A PICARD-MACLAURIN THEOREM FOR INITIAL VALUE PDES

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on n with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset of , we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.


Introduction
In [6] the authors developed a completely explicit notation for presenting ordinary differential equations (ODEs) with polynomial generators that allowed them to make a reasonably transparent proof for the following theorem.
If F is a polynomial from n into n , then the kth Picard iterate for the ODE defined by is the (k −1)st degree Maclaurin polynomial plus a polynomial all of whose terms have degree greater than k − 1.
The notation developed translates directly to implementation of the algorithm arising from the proof of this theorem in either a symbolic or numeric computing environment.
To avoid ambiguity in proving a similar theorem for initial value partial differential equations (PDEs) and to afford direct translation of the resulting algorithm to either a symbolic or numeric computing environment we will again introduce a formal notation.
In order to realize the full impact of the theory on applications it is important to have a precise definition for a polynomial functional F on n (D F = n ) since the components of the generator and the iterates of the solution are polynomial functionals.However, we note that their roles within any computing environment are different.The components of the generator must be stored and accessed; the iterates of the solution must be tracked.
We offer here a definition that translates directly into a (numerical) computing environment.
Definition 1.1.Suppose that n ∈ N. Let n denote {k | k ∈ N and k ≤ n}. (1.3) Elements of E n will define the exponents for the terms of polynomial functionals on n .
Definition 1.2.Suppose that n ∈ N and F : n → .The statement that "F is a polynomial functional on n " means that there is a finite subset, , of E n and A : {0} ∪ → , so that if x ∈ n , then . (1.4) The three main structural differences between the PDE problem and the ODE problem are (1) the initial conditions are analytic functions; (2) the PDE need be autonomous only in the designated component; and (3) the presence of partial derivatives in the domain of the generator for the PDE.For computational reasons it is important to have a formal notation for these partial derivatives.
To illustrate, consider a function u : 2 → .It is common to denote the first component of a typical element of D u as x, the second component as y and, for instance, to write one of the third partial derivatives as u xxy .We prefer d 112 u.To translate to the formalism suggested above, let ν(1) = 1, ν(2) = 1, and ν(3) = 2.The partial derivative is then denoted as d ν u.

G. E. Parker and J. S. Sochacki 49
To set the stage for the presentation of a PDE as our theorem applies to it, consider the following PDE on 3 (from gas dynamics) as it might be typically presented.

Theorems
Since the presentation of the problem is dependent on the designation of a particular component, we adopt, for an element of m , the notation (t, x).t will denote the designated component and x will consist of the other m − 1 components; we will denote the components of x as x 1 , x 2 , . . ., x m−1 .I will denote the identity function in the designated component.We will write the right-hand side of a PDE as P (x, U (t, x), d ν U(t, x)).This is done to emphasize that we demarcate the domain of the generator into components accepting the non-designated components of the domain, components of the solution, and partial derivatives of components of the solution. and and for l > 1 (2.3) Before proving this theorem we note that, by the Cauchy theorem the PDE must have a unique analytic solution.Also, since components of the generator for the PDE and the iterates for the solution are both polynomial functionals, it is important to distinguish between them.We will write the functionals for the generator as indicated in the example.Also note that b r,i (x) and B s,i (x) are power series in x.Using these series as coefficients for powers of t we write the iterates for the solution in powers of the designated component, t.
Proof.The proof will use induction on the powers of the designated component.Let 1 ≤ k ≤ n and for l > 1 consider p k,l .52 A Picard-Maclaurin theorem for initial value PDEs (2.4) Every term from the integral, since the integration is done with respect to the identity in the designated component, must have a power of at least 1 on t.Therefore, the zero degree term in t must be the initial condition, a power series in the components of x.Thus any pair of iterates agree in the zero power term. Let and (2.9) From the induction hypothesis, for any j and i, f j,l−1 The power of the identity in the designated component in any term of g j,l−1 • (I, x) or g j,Q • (I, x) is at least Q + 1.Since the differential operators reduce only powers of components of x, the same is true for d ν(j ) g j,l−1 • (I, x) and d ν(j ) g j,Q • (I, x).Thus, the term having factor t Q+1 must come from integrating powers of f j,l−1 •(I, x) and powers of d ν(j ) f j,l−1 •(I, x) which we have just shown to be identical to the corresponding powers of f j,Q • (I, x) and (2.11) (2.12) From the fundamental theorem of calculus (2.13) If 1 ≤ j ≤ n then p j,l = f j,l + g j,l ; thus (2.14) From the argument in the induction, all terms having degree no greater than l − 1 in t are terms of d 1 f k,l (t, x).All terms of d 1 (V k − f k,l ) have degree at least l and the terms of V j (t, x) of degree no greater than l − 1 are identical to those of f j,l−1 (t, x).Therefore, since arbitrarily high powers of d 1 V k (t, x) satisfy the PDE, V k solves the same PDE as U k , and V = U by the uniqueness in the premise.
If the generator is a polynomial on the appropriate space and the initial conditions are polynomials then, as in the ODE case, the computational implementation follows directly from the proof of the theorem since only polynomial algebra and the power rule are involved.On the other hand, even if the initial conditions are polynomial, if the generator is analytic but not a polynomial, and a polynomial projection is necessary to achieve a polynomial generator, then the initial conditions for the corresponding components of the solution are analytic, but need not be polynomials.For example, consider w in Example 1.4 and suppose that w(0, x, y) = x 2 +y 2 .Then z(0, x, y) = 1/(x 2 +y 2 ) which is not polynomial.In a symbolic computing environment this problem is easily addressed.However, in a numeric computing environment an analytic function cannot be entered.
Even in a numeric computing environment Theorem 2.1 solves PDEs with polynomial generators and polynomial initial conditions, an interesting and extensive class of problems.However, the theorem would carry even greater impact if analytic initial conditions and/or generators that can be projected as polynomials could also be included.Theorem 2.2 addresses this issue.We introduce as the "exponent finder" for the presentation of the solution in order to distinguish it from , the "exponent finder" for the generator.
Theorem 2.2.Suppose U is as in the premise to Theorem 2.1 and if (t, x) ∈ D U and (2.17) Then W k,r+1 = T k,r+1 or the degree of W k,r+1 − T k,r+1 is greater than r + 1.

A Picard-Maclaurin theorem for initial value PDEs
Proof.Let U be as in the premise to Theorem 2.1 and define for r ∈ N and k ∈ n, R k,r so that U k = T k,r + R k,r .Then for (t, x) ∈ m , each term of R k,r (t, x) has degree at least r + M + 1.
Expanding as in the proof of Theorem 2.1 using the binomial theorem, (2.20) Each term of the binomial expansion which has R δ • (•),r as a factor has degree greater than r + M. Any term containing a partial derivative of R δ • (•),r as a factor, since the order of the equation is M, has degree greater than r.Thus any term from the integral arising from these terms has degree greater than r + 1 because of the additional power of t from the power rule.Let a k,r (x) denote the terms of a k (x) of degree less than or equal to r + 1. Thus (2.21) plus terms of degree greater than r + 1.Since (2.22) and T k,r+1 (t, x) is the (r + 1)st degree truncation of U k (t, x), the conclusion is established.
Theorem 2.2 shows that if iterates are truncated properly, then succeeding iterates continue to generate the Maclaurin series.This makes it possible to implement the iteration in a numeric computing environment.

Examples
In this section, we consider four well-studied PDEs and illustrate the usefulness of the theorems and ideas presented in the last section.Our main goal is to show how the results of the last section provide a method for obtaining polynomial solutions in t with coefficients that are functions of x.We use the software package Maple to obtain symbolic results.We stress that all the presented results could have been obtained in a numerical environment using Theorem 2.2.

Example 3.1 (Burger's equation). Consider the equation
It is well known that Burger's equation can produce solutions with discontinuities and that many numerical techniques exhibit a Gibbs phenomenon for sharp fronts.If f is twice differentiable then the fourth modified Picard iterate (the modification is to truncate to all powers of the first component less than the number of the iterate; Theorem 2.2 guarantees that these are terms of the power series for the solution and Theorem 2.1 that they will be carried forward in the iteration) is given by 58 A Picard-Maclaurin theorem for initial value PDEs Higher degree Maclaurin polynomials are easily generated and do have high computational utility, but because of the space occupied by the printout, are not presented here.
To model a sharp front that will propagate over time, the initial condition used was f (x) = 2 + (2/π ) arctan(x).The results from using only the third degree truncation of the fourth Picard iterate with a time step of 0.25 are presented in Figure 3 Of course, there need not be a unique polynomial projection for a given PDE.In [6] the authors show that for ODEs one may be able to choose a polynomial projection that reduces the number of components of the generator or a polynomial projection that reduces the degree of the generator.In that paper it was shown by example that using a polynomial projection that reduces the degree may give more accurate numerical results.However, using a polynomial projection that reduces the degree of the generator usually increases the number of components of the generator.
The following examples show that one can reduce the number of partial derivatives needed by choosing an appropriate polynomial generator.When working in a symbolic environment, at least for these examples, this reduces the computations and computational time needed to generate the modified Picard iterates.

Example 3.2 (The wave equation). Consider the wave equation
where a and b are functions of x only.This can be converted to a form to which G. E. Parker and J. S. Sochacki 59 our theorem applies in at least two ways.One is by using the polynomial projection v = d 1 u.This projection gives the system If one makes the additional polynomial projection w = d 2 u, the system that results is This second system has more components, but the second system is first order and thus less computationally demanding than the first system.The second system generated the modified Picard iterates significantly faster than the first system.Of course, the modified Picard iterates, being the Maclaurin polynomials for u, are the same for both systems.
The third degree Maclaurin polynomial for u(t, x) in this system is Another projection of (3.3) is obtained by letting w = bd 2 u.This gives the polynomial system To model a sharp front that will cause reflections, b(x) was set to 2 + (2/π) arctan(x) and a(x) was set to 1.A strength of Picard iteration is that one does not have to introduce boundary conditions in wave propagation to obtain numerical results.Therefore, there are no spurious reflections from boundaries.To show this p(x) is set to cos(x) + sin(2x − π/4) and q(x) is set to 0. The graphical results for the data obtained from using only the second degree Maclaurin polynomials in t and updating the coefficients by setting p(x) to u(5/32, x), q(x) to d 1 u(5/32, x) and w(0, x) to the update of p (x) are presented in Figure 3.2.
Example 3.3 (The Sine-Gordon equation).The Sine-Gordon equation has the form  G. E. Parker and J. S. Sochacki 61 In order to use the theorems from Section 2 the polynomial projection we use is v = d 1 u, y = sin •u, and z = cos •u.Again, in order to reduce the amount of differentiation, we also let w = d 2 u.The resulting system is (3.9) This system is only a second degree polynomial system.Thus it is straightforward to program and requires little computing time.The second degree Maclaurin polynomial for u(t, x) is 1 2 p (x) + 1 2 sin p(x) t 2 + q(x)t + p(x).
(3.10) Again boundary conditions do not have to be introduced to obtain numerical results.The graphical results in Figure 3.3 are from updating p and q as explained in Example 3.2 with p(x) set to sin(x) and q(x) set to 0, but using a time step of 1.25.(3.12) Of course other polynomial projections are possible.For example, one could let z = ρv.It is well known that "shocks" can develop in the above equations.The modified Picard process presented here does not converge at the shocks, thereby giving numerical evidence of the development of the shock.The third degree Maclaurin polynomial for v(t, x) is p(x) − p(x)q (x) + p (x)q(x)p(x) t + 6 p(x)q(x)q (x) + 6 p(x)q (x) 2 + 3 c 2 p (x) + 12 p(x) 2 p (x)q(x)q (x) + 3 p(x) 2 q(x) 2 p (x) t 2 + 2 p (x) 3 q(x)c 2 − 2 p(x) 2 p (x)q(x) 2 q (x) − 2 p(x) 3 q (x) 3 − 4 p(x) 3 q (x)q(x)q (x) − 2 p(x)q (x)c 2 p (x) 2 − 6 p (x)q(x)q (x)p(x) 2 − 4 p (x)q(x)p(x)c 2 p (x) − 2 p (x)q(x) 2 p(x) 2 q (x) − 2 p(x) 2 q (x)c 2 p (x) − 2 p(x) 2 q (x)c 2 p (x) /6p(x) 2 t 3 . (3.13) (2.7) G. E. Parker and J. S. Sochacki 53 Differentiating over sums and expanding the resulting integrands using the binomial theorem (C b a = b!/a!(b − a)!), we get

2 c
The graph of the parameter b for the wave equation runs.The graph of the initial condition for the wave equation runs.Graph of second degree Maclaurin polynomials for the wave equation using modified Picard iteration at t = 0(•), 4h(•), 8h(+) and 11h(✷) for h = 5/32.

b
The graph of the initial condition for the Sine-Gordon equation runs.Graph of second degree Maclaurin polynomials for the Sine-Gordon equation using modified Picard iteration at t = h(•), 2h(•), 3h(+), and 3.25h(✷) for h = 1.25.