<Text-field style="Heading 1" layout="Heading 1">1. Introduction</Text-field>

From a rigorous point of view power series can be used for approximate solution of differential equations if we are able to perform some necessary steps.

First of all, we have to calculate the coefficients of the series, formally satisfying given differential equation (or system of them). The convergence of the resulting series is not guaranteed (formal power series).

Second, in order to be sure that the resulting series represent a solution, it is necessary to know that a power series convergent and could be differentiated termwise.

Third, to consider partial sums of a power series as approximation to the solution it is necessary to estimate of the magnitude of the error resulting from truncating the series.

The present contribution concerns mainly the first step - finding of formal power series to solution of differential equation or system of them in so-called normal form with initial conditions assigned on a hyperplane L0kidEc2IkkiYUdGJA== (Cauchy problem). We give the procedure for finding formal power series expantion of the solution about a value L0kidEc2IkkiYUdGJA== . The famous Cauchy-Kovalevskaya theorem gives the answer for the second step, but an automatic verification of conditions of the Cauchy-Kovalevskaya theorem to given DEs is not implemented yet.

There are some known methods for obtaining the upper bound for remainder of truncating series, but it is not implemented yet as well.

The computations of formal power series to solution of differential equation (or system of them) based on the following operator solution of Cauchy problem.

System of SSJuRzYi PDEs for SSJuRzYi unknown functions LSZJInVHNiI2I0kiakdGJTYkSSJ0R0YlSSJ4R0Ylof the following form

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

where t \342\210\210 SSVyZWFsRzYi, 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, JkkmYWxwaGFHNiI2I0kiaUdGJA== \342\210\210SSpub25uZWdpbnRHJSpwcm90ZWN0ZWRH, 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 ... LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYjNiVGKy1JJW1zdWJHRiQ2JS1GLDYlUSJERidGL0YyLUYjNiVGKy1GOzYlLUYsNiZRInhGJy8lJXNpemVHUSMxMEYnRi9GMi1GLDYmUSJtRidGR0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRk1GKy1GIzYlRistRiM2JUYrLUY7NiUtRiw2JVEoJmFscGhhO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZKRk1GK0YrLyUxc3VwZXJzY3JpcHRzaGlmdEdGT0Yr , 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

with initial conditions (Cauchy problem)

Ly1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjSSJqR0YqNiRJInRHRipJInhHRiovRi5JImFHRiotJkkidkdGKkYrNiNGLw==

has the following formal exact solution in an operator form (as it follows from [1,2]) for MUkiYUc2IkkidEdGJA== 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

where 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 , KiQtJkkiVkc2IjYjSSJpR0YmNiNJJVpldGFHRiYhIiI= is the variational derivative.

If now expand the operator exponent into Taylor series and execute operations, we obtain the conventional formal power series to solution of given system of differential equations. Extension of this approach to systems of higher differential order with respect to SSJ0RzYi is obvious as we always can reduce such system to the above considered type by adding some new dependent variables. The procedure for finding formal series solutions to non-linear DEs works by the following way. First it verify input data (DEs system, dependent and independent variables, initial conditions) to be valid for the problem under solution. Then it tries to solve DEs system for highest derivatives for dependent variables with respect to "selected" independent variable "SSJ0RzYi". The procedure does not try any (affine) chage of variables to reduce the given system to normal form. For "normalized" system it then forms operator - argument of operator exponent accordingly the above mentioned expression and finally computes recursively the coefficients of series up to Order (see help on Order), formally satisfying given DEs system. I avoid adding something like LUkiT0clKnByb3RlY3RlZEc2IyksJkkidEc2IiIiIkkiYUdGKSEiIkkia0dGKQ== to the procedure output in the situation when steps 2-3 mentioned above are unfulfiled. As for high order DEs or for complicated systems the procedure can hang indefinably, the infolevel[desolve_series] :=2 can be used to know which order are performed. Examples given below only demonstrate the usage of proposed procedure.

P.S. I would be very grateful if you find an opportunity to send me comments of any type, point out my errors and examples of practical problems solved (or not) by the procedure. It would maintain further directions of my project. Thank you in advance.

References:

1. Yu. N. Kosovtsov "The Introduction to the Operator Method for Solving Differential Equations.First-order DE" http://arxiv.org/abs/math-ph/0202040.

2. Yu. N. Kosovtsov "The Chronological Operator Algebra and Formal Solutions of Differential Equations" http://arxiv.org/abs/math-ph/0409035 .

<Text-field style="Heading 1" layout="Heading 1">PDE Taylor Procedure</Text-field>

desolve_series:=proc(DEsys::set,inits::set(`=`),vars::set) local _a_, _t_, _Y_,deq1,deq2,A,M,m,m0,mm,gs,j,i,h,r,q,qi,li,i5,inco,Inits,M0,MM,S,S0,S1,S2,SR,SSR,KK,KR,KKR,L0,LP,NN,RP,PP,SUB,SUB0,SUB01,ORD,EQ01,n,ordn,ivs,linc,vars1,vars2,k,EQ1,EQ0,ord,inS,L,sol1,soln0,soln01,SOLN,p,sol,W1,W3,WW,WWW,inits0,inits1; option `Copyright (c) 2005 by Yuri N. Kosovtsov. All rights reserved.`; if nops(DEsys)<>nops(vars) then error "The number of DEs must be equal to the number of unknown dependent variables" fi; deq1 :={}; for j from 1 to nops(DEsys) do if hastype(op(j,DEsys),`=`)=true then deq1 :=deq1 union {convert(lhs(op(j,DEsys))-rhs(op(j,DEsys)),diff)} else deq1 :=deq1 union {convert(op(j,DEsys),diff)} fi; od; vars1 :={}; A :=indets(deq1,function); for i from 1 to nops(vars) do for j from 1 to nops(A) do if type( op(i,vars), function )=true then if op(0,op(j,A))=op(0,op(i,vars)) then vars1 :=vars1 union {op(j,A)} fi; else if op(0,op(j,A))=op(i,vars) then vars1 :=vars1 union {op(j,A)} fi; fi; od; od; SUB0 :={}; SUB01 :={}; for i from 1 to nops(vars1) do SUB0 :=SUB0 union {op(i,vars1)=_Y_||i(op(op(i,vars1)),_a_)}; SUB01 :=SUB01 union {op(0,op(i,vars1))=_Y_||i}; od; vars2 :={seq(_Y_||j(op(op(j,vars1)),_a_),j=1..nops(vars1))}; inits0 :=subs(SUB01,inits); inits1 :={}; for i from 1 to nops(inits0) do inits1:=inits1 union {op(0,lhs(op(i,inits0)))(op(lhs(op(i,inits0))),_a_)=rhs(op(i,inits0))}; od; deq1 :=subs(SUB0,deq1); deq2 :=convert(deq1,D); if vars1 intersect indets(vars,function)<>indets(vars,function) then error "sets of independent variables must be the same in DEs system and in set of dependent variables" fi; for i from 2 to nops(vars1) do if {op(op(i,vars1))} union {op(op(1,vars1))}<>{op(op(1,vars1))} then error "sets of independent variables must be the same for all dependent variables" fi; od; for r from 1 to nops(deq1) do m[r] :={}; M[r] :={}; gs[r] :={}; Inits[r] :={}; for i from 1 to nops(inits1) do if has(op(i,inits1),op(0,op(r,vars2)))=true then Inits[r] :=Inits[r] union {op(i,inits1)} fi; od; for j from 1 to nops(Inits[r]) do for i from 1 to nops([op(op(r,vars2))]) do if op(i,[op(op(r,vars2))])-op(i,[op(lhs(op(j,Inits[r])))])<>0 then m[r] :=m[r] union {i}; inco[r] := op(i,[op(op(r,vars2))])=op(i,[op(lhs(op(j,Inits[r])))]); M[r] :=M[r] union {op(i,[op(lhs(op(j,Inits[r])))])} fi; od; gs[r]:= gs[r] union select(type,indets(rhs(op(j,Inits[r]))),name); od; if nops(M[r])<>1 then error ("invalid initial conditions - initial conditions must be set at the same plane for variable %1", op(r,vars1)) else M0[r] :=op(1,M[r]) fi; if nops(m[r])<>1 then error ("invalid initial conditions - initial conditions must be set at one coordinate for variable %1", op(r,vars1)) else m0[r] :=op(1,m[r]); fi; S0[r] :={}; S[r] :=select(has,indets(select(has,indets(Inits[r]),op(0,op(r,vars2)))), D ); for i from 1 to nops(S[r]) do KK[r] :=op(remove(has, [op(op(0,op(0,op(i,S[r]))))], m0[r])); S0[r] :=S0[r] union {KK[r]} od; if gs[r] intersect {lhs(inco[r])}<>{} then error ("invalid initial conditions - initial conditions must not depend on initial coordinate %1 for variable %2" , lhs(inco[r]),op(r,vars1)) fi; if S0[r]<>{} then error("invalid initial conditions - initial conditions must describe the normal derivatives on %1 for variable %2", inco[r],op(r,vars1)) fi; S1[r] :={}; for i from 1 to nops(Inits[r]) do S1[r] :=S1[r] union {PDEtools[difforder](lhs(op(i,Inits[r])))}; od; if nops(S1[r])<> nops(Inits[r]) then error "invalid initial conditions - erroneous doubling of an initial condition" fi; ord[r] :=PDEtools[difforder](select(has,indets(deq1),op(r,vars2)),lhs(inco[r])); ivs[r] :=op(remove(has,[op(op(r,vars1))],lhs(inco[r]))); if r>1 then if {ivs[r]} intersect {ivs[r-1]}<> {ivs[r-1]} then error "dependent variables of the problem must have identical list of independent variables" fi; fi; if ivs[r]<>NULL then linc :=[seq(v[r,i]=_c[r,i](ivs[r]),i=0..ord[r]-1)]; S2 :={}; for i from 1 to nops(Inits[r]) do k :=PDEtools[difforder](lhs(op(i,Inits[r]))); S2 :=S2 union {_c[r,k](ivs[r])=rhs(op(i,Inits[r]))} od; linc :=subs(S2,linc); if nops(S2) < ord[r] then WARNING("some initial conditions are added in the folloving form %1",D[`$`(m0[r],`n`)](op(0,op(r,vars1)))=_c[r,`n`](ivs[r])); fi; else linc :=[seq(v[r,i]=_c[r,i],i=0..ord[r]-1)]; S2 :={}; for i from 1 to nops(Inits[r]) do k :=PDEtools[difforder](lhs(op(i,Inits[r]))); S2 :=S2 union {_c[r,k]=rhs(op(i,Inits[r]))} od; linc :=subs(S2,linc); if nops(S2) < ord[r] then WARNING("some initial conditions are added in the folloving form %1",D[`$`(m0[r],`n`)](op(0,op(r,vars1)))=_c[r,`n`]); fi; fi; inS[r] :=convert(linc,set); SR :=select(has, indets(select(has, indets(deq2), D)), op(0,op(r,vars2))); SSR :={}; for i from 1 to nops(SR) do KR :=PDEtools[difforder](op(i,SR),op(m0[r],[op(op(r,vars2))])); KKR :=op(remove(has, [op(op(0,op(0,op(i,SR))))], m0[r])); SSR :=SSR union remove(has,{op(i,SR)=D[KKR](P[r,KR])(op(op(r,vars2)))},symbol) od; deq2 :=subs(op(r,vars2)=P[r,0](op(op(r,vars2))),subs(SSR,deq2)); if PDEtools[difforder](select(has,indets(deq2),P[r,PDEtools[difforder](deq2,op(m0[r],[op(op(r,vars2))]))]),op(m0[r],[op(op(r,vars2))]))<>0 then error "unable to convert geven set of DEs to an explicit first order system" fi; od; mm :={seq(m0[r],r=1..nops(vars2))}; MM :={seq(M0[r],r=1..nops(vars2))}; if nops(MM)<>1 then error ("invalid initial conditions - initial conditions for all variables %1 must be set at the same plane", op(vars1)) fi; if nops(mm)<>1 then error ("invalid initial conditions - initial conditions must be set at one coordinate for all variables %1", op(vars1)) fi; EQ1 :={solve(deq2,{seq(P[r,PDEtools[difforder](select(has,indets(deq1),op(0,op(r,vars2))),op(m0[1],[op(op(1,vars2))]))](op(op(1,vars2))),r=1..nops(deq2))})}; if EQ1=NULL then error "unable to convert given set of DEs to normal form" fi; EQ0 :={}; for q from 1 to nops(EQ1) do for qi from 1 to nops(op(q,EQ1)) do LP :=op(0,lhs(op(qi,op(q,EQ1)))); RP :=indets(rhs(op(qi,op(q,EQ1)))); if has(RP,LP)=true then error "unable to convert given set of DEs to normal form" fi; od; EQ0 :=EQ0 union {convert(op(q,EQ1),Diff)}; for i from 1 to nops(op(q,EQ1)) do ord[q,op(1,op(0,lhs(op(i,op(q,EQ1)))))] :=op(2,op(0,lhs(op(i,op(q,EQ1))))); od; od; SUB :={seq(op(inS[r]),r=1..nops(vars2))}; sol :={}; for q from 1 to nops(EQ0) do if min(seq(ord[q,i],i=1..nops(vars2)))=0 then error "unable to convert geven set of DEs to an explicit first order DE system" fi; ORD :=max(seq(ord[q,op(1,op(0,lhs(op(rr,op(q,EQ0)))))],rr=1..nops(vars2))); EQ01 :=subs(lhs(inco[1])=s,subs({seq(seq(P[i0,j0](op(op(1,vars2)))=v[i0,j0],j0=0..ORD),i0=1..nops(vars2))},op(q,EQ0))); n :=[seq(op(1,lhs(op(N,EQ01))),N=1..nops(vars2))]; ordn :=[seq(op(2,lhs(op(N,EQ01))),N=1..nops(vars2))]; L0 := [seq(rhs(op(N0,EQ01)),N0=1..nops(vars2)),seq(seq(v[op(N0,n),jl+1],jl=0..op(N0,ordn)-2),N0=1..nops(vars2)),1]; L :=(U)->[seq(diff(U,v[op(N0,n),op(N0,ordn)-1]),N0=1..nops(vars2)),seq(seq(diff(U,v[op(N0,n),jl]),jl=0..op(N0,ordn)-2),N0=1..nops(vars2)),diff(U,s)]; try for h from 1 to nops(vars2) do sol1 :={}; soln0 :=subs(SUB,v[h,0]); soln01 :=v[h,0]; for p from 1 to Order do MM :=L(soln01); SOLN :={}; for li from 1 to nops(MM) do NN :=indets(op(li,MM), function); PP :={}; for i from 1 to nops(NN) do if op(1,op(i,NN))=1 or hastype(op(1,op(i,NN)),Diff(1,anything))=true then PP :=PP union {op(i,NN)} fi; od; WW :=[coeffs]( collect(op(li,MM),PP),PP ,'_t_' ); W3 :=add(op(u,WW)*op(u,[_t_]),u=1..nops(WW)); WWW :={}; for i5 from 1 to nops(WW) do WWW :=WWW union {op(i5,WW)*subs(1=op(li,L0),op(i5,[_t_]))} od; W1:=simplify(convert(value(subs({seq(seq(v[j0,jj]=v[j0,jj](op(op(1,vars2))),jj=0..ORD),j0=1..nops(vars2))},convert(WWW,`+`))),Diff)); SOLN :=SOLN union {W1}; od; soln01 := (subs({seq(seq(v[k0,kk](op(op(1,vars2)))=v[k0,kk],kk=0..ORD),k0=1..nops(vars2))},convert(convert(SOLN,`+`),Diff))); soln0 :=soln0 +1/p!*simplify(subs(SUB, soln01))*(lhs(inco[1])-rhs(inco[1]))^p; userinfo(2,{desolve_series},`calculations of`, p,`th order are performed for`,op(0,op(h,vars1))); od; sol1 :=sol1 union {op(h,vars1)=eval(value(soln0),s=rhs(inco[1]))}; sol :=sol union {op(sol1)}; od; catch :error "unable to obtain solution under given IC, please try another IC"; end try; od; userinfo(2,{desolve_series},`calculations are performed; processing answer ...`); RETURN(sol) end proc:

<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field>

We stress again that formal series solutions have a conventional (informal) sense if only given PDEs satisfy the requirements of the Cauchy-Kovalevskaya theorem.

<Text-field style="Heading 1" layout="Heading 1">General type example</Text-field>

This example demonstrates an anatomy of series solutions by desolve_series procedure. We set the small Order to obtain observable output

Order :=2;

IiIj

deq:=diff(Y(x,y,z),x)+F(x,y,z,diff(Y(x,y,z),y),diff(Y(x,y,z),z));

LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiWUc2IjYlSSJ4R0YpSSJ5R0YpSSJ6R0YpRisiIiItSSJGR0YpNidGK0YsRi0tRiQ2JEYnRiwtRiQ2JEYnRi1GLg==

var :=Y(x,y,z);

LUkiWUc2IjYlSSJ4R0YkSSJ5R0YkSSJ6R0Yk

inits:={Y(0,y,z)=h(y,z)};

PCMvLUkiWUc2IjYlIiIhSSJ5R0YmSSJ6R0YmLUkiaEdGJjYkRilGKg==

ans :=desolve_series({deq},inits,{var});

PCMvLUkiWUc2IjYlSSJ4R0YmSSJ5R0YmSSJ6R0YmLCgtSSJoR0YmNiRGKUYqIiIiKiYtSSJGR0YmNiciIiFGKUYqLUklZGlmZkclKnByb3RlY3RlZEc2JEYsRiktRjY2JEYsRipGL0YoRi8hIiIqKCNGLyIiI0YvLC4qJi0tJkkiREc2JEY3SShfc3lzbGliR0YmNiMiIiU2I0YyNidGNEYpRiotLSZGRDYjRi82I0YtRi4tLSZGRDYjRj5GT0YuRi8tLUZSRklGSkYvRi8qJilGQUY+Ri8tLSZGRDYkRi9GL0ZPRi5GL0YvKipGPkYvRkFGLy0tJkZENiMiIiZGSUZKRi8tLSZGRDYkRi9GPkZPRi5GL0YvKiZGZ25GLy0tJkZENiMiIiRGSUZKRi9GLyomKUZnbkY+Ri8tLSZGRDYkRj5GPkZPRi5GL0YvLS1GTUZJRkpGO0YvKUYoRj5GL0Yv

The solutions here can be checked by the following formal test (If the series solution is valid, the returned result will be 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 , where SSJrRzYiis the order of given PDE with respect to SSJ0RzYi . Note, that in general if the test of such type returns another result, the solution being tested is not necessarily wrong; sometimes further simplifications or manipulations of desolve_series's output are required to obtain 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, and so verify the solution is correct.)

test0 :=simplify(convert(taylor(eval(subs(ans,deq)),x=0),diff));

KyVJInhHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIj

<Text-field style="Heading 1" layout="Heading 1">Non-linear PDEs system example</Text-field>

Order :=6;

IiIn

deq := {diff(Y(t,y,z),t)+t*diff(Y2(t,y,z),y)^2+y*diff(Y3(t,y,z),z)+F(t,y,z) , diff(Y3(t,y,z),t)^2+diff(Y2(t,y,z),t,t)+y*diff(Y(t,y,z),z) ,diff(Y2(t,y,z),t)+diff(Y3(t,y,z),t,t)+y*diff(Y(t,y,z),z)^2};

PCUsKi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJZRzYiNiVJInRHRipJInlHRipJInpHRipGLCIiIiomRixGLyktRiU2JC1JI1kyR0YqRitGLSIiI0YvRi8qJkYtRi8tRiU2JC1JI1kzR0YqRitGLkYvRi8tSSJGR0YqRitGLywoLUYlNiRGNEYsRi8tRiU2JC1GJTYkRjpGLEYsRi8qJkYtRi8pLUYlNiRGKEYuRjZGL0YvLCgqJClGQ0Y2Ri9GLy1GJTYkRj9GLEYvKiZGLUYvRkdGL0Yv

vars :={Y,Y2(t,y,z),Y3};

PCUtSSNZMkc2IjYlSSJ0R0YlSSJ5R0YlSSJ6R0YlSSJZR0YlSSNZM0dGJQ==

inits:={Y(a,y,z)=y+z,Y2(a,y,z)=z,Y3(a,y,z)=y,D[$(1,1)](Y2)(a,y,z) = 0,D[$(1,1)](Y3)(a,y,z) = 0};

PCcvLUkiWUc2IjYlSSJhR0YmSSJ5R0YmSSJ6R0YmLCZGKSIiIkYqRiwvLUkjWTJHRiZGJ0YqLy1JI1kzR0YmRidGKS8tLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YsNiNGL0YnIiIhLy0tRjY2I0YyRidGPQ==

ans :=desolve_series(deq,inits,vars);

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

test0 :=seq(simplify(convert(taylor(simplify(subs(ans,op(i,deq))),t=a,Order+1-PDEtools[difforder](deq,t)),diff)),i=1..nops(deq));

NiUrJSwmSSJ0RzYiIiIiSSJhR0YmISIiLUkiT0clKnByb3RlY3RlZEc2I0YnIiImRiNGIw==

<Text-field style="Heading 1" layout="Heading 1">Highly non-linear PDE Example</Text-field>

Order :=4;

IiIl

deq:=diff(Y(x,y),x,x)+f(x,y)*exp(diff(Y(x,y),y)+sin(Y(x,y)));

LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJZRzYiNiRJInhHRitJInlHRitGLUYtIiIiKiYtSSJmR0YrRixGLy1JJGV4cEc2JEYlSShfc3lzbGliR0YrNiMsJi1GJDYkRilGLkYvLUkkc2luR0Y1NiNGKUYvRi9GLw==

var :=Y(x,y);

LUkiWUc2IjYkSSJ4R0YkSSJ5R0Yk

inits:={Y(0,y)=y};

PCMvLUkiWUc2IjYkIiIhSSJ5R0YmRik=

ans :=desolve_series({deq},inits,{var});

Warning, some initial conditions are added in the folloving form (D[`$`(1, n)])(Y) = (_c[1, n])(y)

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

test0 :=simplify(convert(taylor(eval(subs(ans,deq)),x=0,Order-1),diff));

KyVJInhHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIk