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

Accepted ManuscriptA finite difference scheme for nonlinear ultra-parabolic equations Vo Anh Khoa, Tuan Nguyen, Le Trong Lan, Nguyen Thi Yen Ngoc DOI: http://dx.doi.org/10.1016/j.aml.201

Accepted Manuscript

A finite difference scheme for nonlinear ultra-parabolic equations

Vo Anh Khoa, Tuan Nguyen, Le Trong Lan, Nguyen Thi Yen Ngoc

DOI: http://dx.doi.org/10.1016/j.aml.2015.02.007

To appear in: Applied Mathematics Letters

Received date: 23 December 2014

Revised date: 9 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

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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.

b Applied 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.

Abstract

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function Math-ematically, 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 non-linear case, we use an iterative scheme by non-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

1 Introduction

Let H be a Hilbert space with the inner product h·, ·i and the norm k·k In this paper, we consider the problem of finding

u: [0, T ] × [0, T] → H satisfies the following ultraparabolic equation











∂

∂tu (t, s) +

∂

∂su (t, s) + Au (t, s) = f (u (t, s) , t, s) , (t, s) ∈ (0, T) × (0, T) ,

u (0, s) = ϕ (s) , s ∈ [0, T],

u (t,0) = ψ (t) , 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 multi-parameter 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

∂t +

∂u

∂s −∆u + h (x, t, s) u + Au = f (x, 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

∂t +

∂u

∂s − ∇ ·(k∇u − qu) = −µu, 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

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 ≡ 0 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 ≡ 0 for strongly positive operator and obtained some fundamental stability results On the other hand, Marcozzi et al [14] developed an adaptive method-of-lines extrapolation discontinuous Galerkin method for an ultraparabolic model equation given by∂u

∂t+a (x)∂u∂s−∆u = f (x, t, s) , 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 3 to verify the effectiveness of the method

2 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

k f (u, t, s) − f (v, t, s)k ≤ K ku − vk , (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≥1in H, associated with the eigenvalues such that

0 < λ1≤ λ2≤ λ3≤ lim

n→∞λn=∞

With Gn(t, s) = exp λn

2 (t + s)

 , by a simple computation, the problem (1.1) is transfomed into the following problem











∂

∂t +∂s∂  D

u (t, s) , φn

E

Gn(t, s)=h f (u (t, s) , t, s) , φni Gn(t, s) ,

hu (0, s) , φni Gn(t, s) = hϕ (s) , φni Gn(t, s) , D

u (t,0) , φn

E

Gn(t, s) = hψ (t) , φni Gn(t, s)

(2.3)

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

D

u(tk,sm), φn

E

Gn(tk,sm) −Du(tk−1,sm), φn

E

Gn(tk−1,sm)

D u(tk−1,sm), φn

E

Gn(tk−1,sm) −Du(tk−1,sm−1), φn

E

Gn(tk−1,sm−1) ω

=D

f (u(tk,sm), tk,sm) , φnEGn(tk,sm) , (2.4) and

D

u (0, sm) , φnEGn(tk,sm) =Dϕ (sm) , φnEGn(tk,sm) , Du (tk,0) , φnEGn(tk,sm) =Dψ (t) , φn

E

Gn(tk,sm) (2.5)

By induction, it follows from (2.5) that

D

u(tk,sm), φnEGn(tk,sm) = ω

p

X

l=1

D

fu(tk−p+l,sm−p+l), tk−p+l,sm−p+l, φnE

Gntk−p+l,sm−p+l +D

u(tk−p,sm−p), φnEGntk−p,sm−p

2

for all p ∈ N Hence, we obtain that the explicit form of discrete solution of (1.1)

u (tk,sm) =

∞

X

n=1

D

u (tk,sm) , φnE

whereDu (tk,sm) , φnEis given by

D

u (tk,sm) , φnE=











ωPm l=1

D

f (u(tk−m+l,sl), tk−m+l,sl) , φnEGn(tk−m+l,sl) +Dψ(tk−m), φn

E

Gn(tk−m,s0)

ωPk l=1

D

f (u(tl,sm−k+l), tl,sm−k+l) , φn

E

Gn(tl,sm−k+l) +Dϕ(sm−k), φn

E

Gn(t0,sm−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) =X∞

n=1

D

whereDuq(tk,sm) , φn

E

is defined by

D

uq(tk,sm) , φnE

=











ωPm l=1

D

fuq−1(tk−m+l,sl), tk−m+l,sl, φnE

Gn(tk−m+l,sl) +Dψ(tk−m), φnEGn(tk−m,s0)

ωPk l=1

D

fuq−1(tl,sm−k+l), tl,sm−k+l, φn

E

Gn(tl,sm−k+l) +Dϕ(sm−k), φnEGn(t0,sm−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 fuq−1

Theorem 2.1 Letnuq(tk,sm)o

q≥1be the iterative sequence defined by (2.7) Then, it satisfies the a priori estimate

sup

1≤k,m≤M q(tk,sm) 2≤ CT sup

1≤k,m≤M q−1(tk,sm) 2+ sup

1≤k,m≤Mk f (0, tk,sm)k2+ sup

0≤m≤Mkϕ(sm)k2+ sup

0≤k≤Mkψ(tk)k2!

, where CTis 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

q(tk,sm) 2 ≤ 2ω2

∞

X

n=1

m

X

l=1

Gn(tk−m+l,sl)

Gn(tk,sm)

D

fuq−1(tk−m+l,sl), tk−m+l,sl

 , φn

E

2

+ 2

∞

X

n=1

Gn

(tk−m,s0)

Gn(tk,sm)

D ψ(tk−m), φnE 2

≤ 2ω2m2 sup

1≤k,m≤M

∞

X

n=1

Dfuq−1(tk,sm), tk,sm

 , φnE 2

+ 2 sup

0≤k≤M

∞

X

n=1

Dψ(tk), φnE 2

≤ 2T2 sup

1≤k,m≤M



uq−1(tk,sm), tk,sm

2

+ 2 sup

Case 2: m > k Similarly, we can deduce for the case k < m that

q(tk,sm) 2≤ 2T2 sup

1≤k,m≤M



uq−1(tk,sm), tk,sm 2+ 2 sup

Moreover, from the condition (2.2), we have



uq−1(tk,sm), tk,sm 

uq−1(tk,sm), tk,sm

− f (u0(tk,sm), tk,sm) 0(tk,sm), tk,sm)k

≤ K q−1(tk,sm) − u0(tk,sm) k f (0, tk,sm)k (2.10)

Combining (2.8)-(2.10) and putting CT = maxn

2T2

K2+ 1

; 2o, we get

sup

1≤k,m≤M q(tk,sm) 2 ≤ 2T2K2+ 1

sup

1≤k,m≤M q−1(tk,sm) 2+ sup

1≤k,m≤Mk f (0, tk,sm)k2!

+ 2 sup

0≤m≤Mkϕ(sm)k2+ sup

0≤k≤Mkψ(tk)k2!

≤ CT sup

1≤k,m≤M q−1(tk,sm) 2+ sup

1≤k,m≤Mk f (0, tk,sm)k2+ sup

0≤m≤Mkϕ(sm)k2+ sup

0≤k≤Mkψ(tk)k2!

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

ku (tk,sm)k2 ≤ 2ω2

∞

X

n=1

Pm l=1

D

f (tk−m+l,sl) , φnEGn(tk−m+l,sl)

Gn(tk,sm)

2

+ 2

∞

X

n=1

D ψ(tk−m), φn

E

Gn(tk−m,s0)

Gn(tk,sm)

2

≤ 2m2ω2 sup

1≤k,m≤M

∞

X

n=1

Df (tk,sm) , φnE 2

+ 2

∞

X

n=1

Dψ(tk−m), φnE 2

≤ 2T2 sup

1≤k,m≤Mk f (tk,sm)k2+ 2kψ(tk−m)k2≤ 2 max{T2,1} sup

1≤k,m≤Mk f (tk,sm)k2+ sup

0≤k≤Mkψ(tk)k2!

≤ 2 max{T2,1} sup

1≤k,m≤Mk f (tk,sm)k2+ sup

0≤m≤Mkϕ(sm)k2+ sup

0≤k≤Mkψ(tk)k2!

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 sequencenuq(tk,sm)odefined by (2.7) strongly converges to the discrete solution u(tk,sm) (2.6) of (1.1) in norm H in the sense of

sup

1≤k,m≤M q(tk,sm) − u(tk,sm) ≤ κ

q T

1 − κT sup

1≤k,m≤Mku1(tk,sm)k , where κT <1 is a positive constant depending only on T

Proof For short, we denote fk,m

uk,m

= f (u(tk,sm), tk,sm) Putting wk,m

q = uq+1(tk,sm) − uq(tk,sm), it follows from (2.7) that for k > m we have

q+1(tk,sm) − uq(tk,sm) 2 ≤ ω2X∞

n=1

m

X

l=1

D

fuq(tk−m+l,sl), tk−m+l,sl



− fuq−1(tk−m+l,sl), tk−m+l,sl

 , φn

E

2

≤ ω2m2 sup

1≤k,m≤M



uq(tk,sm), tk,sm

 ) − fuq−1(tk,sm), tk,sm

2

≤ T2K2 sup

1≤k,m≤M q(tk,sm) − uq−1(tk,sm) 2 Thus, we get q+1(tk,sm) − uq(tk,sm) ≤ T K sup1≤k,m≤M q(tk,sm) − uq−1(tk,sm)

enough such that κT := T K < 1 Then, we have

q+r(tk,sm) − uq(tk,sm) ≤ q+r(tk,sm) − uq+r−1(tk,sm) q+1(tk,sm) − uq(tk,sm)

≤ κq+r−1T sup

1≤k,m≤Mku1(tk,sm) − u0(tk,sm)k + + κqT sup

1≤k,m≤Mku1(tk,sm) − u0(tk,sm)k

≤ κqT



κr−1T + κr−2

T + + 1

sup

1≤k,m≤Mku1(tk,sm)k

≤ κ

q T



1 − κr T



1 − κT sup

1≤k,m≤Mku1(tk,sm)k Therefore, we obtain

q+r(tk,sm) − uq(tk,sm) ≤ κ

q T

1 − κT sup

4

which leads to the claim thatnuq(tk,sm)ois a Cauchy sequence in H and then, there exists uniquely u(tk,sm) ∈ H such that n

uq(tk,sm)o → {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 fuq(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

q(tk,sm) − u(tk,sm) ≤ κ

q T

1 − κT sup

1≤k,m≤Mku1(tk,sm)k For m > k, we also have a similar proof Hence, we complete the proof of the theorem

3 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) + us(x, t, s) − uxx(x, t, s) + u (x, t, s) = f (u, x, t, s) , (x, t, s) ∈ (0, π) ×0,1

4



×0,1 4

 ,

u (0, t, s) = ux(π, t, s) = 0 , (t, s) ∈h0,1

4

i

×h0,1 4

i ,

u (x,0, s) = ϕ (x, s) , (x, s) ∈ [0, π] ×h0,1

4

i ,

u (x, t,0) = ψ (x, t) , (x, t) ∈ [0, π] ×h0,1

4

i , where

f (u, x, t, s) = 1

4

 sin (u) + 49h (x, t, s) − sin (h (x, t, s)), ϕ (x, s) = 1

4 1 + e−s
sin 7x

2

! , ψ (x, t) = 1

4



e−t+ 1 sin 7x 2

!

and h (x, t, s) = 1

4 e−t+ e−s
sin 7x

2

! With the operator L = −∂x∂2+I andD (L) =nv ∈ H1(0, π) ∩ H2(0, π) : v (0) = vx(π) = 0o,

we get φn=

r

2

πsin

n +1 2

! x

! and λn= n + 1

2

!2

+ 1 We shall give the iterative scheme (2.8) to get the approximate solu-tion uk,m

q = uq(tk,sm) by the following steps

Step 1 With q = 0: u0(tk,sm) = 0, 1 ≤ k, m ≤ M

Step 2 Let proceed to the (q − 1) - time, we get uq−1(tk,sm), 1 ≤ k, m ≤ M Then we shall obtain uq(tk,sm), 1 ≤ k, m ≤ M as follows

For k > m:

uq(tk,sm) (x) = ω

e538 (t k +s m )

m

X

l=1

e538 (tk−m+l+s l )Rk−m+l,l

q−1 sin 7x

2

! +1

4e−

53

8 (t k +s m )e538 tk−m

e−t k−m+ 1

sin 7x 2

! , (3.13) where

Rk−m+l,l

1 2π

Z π 0

 sinuk−m+l,l q−1 (x)− sinhk−m+l,l(x)sin 7x

2

! dx

! +49 16



e−t k−m+l+ e−sl

For m > k:

uq(tk,sm) (x) = ω

e538 (t k +s m )

k

X

l=1

e538 (t l +s m−k+l )Rl,m−k+lq−1 sin 7x

2

! +1

4e−

53

8 (t k +s m )e538 sm−k 1 + e−s m−k
sin 7x

2

! , (3.15) where

Rl,m−k+lq−1 = 1

2π

Z π 0

 sinul,m−k+lq−1 (x)− sinhl,m−k+l(x)sin 7x

2

! dx

! +49 16



e−t l+ e−sm−k+l

Since (3.14) and (3.16) are hard to compute, we shall approximate them by using Gauss-Legendre quadrature method (see e.g [10]) Particularly, they can be determined in the following formR0πH (x)sin7x

2



dx = Pj0

j=0wjHxj

sin7xj

2

 ,where xj

are abscissae in [0, π] and wjare corresponding weights, j0 ∈ N is a given constant Denoting E = uex− u, we compute the discrete l2-norm and l∞-norm of E by

kEkl 2=

s 1

|G|

X

χ G ∈G

|E (χG)|2, kEkl∞ = max

(a) (b)

Figure 1: The exact solution uex(x, t, s) = 1

4 e−t+ e−s
sin 7x

2

! shown in (a) in comparison with the approximate solution (3.13)-(3.15) shown in (b) at t = 1

4.

where G = {χG} is a set of (L + 1) M2points on uniform grid [0, π] × (0, T] × (0, T] and |G| cardinality of G

In our computations, we always fix j0 = 5 and L = 20 The comparison between the exact solutions and the approximate solutions for the examples respectively are shown in Figure 1 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

q = 2, M = 50 q = 3, M = 100 q = 4, M = 200 q = 5, M = 400

kEkl 2 5.8273E-3 2.9003E-3 1.4417E-3 7.1870E-4

kEkl ∞ 1.1642E-2 5.8511E-3 2.9184E-3 1.4572E-3

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6

− f (u0(tk,sm), tk,sm) 0(tk,sm), tk,sm)k...

Combining (2.8)-(2.10) and putting CT = maxn

2T2

K2+... tk,sm)k

≤ K q−1(tk,sm) − u0(tk,sm) k f (0, tk,sm)k