The Method of Inverse Differential Operators Applied for the Solution of PDEs Robert Kragler Weingarten University of Applied Sciences Weingarten, Germany
[email protected] Abstract. In this paper the method of inverse differential operators for solving PDEs as given in [1] is implemented into Mathematica. A wide class of PDEs for which the differential polynomial can be decomposed into linear factors and for nonhomogenuities which comprise certain combinations of exponential, trigonometric and hyperbolic functions can be solved. In most cases the built-in Mathematica routine DSolve cannot find a solution.
1. Introduction The method of inverse differential operators which is well established for ordinary differential equations (ODEs) can be applied to certain classes of partial differential equations (PDEs). The inverse operator method to solve homogeneous and nonhomogeneous PDEs with constant coefficients was extended to PDEs by P.K. Kythe, P. Puri and M.R. Schaferkotter [1]. This method which was originally developed for solving ODEs turns likewise useful for finding general solutions of PDEs with constant coefficients and an nonhomogenuity Φ which could either be of exponential Φ1 = ãarg , trigonometric HΦ2 = sinHargL cosHargL) or hyperbolic (Φ2 = sinhHargL coshHargL ) type or any linear combination of an exponential Φ1 multiplied by resp. added to trigonometric or hyperbolic functions Φ2 such that Φ = Φ1 *Φ2 or Φ = Φ1 + Φ2 where arg could be any linear combination of (in principle) arbitrary selection of variables {t, x, y, z, Ξ, Η, Ζ}. In §2 the general solutions of homogeneous PDEs are investigated. It is essential for the method used that the differential polynomial ΧHD x , D y , ...L can be factorized into linear factors Hai Dx + bi D y + + ci L Κ with multiplicity Κ. In §3 a general scheme for the initial value problem of 2nd order PDEs with auxiliary contions u(x,0) = Φ(x) and ut Hx, 0L = Ψ’ HxL is discussed. The PDEs in question are homogeneous/nonhomogeneous 1d Wave equations and the 2d Laplace equation with various initial conditions given. In principle this technique should be applicable to any 2nd order PDE for which the differential polynomial ΧIDx 2 , D y 2 , ...M can be factorized into two linear factors 1 f HDx , D y , ...L * f HD x , D y , ...L conjugate to each other.
In §3 a general scheme for the initial value problem of 2nd order PDEs with auxiliary contions u(x,0) = Φ(x) and ut Hx, 0L = Ψ’ HxL is discussed. The PDEs in question are homogeneous/nonhomogeneous 1d Wave equations and the 2d Laplace equation with various initial conditions given. In principle this technique should be applicable to any 2nd order PDE for which the differential polynomial ΧIDx 2 , D y 2 , ...M can be factorized into two linear factors f HDx , D y , ...L * f HD x , D y , ...L conjugate to each other.
§4 is devoted to find the particular solution u p of nonhomogeneous PDEs. The method of inverse differential operators, although basically developed for solving nonhomogeneous ODEs, turns out to be likewise useful to determine the particular solutions of PDEs with constant coefficients, however, the functional form of the nonhomogenuity Φ is subject to certain restrictions which are causative to this technique - as will become obvious from the examples discussed later. As general remark it is noted that a step-by-step usage of Mathematica commands to achieve a particular solution is not too complicated. However, with the (ambitious) aim to implement a kind of "black box" with input only of the lhs of the PDE, i.e. the differential polynomial Χ HDx , D y , ...L and the rhs, i.e. the nonhomogenuity Φ = Φ1 × Φ2 or Φ = Φ1 + Φ2 and no further user intervention requires very sophisticated tools for pattern recognition as regards to Mathematica. The Mathematica version used is V7.0 but the ample toolbox of procedures and auxiliary routines developed should run under V5.2 and V8.0 likewise and has to be loaded before any calculations. In a toolbox given in §1.1 all the procedures and auxilary routines are collected. This is not a Mathematica package because there the notation with respect to indexed quantities etc. is limited. In principle any kind of symbol can be used as a variable for the differential operator related too. But the symbols used are restricted to the following reference list {t,x,y,z,Ξ,Η,Ζ} and the corresponding list for differentials 8Dt , Dx , Dy , Dz , DΞ , DΗ , DΖ
(rules
n odd
7,8); the + sign applies for hyperbolic, the - sign for trigonometric functions. Another important auxiliary routine is rationalizeX[X,subDD,onoff] which ’rationalizes’ the denominator of the inverse differential polynomial Χ-1 HDi , D j , ) . In analogy to complex conjugation z with the help of which the denominator of a complex number
1 z
containing
the
1 a+Di
=
a-Di a2 -Di 2
factors subDD
Þ
linear a-Di , a2 ± ci 2
in
=
1 a+ä b
=
a-ä b
a2 +b2
differential
becomes real, an expression operators
Di
,
i.e.
is simplified by application of replacement rules 7,8 .
However, if products such as Di × D j appear in the denominator the rationalizing (6) process has to be repeated until all differential operators Di , D j , are squared and thus can be replaced by I±ci 2 M, I±c j 2 M , ... . 10
denominator of a complex number
1 z
containing
the
1 a+Di
=
a-Di a2 -Di 2
factors subDD
Þ
linear a-Di , a2 ± ci 2
in
=
1 a+ä b
=
a-ä b a2 +b2
becomes real, an expression
differential
operators
Di
,
i.e.
is simplified by application of replacement rules 7,8 .
However, if products such as Di × D j appear in the denominator the rationalizing process has to be repeated until all differential operators Di , D j , are squared and 2 2 thus can be replaced by I±ci M, I±c j M , ... . Finally, the resulting particular function u p is checked with testPDE[Χ,Φ,up] whether it is a solution of the PDE given. By means of D2DRule the differential polynomial ΧIDi , D j , M is rewritten into a pure function polynomial of differentials (a ¶8x,n< ð + b ¶8y,m< ð + ... + cL & applied to u p @x, y, D which constitutes the lhs of the PDE, the rhs is simply Φ(x,y, ) . Auxiliary routines for the replacement rules where differential operators Di are converted into differential ¶8x,n< ð are ruleD2D, concatDRule1, uvPairs, pairsDiDj, Duv2Dxy and D2DMultiRule . In addition to the implemented solver it is investigated whether the built-in Mathematica procedure DSolve is able to find a solution for the PDE given. However, for most of the examples treated below this is NOT the case, hence our implementation of the method of inverse differential operators provides solutions for a special class of PDEs not generally covered by DSolve . Subsequent examples are investigated with the procedure nonhomogeneousPDEsolutions[Χ,Φ,onoff]. This is the essential procedure to obtain particular solutions of nonhomogeneous PDEs. This procedure works as a kind of ’black-box’.The only input is the differential polynomial Χ (written in terms of the Di operators) and the nonhomogenuity Φ = Φ1 Φ2 or Φ = Φ1 × Φ2 resp. Φ = Φ1 + Φ2 . It should be pointed out that the coefficients of Χ and Φ can either be numbers, e.g. {1,-3}, or symbols, e.g. {Α,Β}, or a mixture of both types. Exponential nonhomogenuity :
Φ = Φ1 = ãH
L
Example 1 : ( 3 D x 2 + 4 Dx D y - D y M uHx, yL = ã x-3 y Þ up = - 16 ãx-3 y
Example 2 : ID x 2 + 2 Dx D y 3 + 3 D y 5 M uHx, yL = A ãΑ x+ Β y Þ
up =
A ãx Α+y Β Α Β3 +3 Β5
Α2 +2
Trigonometric nonhomogenuity : Φ = Φ2 = sin cosH
11
L
Example 3 : I3 Dx 2 - D y M uHx, yL = sin H a x + b yL Þ
up =
b Cos@a x+b yD-3 a2 Sin@a x+b yD 9 a4 +b2
Example 4 : I3 Dx 2 - D y + 4 Dz M uHx, yL = sin H a x + b y + c zL Þ
up =
Hb-4 cL Cos@a x+b y+c zD-3 a2 Sin@a x+b y+c zD 9 a4 +Hb-4 cL2
Example 5 : H3 D x - D y L uHx, yL = sin H a x + b yL Þ
up = -
Cos@a x+b yD 3 a-b
This is one of the rare cases where DSolve is able to find the complete solution of this 1st order PDE Example 6 : I3 Dx 2 - D y 2 M uHx, yL = sin H a x + b yL Þ
up =
Sin@a x+b yD -3 a2 +b2
Following two examples require a special treatment of the prefactor Ρ , otherwise analyzeΦ would determine as header "Times" which would be misleading. Example 7 : ID x 4 - 10 Dx 2 D y 2 + 9 D y 4 M uHx, yL = 135 sin H3 x + 2 yL Þ u p = -Sin@3 x + 2 y D The correct u = u h + up =
answer
for
the
complete
solution
is
:
f10 H3 x + yL + f20 H3 x - yL + f30 Hx + yL + f40 Hx - yL - sinH3 x + 2 yL
Example 8 : ID x 4 - 10 Dx 2 D y 2 + 9 D y 4 M uHx, yL = 17 cos J x + Þ
u p = CosBx +
2 yF
Example 9 : I3 Dx 3 - D y M uHx, yL = sin H a x + b yL Þ
up =
1 5
Cos@x + 2 yD
Hyperbolic nonhomogenuity :
Φ = Φ2 = sinh coshH
12
L
2 yN
Example 10 : ( 3 D x 2 - D y ) u( x, y) = cosh( a x + b y) 3 a2 Cosh@a x+b yD+b Sinh@a x+b y D Þ up = 9 a4 -b2 Example 11 : IDx 4 + 3 Dx 2 D y 2 + 2 Dz M uHx, y, zL = cosh H a x + b y + 11 zL Þ a2 Ia2 +3 b2 M Cosh@a x+b y+11 zD-22 Sinh@a x+b y+11 zD up = 2 -484+a4 Ia2 +3 b2 M Example 12 : ID x 4 + 3 Dx 2 D y + 2 Dt M uHx, y, tL = sinh H x - 2 y + 3 t L Þ u p = Sinh@3 t + x - 2 y D Multiplicative nonhomogenuity : Φ = Φ1 × Φ2 = ãH
L
× sin cosH
L
Example 18 : I3 Dx 2 - D y M uHx, yL = ã x sinH x + yL Þ
u p = - 15 ãx Cos@x + yD
Example 19 : I3 Dx 2 - D y M uHx, yL = ãΑ x+ Β y sinHa x + b yL Þ up =
ãx Α+y Β IHb-6 a ΑL Cos@a x+b yD-I3 a2 -3 Α2 +ΒM Sin@a x+b yDM 9 a4 +b2 -12 a b Α+I-3 Α2 +ΒM +6 a2 I3 Α2 +ΒM 2
Example 20 : IDx 2 - D y L uHx, yL = 17 ã x+ y sinHx - 2 yL Þ u p = -ãx+y H4 Cos@x - 2 y D + Sin@x - 2 yDL
Example 21 : IDx 2 + D y 2 - Dx M uHx, yL = 37 ã5 y cosH3 x + 4 yL Þ u p = ã5 y sinH3 x + 4 yL
Example 22 : IDx 2 + D y 2 - Dx M uHx, yL = 37 ã5 y coshH3 x + 4 yL Þ
up =
37 ã-3 x+y I87+13 ã6 x+8 y M 2262
Example 23 : IDt - c Dx L HDt + c Dx L uHx, tL = ã- x sinHtL equation with initial conditions {u Hx, 0L = 0, ut Hx, 0L = ã- x } Þ u p = - 1 2 ã- x sin@tD
Wave
1+c
The last two examples 24 and 25 give rise to a problem when generating replacement lists where the original lists for the differential operators. e.g. 8D x , D y , Dz
