DSpace at VNU: A finite difference scheme for nonlinear ultra-parabolic equations

Accepted Manuscript A finite difference scheme for nonlinear ultra-parabolic equations Vo Anh Khoa, Tuan Nguyen, Le Trong Lan, Nguyen Thi Yen Ngoc PII: DOI: Reference: S0893-9659(15)00056-7 http://dx.doi.org/10.1016/j.aml.2015.02.007 AML 4726 To appear in: Applied Mathematics Letters Received date: 23 December 2014 Revised date: February 2015 Accepted date: 10 February 2015 Please cite this article as: V.A Khoa, T Nguyen, L.T Lan, N.T.Y Ngoc, A finite difference scheme for nonlinear ultra-parabolic equations, Appl Math Lett (2015), http://dx.doi.org/10.1016/j.aml.2015.02.007 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain A finite difference scheme for nonlinear ultra-parabolic equations Vo Anh Khoaa , Tuan Nguyenb,∗, Le Trong Lanc , Nguyen Thi Yen Ngocc a Mathematics and Computer Science Division, Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam c Department of Mathematics, University of Science, Vietnam National University, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Viet Nam b Applied Abstract In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function Mathematically, the bibliography on initial-boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates A numerical example is given to justify the theoretical analysis Keywords and phrases: ultraparabolic equation, finite difference scheme, Fourier series, linear approximation, stability Mathematics subject Classification 2000: 65L12; 65L80; 34A45; 34G20 Introduction Let H be a Hilbert space with the inner product ·, · and the norm · In this paper, we consider the problem of finding u : [0, T ] × [0, T ] → H satisfies the following ultraparabolic equation  ∂ ∂    u (t, s) + u (t, s) + Au (t, s) = f (u (t, s) , t, s) ,    ∂t ∂s    u (0, s) = ϕ (s) , s ∈ [0, T ],      u (t, 0) = ψ (t) , t ∈ [0, T ] (t, s) ∈ (0, T ) × (0, T ) , (1.1) where A : D(A) ⊂ H → H is a positive-definite, self-adjoint operator with compact inverse on H and ϕ, ψ are known smooth functions satisfying ϕ(0) = ψ(0) for compatibility at (t, s) = (0, 0) and f is a source function which is defined later The problem (1.1) involving multi-dimensional time variables is called the initial-boundary value problem for ultraparabolic equation The ultraparabolic equation has many applications in mathematical finance (e.g [6]), physics (such as multiparameter Brownian motion [16]) and biological models Among many applications, the ultraparabolic equation arises as a mathematical model of population dynamics The study of ultraparabolic equation for population dynamics can be found in some papers such as [5, 7] In particular, Kozhanov [7] studied the existence and uniqueness of regular solutions and its properties for an ultraparabolic model equation in the form of ∂u ∂u + − ∆u + h (x, t, s) u + Au = f (x, t, s) , ∂t ∂s where ∆ is Laplace operator, A is a nonlocal linear operator In the same work, Deng and Hallam in [5] considered the age structured population problem in the form of ∂u ∂u + − ∇ · (k∇u − qu) = −µu, ∂t ∂s associated with non-locally integro-type initial-bounded conditions The ultraparabolic equation is also studied in many other aspects In the study of inverse problems, Lorenzi [19] studied the well-posedness of a class of forward problems for ultraparabolic partial integrodifferential equations Very recently, Zouyed and Rebbani [3] proposed the modified quasi-boundary value method to regularize a final value problem for (1.1) For other studies regarding the properties of solutions of abstract ultraparabolic equations, we can find many papers and some of them are [8, 9, 11, 12, 13, 15] Numerical methods for ultraparabolic equation are studied long time ago, for example [1, 2, 4, 14] Since 1996, Akrivis, Crouzeix and Thome [2] investigated a backward Euler scheme and second-order box-type finite difference procedure to ∗ Corresponding Author Email: nguyenhuytuan@tdt.edu.vn Preprint submitted to Elsevier February 23, 2015 numerically approximate the solution to the Dirichlet problem for the ultraparabolic equation (1.1) in two different time intervals with the Laplace operator A = −∆ and the source function f ≡ As opposed to the method-of-lines approach developed in [2], the well-posedness of the ultraparabolic problem, utilizing the directional derivative and demonstrated in Tersenov et al [1], yields a method-of-characteristics numerical scheme Recently, Ashyralyev and Yilmaz [4] constructed the first and second order difference schemes to approximate the problem (1.1) in case that f ≡ for strongly positive operator and obtained some fundamental stability results On the other hand, Marcozzi et al [14] developed an adaptive method-of(x) ∂u (x, t, s) , lines extrapolation discontinuous Galerkin method for an ultraparabolic model equation given by ∂u ∂t +a ∂s −∆u = f with a certain application to the price of an Asian call option However, most of papers for numerical methods aim to study linear cases Equivalently, numerical methods for nonlinear equations are still limited Motivated by this reason, in this paper, we develop a finite difference method for a nonlinear ultraparabolic equation Our method comes from the idea of Ashyralyev et al [4, 17], but it is different to their method We remark that the numerical approach by Ashyralyev et al [4] is computationally expensive, especially using finite difference scheme in space Normally, the matrices generated by space-discretization are very big if the spatial dimension is high Therefore, in this paper we shall study the model problem (1.1) in the numerical angle for the smooth solution by a different approach in space Theoretically, paying attention to the idea of finite difference scheme in time, studied in e.g [4, 17, 18] by Ashyralyev et al., and conveying a fundamental result in operator theory, we construct an approximate solution for problem (1.1) in terms of Fourier series The rest of the paper is organized as follows In Section 2, a finite difference scheme is propsed Furthermore, stability and convergence of the proposed scheme is established Finally, a numerical example is implemented in Section to verify the effectiveness of the method Finite difference scheme for nonlinear ultra parabolic : stability analysis In this section, we consider a numerical method for Problem (1.1) in the case that f satisfies the global Lipschitz condition f (u, t, s) − f (v, t, s) ≤ K u − v , (2.2) where K is a positive number independent of u, v, t, s By simple calculation analogous to the steps in linear nonhomogeneous case, we get the discrete solution, then use linear approximation to get the explicit form of the approximate solution From now on, suppose that A : D(A) ⊂ H → H is a positive-definite, self-adjoint operator with a compact inverse in H As a consequence, A admits an orthonormal eigenbasis {φn }n≥1 in H, associated with the eigenvalues such that < λ1 ≤ λ2 ≤ λ3 ≤ lim λn = ∞ n→∞ With Gn (t, s) = exp λn (t + s) , by a simple computation, the problem (1.1) is transfomed into the following problem  ∂ ∂   u (t, s) , φn Gn (t, s) = f (u (t, s) , t, s) , φn Gn (t, s) ,  ∂t + ∂s    (2.3) u (0, s) , φn Gn (t, s) = ϕ (s) , φn Gn (t, s) ,       u (t, 0) , φn Gn (t, s) = ψ (t) , φn Gn (t, s) For the numerical solution of this problem by finite difference scheme as introduced in the above section, a uniform grid of mesh-points (t, s) = (tk , sm ) is used Here tk = kω and sm = mω, where k and m are integers and ω the equivalent mesh-width in time t and s We shall seek a discrete solution uk,m = u (tk , sm ) determined by an equation obtained by replacing the time derivatives in (2.3) by difference quotients The equation in (2.3) becomes u(tk , sm ), φn Gn (tk , sm ) − u(tk−1 , sm ), φn Gn (tk−1 , sm ) ω = f (u(tk , sm ), tk , sm ) , φn Gn (tk , sm ) , u(tk−1 , sm ), φn Gn (tk−1 , sm ) − u(tk−1 , sm−1 ), φn Gn (tk−1 , sm−1 ) + ω (2.4) and u (0, sm ) , φn Gn (tk , sm ) = ϕ (sm ) , φn Gn (tk , sm ) , u (tk , 0) , φn Gn (tk , sm ) = ψ (t) , φn Gn (tk , sm ) By induction, it follows from (2.5) that p u(tk , sm ), φn Gn (tk , sm ) = f u(tk−p+l , sm−p+l ), tk−p+l , sm−p+l , φn Gn tk−p+l , sm−p+l ω l=1 + u(tk−p , sm−p ), φn Gn tk−p , sm−p (2.5) for all p ∈ N Hence, we obtain that the explicit form of discrete solution of (1.1) ∞ u (tk , sm ) = u (tk , sm ) , φn φn (2.6) n=1 where u (tk , sm ) , φn is given by u (tk , sm ) , φn   ω         =    ω      m l=1 f (u(tk−m+l , sl ), tk−m+l , sl ) , φn Gn (tk−m+l , sl ) + ψ(tk−m ), φn Gn (tk−m , s0 ) Gn (tk , sm ) k l=1 f (u(tl , sm−k+l ), tl , sm−k+l ) , φn Gn (tl , sm−k+l ) + ϕ(sm−k ), φn Gn (t0 , sm−k ) Gn (tk , sm ) , if k > m , if m > k From now on, we shall give an iterative scheme by linear approximation Choosing u0 (tk , sm ) = 0, from uq (tk , sm ) ∈ H, we shall use the Fourier series to express it in the form ∞ uq (tk , sm ) = uq (tk , sm ) , φn φn (2.7) n=1 where uq (tk , sm ) , φn is defined by uq (tk , sm ) , φn   ω         =    ω      m l=1 f uq−1 (tk−m+l , sl ), tk−m+l , sl , φn Gn (tk−m+l , sl ) + ψ(tk−m ), φn Gn (tk−m , s0 ) Gn (tk , sm ) k l=1 f uq−1 (tl , sm−k+l ), tl , sm−k+l , φn Gn (tl , sm−k+l ) + ϕ(sm−k ), φn Gn (t0 , sm−k ) Gn (tk , sm ) , if k > m , if m > k Here uq (tk , sm ) is called the approximate solution for the problem (1.1) Our results are to prove that this solution approach to the discrete solution u(tk , sm ) in norm H as q → ∞ and study the stability estimate of uq (tk , sm ) in norm H with respect to the initial data and the right hand side f uq−1 Theorem 2.1 Let uq (tk , sm ) sup 1≤k,m≤M uq (tk , sm ) q≥1 be the iterative sequence defined by (2.7) Then, it satisfies the a priori estimate ≤ CT sup 1≤k,m≤M uq−1 (tk , sm ) + sup 1≤k,m≤M f (0, tk , sm ) + sup 0≤m≤M ϕ(sm ) + sup ψ(tk ) , 0≤k≤M where CT is a positive constant depending only on T Proof We divide two cases Case 1: k > m Using Parseval’s identity in (2.7) for the case k > m, we have uq (tk , sm ) ≤ ≤ ≤ 2ω m ∞ n=1 l=1 Gn (tk−m+l , sl ) f uq−1 (tk−m+l , sl ), tk−m+l , sl , φn Gn (tk , sm ) ∞ 2ω2 m2 sup 1≤k,m≤M n=1 2T sup 1≤k,m≤M f uq−1 (tk , sm ), tk , sm , φn f uq−1 (tk , sm ), tk , sm 2 + sup +2 ∞ n=1 ∞ 0≤k≤M n=1 Gn (tk−m , s0 ) ψ(tk−m ), φn Gn (tk , sm ) ψ(tk ), φn 2 + sup ψ(tk ) (2.8) 0≤k≤M Case 2: m > k Similarly, we can deduce for the case k < m that uq (tk , sm ) ≤ 2T sup 1≤k,m≤M f uq−1 (tk , sm ), tk , sm + sup 0≤m≤M ϕ(sm ) (2.9) Moreover, from the condition (2.2), we have f uq−1 (tk , sm ), tk , sm ≤ ≤ f uq−1 (tk , sm ), tk , sm − f (u0 (tk , sm ), tk , sm ) + f (u0 (tk , sm ), tk , sm ) K uq−1 (tk , sm ) − u0 (tk , sm ) + f (0, tk , sm ) (2.10) Combining (2.8)-(2.10) and putting CT = max 2T K + ; , we get sup 1≤k,m≤M uq (tk , sm ) ≤ 2T K + ≤ CT sup sup uq−1 (tk , sm ) 1≤k,m≤M uq−1 (tk , sm ) 1≤k,m≤M f (0, tk , sm ) + sup 1≤k,m≤M f (0, tk , sm ) + sup 1≤k,m≤M + sup 0≤m≤M +2 ϕ(sm ) sup 0≤m≤M ϕ(sm ) + sup ψ(tk ) 0≤k≤M + sup ψ(tk ) 0≤k≤M Remark 2.1 If f (u(t, s), t, s) ≡ f (t, s), then for k > m, since Gn (tk−m+l , sl ) ≤ Gn (tk , sm ) and Gn (tk−m , s0 ) ≤ Gn (tk , sm ) we have following estimate u (tk , sm ) ≤ 2ω2 m l=1 ∞ f (tk−m+l , sl ) , φn Gn (tk−m+l , sl ) Gn (tk , sm ) n=1 ≤ 2m2 ω2 sup ≤ 2T sup ∞ ≤ max{T , 1} +2 +2 ∞ Gn (tk , sm ) ψ(tk−m ), φn n=1 f (tk , sm ) 1≤k,m≤M ψ(tk−m ), φn Gn (tk−m , s0 ) ∞ n=1 f (tk , sm ) , φn 1≤k,m≤M n=1 2 f (tk , sm ) sup 1≤k,m≤M + ψ(tk−m ) ≤ max{T , 1} + sup 0≤m≤M ϕ(sm ) sup 1≤k,m≤M f (tk , sm ) + sup ψ(tk ) 2 + sup ψ(tk ) 0≤k≤M (2.11) 0≤k≤M For m > k, we have a similar proof Theorem 2.2 Let the source function f of the problem (1.1) satisfying the Lipschitz condition (2.2) Then, the iterative sequence uq (tk , sm ) defined by (2.7) strongly converges to the discrete solution u(tk , sm ) (2.6) of (1.1) in norm H in the sense of κTq sup uq (tk , sm ) − u(tk , sm ) ≤ sup u1 (tk , sm ) , − κT 1≤k,m≤M 1≤k,m≤M where κT < is a positive constant depending only on T Proof For short, we denote f k,m uk,m = f (u(tk , sm ), tk , sm ) Putting wqk,m = uq+1 (tk , sm ) − uq (tk , sm ), it follows from (2.7) that for k > m we have uq+1 (tk , sm ) − uq (tk , sm ) 2 ≤ ω ≤ ω2 m2 sup ≤ m ∞ n=1 l=1 f uq (tk−m+l , sl ), tk−m+l , sl − f uq−1 (tk−m+l , sl ), tk−m+l , sl , φn f uq (tk , sm ), tk , sm ) − f uq−1 (tk , sm ), tk , sm 1≤k,m≤M T K sup 1≤k,m≤M uq (tk , sm ) − uq−1 (tk , sm ) 2 Thus, we get uq+1 (tk , sm ) − uq (tk , sm ) ≤ T K sup1≤k,m≤M uq (tk , sm ) − uq−1 (tk , sm ) We can always choose T > small enough such that κT := T K < Then, we have uq+r (tk , sm ) − uq (tk , sm ) ≤ uq+r (tk , sm ) − uq+r−1 (tk , sm ) + + uq+1 (tk , sm ) − uq (tk , sm ) ≤ κTq+r−1 sup 1≤k,m≤M u1 (tk , sm ) − u0 (tk , sm ) + + κTq ≤ κTq κTr−1 + κTr−2 + + ≤ Therefore, we obtain κTq − κTr − κT sup 1≤k,m≤M sup 1≤k,m≤M sup 1≤k,m≤M u1 (tk , sm ) − u0 (tk , sm ) u1 (tk , sm ) u1 (tk , sm ) κTq uq+r (tk , sm ) − uq (tk , sm ) ≤ − κT sup 1≤k,m≤M u1 (tk , sm ) , (2.12) which leads to the claim that uq (tk , sm ) is a Cauchy sequence in H and then, there exists uniquely u(tk , sm ) ∈ H such that uq (tk , sm ) → {u(tk , sm )} as q → ∞ Because of this convergence and Lipschitz property (2.2) of nonlinear source term f , it is easy to prove that f uq (tk , sm ) → f (u(tk , sm )) as q → ∞ Therefore, u(tk , sm ) is the discrete solution of the problem (P).(1.1) When r → ∞, it follows (2.12) from that uq (tk , sm ) − u(tk , sm ) ≤ κTq − κT sup 1≤k,m≤M u1 (tk , sm ) For m > k, we also have a similar proof Hence, we complete the proof of the theorem Numerical example In this section, we are going to show a numerical example in order to validate the efficiency of our scheme It will be observed by comparing the results between numerical and exact solutions We shall choose given functions in such a way that they lead to a given exact solution The example is involved with the Hilbert space H = L2 (0, π) and associated with homogeneous boundary conditions On the other hand, numerical results with many 3-D graphs shall be discussed in the last subsection Now we take the following problem as an example for the nonlinear case with f the Lipschitz function   ut (x, t, s) + u s (x, t, s) − u xx (x, t, s) + u (x, t, s) = f (u, x, t, s) , (x, t, s) ∈ (0, π) × 0, 41 × 0, 14 ,        , (t, s) ∈ 0, 41 × 0, 14 , u (0, t, s) = u x (π, t, s) =    u (x, 0, s) = ϕ (x, s) , (x, s) ∈ [0, π] × 0, 14 ,      u (x, t, 0) = ψ (x, t) , (x, t) ∈ [0, π] × 0, 14 , where f (u, x, t, s) = 1 7x sin (u) + 49h (x, t, s) − sin (h (x, t, s)) , ϕ (x, s) = + e−s sin , 4 ψ (x, t) = −t 7x e + sin 7x −t ∂ e + e−s sin With the operator L = − +I and D (L) = v ∈ H (0, π) ∩ H (0, π) : v (0) = v x (π) = , ∂x 1 sin n + x and λn = n + + We shall give the iterative scheme (2.8) to get the approximate soluwe get φn = π 2 tion uk,m q = uq (tk , sm ) by the following steps Step With q = 0: u0 (tk , sm ) = 0, ≤ k, m ≤ M Step Let proceed to the (q − 1) - time, we get uq−1 (tk , sm ), ≤ k, m ≤ M Then we shall obtain uq (tk , sm ), ≤ k, m ≤ M as follows For k > m: and h (x, t, s) = uq (tk , sm ) (x) = where Rk−m+l,l = q−1 m ω e 53 (tk +sm ) π 2π 53 e (tk−m+l +sl ) Rk−m+l,l sin q−1 l=1 53 7x 53 7x + e− (tk +sm ) e tk−m e−tk−m + sin , (x) − sin hk−m+l,l (x) sin sin uk−m+l,l q−1 7x 49 −tk−m+l dx + e + e−sl 16 (3.13) (3.14) For m > k: uq (tk , sm ) (x) = k ω e 53 (tk +sm ) 53 e (tl +sm−k+l ) Rl,m−k+l sin q−1 l=1 53 7x 53 7x + e− (tk +sm ) e sm−k + e−sm−k sin , (3.15) where Rl,m−k+l = q−1 2π π l,m−k+l (x) − sin hl,m−k+l (x) sin sin uq−1 7x 49 −tl dx + e + e−sm−k+l 16 (3.16) Since (3.14) and (3.16) are hard to compute, we shall approximate them by using Gauss-Legendre quadrature method (see π 7x j j0 e.g [10]) Particularly, they can be determined in the following form H (x) sin 7x j=0 w j H x j sin , where x j dx = are abscissae in [0, π] and w j are corresponding weights, j0 ∈ N is a given constant Denoting E = uex − u, we compute the discrete l2 -norm and l∞ -norm of E by E l2 = |G| χ G ∈G |E (χG )|2 , E l∞ = max |E (χG )| , χG ∈G (3.17) (a) (b) Figure 1: The exact solution uex (x, t, s) = shown in (b) at t = 7x −t e + e−s sin shown in (a) in comparison with the approximate solution (3.13)-(3.15) where G = {χG } is a set of (L + 1) M points on uniform grid [0, π] × (0, T ] × (0, T ] and |G| cardinality of G In our computations, we always fix j0 = and L = 20 The comparison between the exact solutions and the approximate solutions for the examples respectively are shown in Figure in graphical representations As this figure, we can see that the exact solution and the approximate solution are close together Furthermore, convergence is observed from the computed errors in Table 1, respectively, which is reasonable for our theoretical results Table 1: Numerical results (3.17) with j0 = 5, L = 20 E E l2 l∞ q = 2, M = 50 5.8273E-3 1.1642E-2 q = 3, M = 100 2.9003E-3 5.8511E-3 q = 4, M = 200 1.4417E-3 2.9184E-3 q = 5, M = 400 7.1870E-4 1.4572E-3 References [1] A Tersenov, Basic boundary value problems for one ultraparabolic equation, Siberian Mathematical 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finance (e.g [6]), physics (such as multiparameter Brownian motion [16]) and biological models Among many applications, the ultraparabolic equation arises as a mathematical model... 2014 [4] A Ashyralyev and S Yilmaz, An Approximation of Ultra-Parabolic Equations, Abstract and Applied Analysis, vol 2012, Article ID 840621, 14 pages, 2012 [5] Q Deng , T G Hallam , An age structured