Approximate solutions to second order parabolic equations. I: Analytic estimates

Abstract

Articles you may be interested inQuenching phenomena for second-order nonlinear parabolic equation with nonlinear source AIP Conf.Approximation of solutions to an abstract Cauchy problem for a system of parabolic equations AIP Conf.A parabolic approximation method with application to global wave propagation We establish a new type of local asymptotic formula for the Green's function G t ͑x , y͒ of a uniformly parabolic linear operator ‫ץ‬ t − L with nonconstant coefficients using dilations and Taylor expansions at a point z = z͑x , y͒ for a function z with bounded derivatives such that z͑x , x͒ = x R N . Our method is based on dilation at z, Dyson, and Taylor series expansions. We use the Baker-Campbell-Hausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the shorttime limit. We establish mapping properties and precise error estimates in the exponentially weighed, L p -type Sobolev spaces W a s,p ͑R N ͒ that appear in practice.We observe that ␣ uniquely determines k and ᐉ, so that our notation is justified. Let ␣ = ͑␣ j ͒ A k,ᐉ . We remark that if k = n + 1 or some ␣ j = n +1 ͑in which case L ␣ j z stands in fact for L n+1 s,z ͒, then ⌳ ␣,z and ⌳ z ᐉ depend on s, so we shall sometimes denote these terms by ⌳ ␣,s,z and ⌳ s,z ᐉ .Also, in what follows, when no confusion can arise, we will drop the explicit dependence on z. However, in Sec. V, z will be allowed to vary and we will reinstate the full notation. We also observe that each ⌳ z ᐉ or ⌳ s,z ᐉ is well defined as a Riemann integral by Lemma II.9 and by the following lemmas. Let us recall that ͗x͘ w = ͑1+͉x − w͉ 2 ͒ 1/2 . Lemma 3.6: The family ͕͗x͘ z −j L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ defines a bounded subset of L.Proof: This is an immediate consequence of Remark 3.3 if j ഛ n and of directly estimating the remainder in the Taylor series for j = n +1.In the following lemma, we shall use an arbitrary center for our weight. Lemma 3.7: For each given ⑀ Ͼ 0, the family ͕e −⑀͗z−w͘ e −⑀͗x͘ w L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ is a bounded subset of L.Proof: Let us assume first that w = z. We need to prove that the family ͕e −⑀͗x͘ z L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ is bounded in L. Indeed, this follows from Lemma 3.6 and the simple observation that ͗x͘ z j e −⑀͗x͘ z ഛ C, with C independent of z and j.To obtain the statement of the theorem, we then apply the triangle inequality to the vectors ͑0,x͒ , ͑1,z͒ , ͑1,w͒ R 1+N to conclude that ͗x − z͘ − ͗x − w͘ ഛ ͉z − w͉ ഛ ͗z − w͘. This shows that e ⑀͑͗x−z͘−͗x−w͘−͗z−w͒͘ ഛ 1. Hence, the familys ͑0,1͔, z R N , j =0, ... ,n + 1, is bounded in L, as claimed. Lemma 2.9 together with Lemma 3.7 then give the following result.Corollary 3.8: We have ⌳ ␣,z B͑W a s,p , W a−⑀ r,p ͒ for any ␣ A k,ᐉ , z R N , r , s R, 1Ͻ p Ͻϱ, and ⑀ Ͼ 0. Moreover, we have that ʈ⌳ ␣,z ʈ W a,z q,p →W a−⑀,z r,p ഛ C q,r,p,a,⑀ e k⑀͗z−w͘ ...

A Green's Function Numerical Method for Solving Parabolic Partial Differential Equations

“…The resulting solution is periodic on the plane. As in [2,3], our method relies on writing the solution u(t, x) = G(t, x, y)f (y)dy, where f is the initial data and G is the Green function of our problem.…”

“…The main issue in the variable coefficient case is to approximate G. A solution to this problem is proposed in [2,3], where equations on domains without boundaries were considered, and hence no boundary conditions were needed. A natural question is whether the approximation in [2,3] recovers the exact solution in the case of constant coefficients, for which the fundamental solution is known.…”

“…A natural question is whether the approximation in [2,3] recovers the exact solution in the case of constant coefficients, for which the fundamental solution is known.…”

“…These two questions are addressed in this paper in the particular case of a rectangular domain and the approximation of solutions to the heat equation in two space dimensions. It should also be noted that the method is applicable in situations where we have coefficients that are constant at the boundary but may be non-constant within the domain of interest by combining the approximations discussed here with those in [2,3] for the variable-coefficient case. Another general issue, is to estimate the integrals G(t, x, y)f (y)dy numerically, and for this we use the simplest approach of a composite trapezoid rule for periodic functions, since there seems to be no additional advantage in considering some higher order rules.…”

Asymptotic expansions for degenerate parabolic equations

“…From the analytical point of view, this assumption is equivalent to a uniform parabolicity for the pricing PDE. In this regard, we refer to the results in [18,19,3], based on Malliavin calculus techniques, and [6,13] based on purely analytic arguments. Unfortunately, these assumptions are hardly ever satisfied by the financial models used in practice.…”