionDimension, DerivedAlgebra and IsAbelian.
printIsLinearizable
procDEs::equation,X::LieAlgebrasOfVectorFields:-VectorField ≔ NULL, $localn,L,m,Da,S;ifDEs::equationthenS ≔ DEselseS ≔ DEsfi;n ≔ PDEtools[difforder]S;ifn<2thenreturntruefi;tryL ≔ LieAlgebrasOfVectorFields:-SymmetryLAVFS,Xcatch:errorend;m ≔ SolutionDimensionL;ifn=2andm=8thenreturntruefi;ifm=n+4thenreturntruefi;ifm=n+1orm=n+2thenDa ≔ DerivedAlgebraL;returnevalbIsAbelianDaandn=SolutionDimensionDafi;returnfalseend
In the procedure above, for our example, n = 3. SymmetryLAVF sets up the Lie algebra of vector fields, and m is the dimension of the Lie symmetry algebra m = 4 = n + 1 . The LAVF instruction IsAbelian(Da) yields true, without determining a basis for the Derived Algebra Da. Only algorithmic differentiations and eliminations of the determining system for Da were needed, based in an induced Lie algebra on the space of initial data for Da.
Apply MapDE to uux,x,x+3uxux,x−3uux,x+ux2+12uux+x−8xuux+x4u2+x2+1u2+x2=0
The IsLinearizable command in the previous section only determines the existence of a mapping of an input DE to some (unknown) linear (target) DE. Here we illustrate how MapDE determines more information about the mapping, and in some cases can determine the map explicitly.
Applying MapDE to NLODE:
We set the MapDE infolevel to get some information about the main steps of the computation:
infolevelMapDE ≔ 1:
MLODE ≔ MapDENLODE, x,u, xi,eta, ToLinearDE, xh, uh, psi, phi
MapDE: Begin section where we compute sys S' of subalgebra L' of L, containing solution of mapping problem.
MapDE: Single ODE so apply the Lyakhov-Gerdt-Michels ODE Test for Existence of Linearization
MapDE: Input DE is linearizable by Lyakhov-Gerdt-Michels ODE Test: dimL = 4
MapDE: Construct BK map eqns on Tangent Space to Jet Variety.
MapDE: Mapping equations constructed.
MapDE: Set up ranking and mindim specification for differential-elimination.
MapDE: In ToLinear and not CK.
MapDE: Settings for differential-elimination: MinDim = 3 ranking = [rho[1] xi eta] RifStrategy = ezcriteria = [1 .9]
MapDE: Apply differential-elimination (rifsimp) to Mapping Equations.
MapDE: Completed differential-elimination and casesplitting of Mapping System. Time in rifsimp = 0.45e-1
MapDE: Existence of linearization was found earlier. Now attempting to integrate mapping eqns
MapDE: Try to simplify PhiEqSol by setting values for the unknown functions, constants etc.
MapDE: Attempted simplification of PhiEq yielded nonzero Jacobian, now using simplified form
MLODE≔tableIntegratedMap=xh=u2+x2,uh=−x,MapDEPivs=∂∂uψx,u≠0,∂∂uφx,ux−∂∂xφx,uu≠0,TargetDE=ⅆ3ⅆxh3uhxh=−xh+1uhxhxhwhereuhxhⅆⅆxhuhxh≠0,−uhxh2+xh≠0,uhxh≠0,MapExists=true,RedMapDE=∂2∂x2φx,u=−∂2∂u2φx,uux2−2∂2∂u∂xφx,uu2x−∂∂uφx,uu2−∂∂uφx,ux2u3,∂∂xψx,u=∂∂uψx,uxu
Here MapDE has carried out the existence step described in the previous section, to determine that there exists a mapping [xh = psi(x,u), uh = phi(x,u)] of NLODE to a linear ODE. Then MapDE is guaranteed in theory to return canonical differential equations in phi and psi for the mapping MLODE[RedMapDE] subject to the inequality constraints MLODE[MapDEPivs]. MapDE then uses pdsolve to attempt to integrate those equations to obtain an explicit form for the mapping, obtaining in this case:
MLODE[IntegratedMap]
xh=u2+x2,uh=−x
The target linear ODE is also obtained:
MLODETargetDE
ⅆ3ⅆxh3uhxh=−xh+1uhxhxhwhereuhxhⅆⅆxhuhxh≠0,−uhxh2+xh≠0,uhxh≠0
Even though both NLODE and the considerably simpler linear MLODE can not be explicitly solved by dsolve, approximations for a basis of 3 solutions can be found by dsolve/series or dsolve/numeric. In this way the linearity of the target can be exploited to enable a general solution to be constructed without further significant computation.
Nonlinear PDE System linearizable by hodograph transformation vt=v3ux, ut=u5vx
infolevelMapDE ≔ 2:
NPDESys≔diffvx,t,t=vx,t3diffux,t,x,diffux,t,t=ux,t5diffvx,t,x
NPDESys≔∂∂tvx,t=vx,t3∂∂xux,t,∂∂tux,t=ux,t5∂∂xvx,t
We now compute the linearizing mapping and, when possible, also the resulting linear DE system, with MapDE
MPDESys3≔MapDENPDESys,x,t,u,v,ξ,τ,η,β,ToLinearDE,xh,th,uh,vh,ψ,ϒ,ϕ,φ
MapDE: Enter MapDE: Planned to have glossary of symbols and their meaning similar to rifsimp:
MapDE: ID = Initial Data
MapDE: Dim = Dimension
MapDE: DA = Derived Algebra
MapDE: R = input DE or its rif-form, Rh = Target DE or its rif-form
MapDE: S = DetSys Sym of R, S' = DetSys for SubAlg of Sym of S
MapDE: Input DE of order 1 in 2 dependent variable(s) 2 independent variable(s) 2 equation(s)
MapDE: Enter Fast Early Algebraic-Geometric Prop Test
MapDE: About to enter DerivedAlgebraInf
DerivedAlgebraInf: Enter DerivedAlgebraInf
DerivedAlgebraInf: Lie pseudogroup G of sym of DE: dim(G) = infinity So DE is possibly linearizable
DerivedAlgebraInf: Apply rif to commutator DA sys
DerivedAlgebraInf: ID for DA sys = [[] [xi(x[0] t[0] u[0] v) = _F1(v) tau(x[0] t[0] u[0] v) = _F2(v)]]
MapDE: Exited from DerivedAlgebraInf. Time in DerivedAlgebraInf = .588
MapDE: Settings for differential-elimination: MinDim = infinity ranking = [rho[1] rho[2] xi tau eta beta] RifStrategy = ezcriteria = [1 .9]
MapDE: For a detailed description of these settings see Maple's documentation on rifsimp.
MapDE: Completed differential-elimination and casesplitting of Mapping System. Time in rifsimp = 2.815
MapDE: After rif ncases = 46
MapDE: Number consistent cases satisfying minimum dim condns 2
MapDE: Multiple 2 cases found. Analyzing cases.
MapDE: nops(RifMapEqsSys)= 2 ntruecases= 2
MapDE: CaseSelectList = [2]
MapDE: Apply a pure elimination strategy to isolate the target and prepare to find the transformations.
MapDE: PsiPhiEqs have triangular structure wrt phi - psi vars
MPDESys3≔tableIntegratedMap=xh=u,th=v,uh=x,vh=t,MapDEPivs=∂∂xφx,t,u,v≠0,∂∂tϕx,t,u,v≠0,∂∂uψx,t,u,v∂∂vϒx,t,u,v−∂∂vψx,t,u,v∂∂uϒx,t,u,v≠0,TargetDE=∂∂xhvhxh,th=∂∂thuhxh,thxh5,∂∂thvhxh,th=∂∂xhuhxh,thth3,∂2∂xh2uhxh,th=∂2∂th2uhxh,thth3xh5where−∂∂thuhxh,th2th3+∂∂xhuhxh,th2xh5≠0,vhxh,th≠0,MapExists=true,RedMapDE=∂2∂x2φx,t,u,v=0,∂2∂x2ϕx,t,u,v=0,∂2∂t∂xϕx,t,u,v=0,∂2∂t2ϕx,t,u,v=0,∂∂xϒx,t,u,v=0,∂∂xψx,t,u,v=0,∂∂tϒx,t,u,v=0,∂∂tφx,t,u,v=0,∂∂tψx,t,u,v=0
Here MapDE finds an elegant hodograph form for the transformation to a linear system. Such hodograph transformations are important in applications, precisely because they linearize such problems. Notice that during the linearization analysis, MapDE detected highly non-trivial infinite dimensional Lie pseudogroups which were key to the analysis.
Fast certification of nonlinearizability of the KP Equation ux,t+3ux22+3uux,x2+ux,x,x,x4+3uy,y4=0
Consider the KP (Kadomtsev–Petviashvili) equation which arises from nonlinear wave motion, and is a generalization of the KdV (Korteweg–de Vries) equation to two dimensions:
infolevelMapDE ≔ 2
infolevelMapDE≔2
KPDE≔diffdiffux,y,t,t+32ux,y,tdiffux,y,t,x+14diffux,y,t,x,x,x,x+34diffux,y,t,y,y=0
KPDE≔∂2∂t∂xux,y,t+3∂∂xux,y,t22+3ux,y,t∂2∂x2ux,y,t2+∂4∂x4ux,y,t4+3∂2∂y2ux,y,t4=0
MKPDE≔MapDEKPDE,x,y,t,u,ξ,η,τ,β,ToLinearDE,xh,yh,th,uh,ψ,ϒ,ϕ,φ
MapDE: Input DE of order 4 in 1 dependent variable(s) 3 independent variable(s) 1 equation(s)
MapDE: Input R not linearizable since DiffDim(L}= 1 < 2 = DiffDim(R))
MKPDE≔tableMapExists=false
Although the KP equation has an infinite dimensional Lie pseudogroup of symmetries it is not linearizable, since the differential dimension (a measure of size of such pseudogroups) is not large enough (not equal to the differential dimension of solutions of the KP equation).
To solve the equation in cases like this one, note that despite the non-existence of a linearizing mapping, the underlying point symmetries of the equation can be used to compute invariant solutions. The existence of point symmetries is guaranteed when the PDEtools:-DeterminingPDE command returns related infinitesimals different from zero
PDEtools:-DeterminingPDEKPDE
∂2∂y2_ξyx,y,t,u=0,∂∂t_ξtx,y,t,u=3∂∂y_ξyx,y,t,u2,∂∂u_ξtx,y,t,u=0,∂∂x_ξtx,y,t,u=0,∂∂y_ξtx,y,t,u=0,∂∂u_ξxx,y,t,u=0,∂∂x_ξxx,y,t,u=∂∂y_ξyx,y,t,u2,∂∂y_ξxx,y,t,u=−2∂∂t_ξyx,y,t,u3,∂∂u_ξyx,y,t,u=0,∂∂x_ξyx,y,t,u=0,_ηux,y,t,u=−u∂∂y_ξyx,y,t,u+2∂∂t_ξxx,y,t,u3
Indeed, in this example, infinitesimals depending on three arbitrary functions are computable using PDEtools:-Infinitesimals
PDEtools:-InfinitesimalsKPDE
_ξxx,y,t,u=ⅆⅆt_F1tx2−ⅆ2ⅆt2_F1ty23−2ⅆⅆt_F2ty3+_F3t,_ξyx,y,t,u=ⅆⅆt_F1ty+_F2t,_ξtx,y,t,u=3_F1t2+1,_ηux,y,t,u=−uⅆⅆt_F1t+ⅆ2ⅆt2_F1tx3−2ⅆ3ⅆt3_F1ty29−4ⅆ2ⅆt2_F2ty9+2ⅆⅆt_F3t3
With the purpose of using these infinitesimals to compute solutions, this general form (2.4.5) can be specialized
PDEtools:-InfinitesimalsKPDE,specialize_Fn
_ξxx,y,t,u=0,_ξyx,y,t,u=1,_ξtx,y,t,u=1,_ηux,y,t,u=0,_ξxx,y,t,u=1,_ξyx,y,t,u=0,_ξtx,y,t,u=1,_ηux,y,t,u=0,_ξxx,y,t,u=0,_ξyx,y,t,u=0,_ξtx,y,t,u=52,_ηux,y,t,u=0,_ξxx,y,t,u=t,_ξyx,y,t,u=0,_ξtx,y,t,u=1,_ηux,y,t,u=23,_ξxx,y,t,u=−2y3,_ξyx,y,t,u=t,_ξtx,y,t,u=1,_ηux,y,t,u=0,_ξxx,y,t,u=x2,_ξyx,y,t,u=y,_ξtx,y,t,u=3t2+1,_ηux,y,t,u=−u
Using any of the infinitesimals above, for example the first one, we get a solution
PDEtools:-InvariantSolutionsKPDE,1
ux,y,t=−4_C24−4_C2_C3−3_C32cosh_C2x+_C3−y+t+_C12+12_C246cosh_C2x+_C3−y+t+_C12_C22
From the form of this solution one can infer this equation also admits travelling wave solutions
PDEtools:-TWSolutionsKPDE
ux,y,t=_C5,ux,y,t=−2_C22tanh_C2x+_C3y+_C4t+_C12+8_C24−4_C2_C4−3_C326_C22
(1) D.A. Lyakhov, V.P. Gerdt, and D.L. Michels. Algorithmic Verification of Linearizability for Ordinary Differential Equations. Proceedings of ISSAC 2017. ACM Press, 2017.
(2) Z. Mohammadi, G. Reid, and S.-L.T Huang. Introduction of the MapDE algorithm for determination of mappings relating differential equations. Proceedings of ISSAC 2019, pp. 331–338. ACM Press, 2019.
(3) Z. Mohammadi, G. Reid, and S.-L.T. Huang. Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE. Mathematics in Computer Science, 1–24, 2020.
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