On rate of convergence of Tonelli’s and weak approximation methods for loaded equations Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2018, 11(3), 286-294

УДК 517.95

On Rate of Convergence of Tonelli's and Weak Approximation Methods for Loaded Equations

Yuri Ya. Belov* Kirill V.Korshun^

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

*

Received 27.06.2017, received in revised form 05.09.2017, accepted 20.02.2018

We consider the Cauchy problem for a loaded partial differential equation arising in coefficient inverse problem. The convergence of Tonelli's and weak approximation methods for this problem is previously proved. In the article we will prove linear rate of convergence of given methods.

Keywords: differential equation, inverse problem, Cauchy problem, Tonelli's method, weak approximation method, convergence.

DOI: 10.17516/1997-1397-2018-11-3-286-294.

1. Auxiliary symbols and theorems

We will use the following notation.

Q is bounded domain in the En space. x = (x1,... ,xn) is a point in En. dQ is the boundary of Q. QT is the cylindrical domain (0,T) x Q.

Ck(Q) (Ck(Q)) is the set of all k times continuously differentiable functions of Q (Q), having bounded derivatives up to k-th order.

n[QiT] = {(t,x)\t e [0,T],x e En}.

nj^T] = {(t,x)\te [0,T],x e En, \x\ < M,M - const}.

Ckm([0,T], Q) (Ck'm(n[0,T])) is the set of all functions of n +1 variables (t,xu ... ,xri) in n[0 ,T], which are k times continuously differentiable with respect to t and m times continuously differentiable with respect to spartial variables. All the derivatives listed above are bounded in

n[0,T ].

t is a real-valued parameter, t e [0, t0], t0 > 0.

A, B, Cj, i e N, are nonnegative constants depending on the initial data of problems being investigated but do not depending on t.

Mean value theorem for Definite integral. Let f (x) be a continuous function defined for x e [a, b]. Then

* ybelov@sfu-kras.ru 1 kkorshun@sfu-kras.ru © Siberian Federal University. All rights reserved

2. The Cauchy problem

We consider the inverse problem with unknown coefficient *(t):

du d2u . . „, . . .

m = + *(t)f (tx> (1)

u(0, x) = uo(x), (2)

u(t, 0) = <(t), (3)

for (t,x) G n[0jT], n = 1 • We consider

uo(0) = <£>(0), (4)

u0(x) and f (t,x) are bounded and arbitrary smooth functions. By substitution x = 0 into (1) and using (3) we find unknown *(t)

<(t) - -uxx(t, 0) (5)

m = -• (5)

Using (5) we reduce inverse problem (1)-(3) to the Cauchy problem for a loaded equation

du d2 u

dt = dhfi + 4(t>x)(<(t) - uxx(t, 0)), (6)

u(0, x) = uo(x), (7)

where t,x) = f (t,x)/f (t, 0).

3. Lemma 1

Consider the one-dimensional problem dzT d2zT

~dT = - *(t,x)zTx(t, 0)+ Ft(t,x), T G [0,To], (8)

zT (t,x)\-T <t<0 = 0, (9) in domain n^T]. If zT(t,x) G C1,4(n[0jT]) is a solution to (8), (9), t,x) G C0'2(n[0jT]),

dk

dxk FT ^

^ At, 0 ^ k ^ 2, then \zx(t,x)\ ^ Bt, where A, B are

Ft (t,x) G C0'2(U[0,T]),

constants not depending on t.

Proof. Note that (8), (9) have a unique solution [1, p. 65]. By applying the maximum principle [2, p. 16] to (8), (9) in domain n[0ji], 0 <t < T,

\zT(n,x)\ < sup\zT(0,x)\ + n(Ci sup \zTxx(9,x)\ + sup \FT(0,x)M, n G [0,t].

E1 V n[0,t] n[0,t] J

Here, C1 is a constant limiting \^(t, x)\. By applying supn to both sides of previous inequality, sup \zx(n,x)\ < sup\zx(0,x)\ + C21( sup \zTXT(e,x)\ + sup \Fx(0,x)M,C2 =max(C1,1). (10)

n[o,t] E1 \n[o,t] n[o,t] /

We differentiate (8), (9) two times with respect to x and apply the maximum principle to the resulting equation,

suP lzL(n,x)l < sup |zXx(0,x)| + Cstl sup \zlx(e,x)\ + sup

n[0,t] E1 \n[ojtj n[0,t]

d2

dx FT (e'x)

(11)

where C3 = supn[0 \&xx(t,x)\. We denote 7(t) = supn[0t] \zT(e,x)\ + supn[01] \zTXx(e,x)

i[0, T] —) — "-Kii[0, t] lzX(e,x)\ + supn[0, t] \ZXx\

[1, p. 65]. Note, that

zX

xx

E1 E1

7(0) = sup \zT(0, x) + sup \zXx(0, x)\ =0. By summing up (10) and (11),

Y(t) < 7(0) + C\t sup ^(e^ + C5Tt < C6tY(t) + Cfjrt,

n[0, t]

C4 = C2 + C3, C5 = 2C4A, C6 = max(C4,C5).

(1 - C6t)7(t) < C6Tt. (12)

Let t* > 0 be a constant such as 1 - C6t* = S > 0, t*K > T, K is integer. Note that t* and S not depend on r. Thus,

7(t) < S-1C6rt*, 0 < t < t*. (13)

We consider (8), (9) in domain n^*^*]. From (13)

sup \zT(t*,x)\ + sup\zTxx(t*,x)\ < S-1C6rt*.

E1 E1 xx

Similarly to (10), (11), we prove

sup \zx(n,x)\ < sup \zx(t*,x)\ + C2(t - t*) sup \

zxcx(e,x)\ + sup \FX (e,x)\ K

E1 Vn,,, t] n[t*, t] /

i[t*,t] E \n[t» ,t] n[t*,t]

sup \zlx (n,x)\ < sup\zlx(t*,x)\ + C3(t - t*)[ sup \zTxx(e,x)\ + sup

n[t*,t] E1 Vn[t, ,t] n[t*,t]

(1 - C6(t - t*))7(t) < S-1C6Tt* + C6Tt*,

and finally,

7(t) < S-1 (S-1C6Tt* + C6Tt*) , 0 < t < 2t*.

By applying exact same reasoning in domain n[2i* 3i*], we prove

3

d2

dx Fx (e,x)

7(t) < S-1 (S-1 (S-1C6Tt* + C6Tt*) + C6Tt*) = C6Tt*Y^ S-i, 0 < t < 3t*

i=1

Making K steps, we prove

K

sup \z(t,x)\ < 7(T) = C6Tt*Y^ S-i = Bt.

i[0,T ]

)

4. Tonelli's method

Let t G [0, t0] be a constant such that Nt = T, N is integer. We make time shift by t in trace of unknown function:

duT

d2t

-df = dx^ + W, x)(V(t) - uTx(t - t, 0)),

ux(t, x)

T <t<0 = uo(x).

(14)

(15)

The method of solving problems like (6), (7) by approximation (14), (15) is called Tonelli's method [3].

We consider initial data of the Cauchy problem (6), (7) to satisfy the following conditions:

uo(x) G Ck (E1), f (t,x) G C0'k (H[ot]), k > 6, v(t) G C2([0,T]), f (t, 0) > S> 0. (16)

By (16), the problem (14), (15) is a Cauchy problem for a heat equation with continuous and bounded coefficients, which have a solution ux for any t g [0, t0].

Remark. The problem (14), (15) is a particular case of problem (4.1.8), (4.1.9) [1, p. 60] with a(t) = 1, b(t) = c(t) = y = 0. Under Theorem 4.1.1 [1, p. 64], the solution ux (t,x) of problem (14), (15) converges to solution u(t,x) of problem (6), (7) with t ^ 0.

We prove inequalities (4.1.15) [1, p. 61] for k from 1 to 6 and (4.1.16), (4.1.17) [1, p. 61] for k from 0 to 4 exactly same way as in proof of Theorem 4.1.1. Using Theorem 4.1.1,

d

—■ux (t,x)

< C7,

d k

dxk u (t,x)

< C8, k = 0,..., 6.

dk

dxk u(t,x)

< C9, k = 0,..., 4, (t,x) G Pi[0,T].

(17)

(18)

Here and later, Ci are constants (maybe different ones) depending on initial data of the problem (6), (7), but not depending on t.

We denote zT = uT — u. With substraction (6), (7) from (14), (15), function zT is a solution

to

dzT d2zT

Ht = dx2 + ^x)Fx(t),

zx (0,x) = 0,

(19)

(20)

where

Fx (t) =

uxx(t, 0) - u'0(0),

t ^ t,

c(t, 0) - uxxx(t - t, 0), t > t.

We add and substract uxx(t, 0) to Fx:

Fx (t) =

[uxx(t, 0) - uxxx(t, 0)] + [uxxx(t, 0) - u0'(0)] :

t ^ t,

[uxx (t, 0) - VXxx(t, 0)] + [vxxx (t, 0) - Vxxx(t - T, 0)] , t>T

Thus, function z is a solution to

dzx d zx

= dx^ - *(t, x)zTxx(t, 0) + 4>(t, x)Fx(t),

T

u

zT (0, x) = 0,

(22)

where

Ft (t) =

uXx(t, 0) - u0(0),

t ^ T,

uXx(t, 0) - uTxx(t - T, 0), t>T.

Let t < t .By initial condition (15), <x(0,0) = u0(0); by (14) and (17), \uXXxt\ < C9. Thus,

Ft (t)\ = \Kx(t, 0) - u0'(0)\ =

Let t > t . By mean value theorem,

5(0,0) - u0(0) + / uixt(e,0)dd Jo

< C9t, 0 < t < t.

\Ft (t)\ = Kx(t, 0) - ulx (t - T, 0)\ = \ulxt(d, 0)\T < C9T, T<t < T, t - T < e < t.

Thus, FT(t) < C9t. Under Lemma 1,

sup \uT - u\ = max \zT(t,x)\ ^ C10T.

1[0,T ]

1[0,T ]

Theorem 1. If the conditions (16) are valid, then the Tonelli's method converges at linear rate, i.e. max \uT - u\ ^ Ct.

i[0,T ]

5. Weak approximation method

Let t > 0 be a constant such that Nt = T, N is integer. We make a split (see [2,5,6]) of the problem (6), (7) to two fractional steps, making time shift by T/2 in trace of unknown function:

duT ~dt duT

2-

d2u

dx2 '

— = 2^(t, x) [¿(t) - uxx(t -T /2, x)] uT(0, x) = uo(x),

t G (ut, (n +1 /2 )t], (23) t G ((n +1 /2)t, (n +1)t], (24) n = 0,1,...,N - 1. (25)

Remark. The problem (23)-(25) is a particular case of problem (2.2.18) investigated in article [7]. Inequalities (17), (18), and convergence of uT to u are proved. We denote averaging function (see [2, p. 41]) as

1 it+T

uTcp(t,x) = — J uT (e,x)de.

(26)

Note that uTcp(t,x) is defined in n^T-T]. By applying mean value theorem to right-hand side of (26) (uT(t,x) is continuous function of t),

uCP(t,x) = -/ uT(e,x)de = uT(£,x), t < £ < t + t.

1 r

(t,x) = -

T Jt

d2 1 ft+T dx2uCp(t,x) = - J uix(e,x)de = uxx(£,x), t < £ < t + T.

T

u

t

By (17), ux and uxx are

satisfying Lipschitz condition with respect to t, thus

\ux(t,x) - ux(t,x)\ = \ux(£,x) - ux(t,x)\ < C9\£ - t\ < C9t,

d2 x x dx2 ucp(t, x) - uxx(t, x)

\ulx(£,x) - uTxx(t,x)\ < C9 \£ - t\ < C9T.

We apply averaging function to the problem (23)-(25):

duCp dt

d 2

cp

dx2

( , d2 uP \ + 4(t, x) I y'(t) - -d^f (t, 0)1 + Fx (t, x), (t, x) G n[0,T-x]

uCp(0, x) = u0(x),

(27)

(28) (29)

where

Fx(t, x) = 1 ^ + |ai,x(d)uxxx(e, x) + &2,x(0)4(0, x) y (0) - uxx(0 -x /2, x)] -

d2t

cp

dx2

(t, x) - 4(t, x)

d2ux

y (t) - -dJ(t,0)

d0, (30)

ai,x (t)

ft

2, t G (ut, (n +1 /2)t],

^0, t G ((n +1 /2)t, (n + 1)t], Note that the term

a2,x (t)

I*

0, t G (ut, (n +1 /2)t], 2, t G ((n +1 /2)t, (n + 1)t].

*(t,x)

d 2l

dx2

'-(t, x) + 4(t, x)

d2ux

y'(t) - -dJ(t,0)

t+x t+x

in the integrand of Fx(t,x) (30) does not depend on 0, f a\x(0)d0 = f a2,x(0)d0 = 1, thus

tt

t+x

ty(t,x)d0 =

r[

d2l

a1,x(0) dx2 (t,x) + a2,x(0)4(t,x)

d2ux

y'(t) - -jut(t,0

d0.

Interval of integration [t, t + t ] is split by points {jT/2},j =0,..2N — 1 to three parts (or two, if t is exactly jT/2). Let Is,s = 1, 2 be subsets of interval [t, t + t] such that as,T(0) =0,0 G Is. We rewrite (30) as

2

Fx (t,x) = -

2 + -

T

J,. (• /{

d2ux \

,x) - -d^P (t,x)) d0+

x)V (0) - uxxx(0 - t, 0)] - 4(t,x)

v'(t) -

d 2u

dx2

■(t, 0)

d0. (31)

From (31)

d2u_Çv, dx2

9,x) (t,x)

<

uxx(0,x) uxx(t,x)

+

d2uÇp

c(t, x) dx2 (t, x)

<

< uTxxt(£, x)\0 - t\ + C9T < C11T, t < £ < 0 < t + T.

T

t

t

T

T

xx

We estimate the second term in FT (t,x):

02uZp

№,x)w (0) - <x(0 - t, 0)] - 4(t,x)[(^ ' (t) - ^xf (t, 0)]

< \4(0,x)W(0) - uXx(0 - t,0)] - 4(0,x)W(0) - uix(e,0)]| +

+ \4(0,x)[v'(0) - uXx(0, 0)] - t(t,x)[v'(t) - uXx(t, 0)]\ +

+

02uZp

4(t, x)№(t) - uXx(t, 0)] - 4(t, xW(t) - -d^T(t, 0)]

(32)

By applying the mean value theorem to (32), considering (16), (17), (27) we prove \FT(t,x)\ < C12t. By differentiation (31) two times with respect to x and applying same reasoning we prove

FT (t,x) < C13r.

By substraction (6), (7) from (28), (29) (here z — u cp u),

dzT d 2zT

= dx? - 0) + ft(t,x), (t,x) e n[q,t-t],

zT (0, x) — 0.

Under Lemma 1,

\zT(t,x)\ < Cisr, (t,x) G n[ojT_T].

Note that zT(t,x) G C^ and by (18) \zj(t,x)\ < C9, thus \zT(t,x)\ < C13T for (t,x) G n[0jT]. By this, (27), and the triangle inequality,

\uT — u\ < \uT — nTcp \ + \uTcp — u\ < C14T.

We finally proved

Theorem 2. If the conditions (16) are valid, then weak approximation method converges at linear rate in domain n[0 ^].

6. Future research

One can prove the Lemma 1 in multidimensional case, i.e. for problem

dzT

— — AzT (t, x) + B(zT) + Ft (t,x), (33)

dt

zT (0, x) — 0 (34)

in domain n[0^] — {(t,x)\0 ^ t ^ T,x G En,n > 1}, where B(z) is a linear operator with coefficients of C0'2(n[0 T]) class, depending on function z, its first-order partial derivatives, and traces of partial derivatives up to second order. For example,

n

B(zT) — zT(t, x) + sin(tx1x2 ... xn)zT(t, 0) + zTXi (t, 0) + AzT(t, 0).

i=i

This will provide a method for proving linear rate of convergence of weak approximation method for a range of inverse problems.

Example. The inverse problem

n

ut = Au + uxi + n(t)u + f (t, x), (t, x) e n[0jT], i=i

u(0, x) = uo(x), u(t, 0) = $(t),

with unknown n(t), reduces to a Cauchy problem for a loaded equation

Ut =

Au + Uxi +

4'(t) - Au(t, 0) - £n=i uxi (t, 0) - f (t, 0)

m

u(0, x) = uo(x).

l6( TTn\

-u + f (t,x), (35)

(36)

Provided f (t,x) e C0'6(n[0jT]), u0(x) e C6(En), <fr(t) e C2([0,T]), one can use weak approximation method to above problem, by splitting (35), (36) to fractional steps

= aiT (t)

AuT + J2 uXi + f (t,x)

+

+ a2,T (t)

4'(t) - AuT(t - 2, 0) - £n=i uTxi (t - 2, 0) - f (t, 0)

4(t)

uT (0, x) = uo(x),

(37)

(38)

proving boundedness [4] of derivatives of uT with respect to xi up to sixth order, proving convergence of uT to u. As in Section 4, by subtraction of (35), (36) from the split-problem and applying averaging function, one can write down the problem for function zT = uT — u:

zj = AzT + B(zT) + FT(t, x),

zT (0, x) = 0,

where

B( T) v- T (t )+ 4'(t) - f (t, 0) T( ) , AuT(t, 0) -£ni=i uXi (t, 0) T( ) B(z ) = 2_zXi (t,x)+--—-z (t,x)+--—-i-z (t,x)-

i=1

4(t)

4(t)

u(t,x)-AzT 10) - utgË zXx„, t 0),

4(t)

4(t)

and F(t,x) G C°'2(n\pTT]) is a function satisfying the conditions of Lemma 1. Thus one can prove linear rate of convergence of weak approximation method in this case.

T

u

T

References

[1] Yu.Ya.Belov et al., Nonclassical and inverse boundary-value problems, Nauchnye zametki, Krasnoyarsk, 2007 (in Russian).

[2] Yu.Ya.Belov, S.A.Cantor, Weak appromixation method, Krasnoyarsk, Krasnoyarsk Gos. Univ., 1999 (in Russian).

[3] L.Tonelli, Opere scelte. A cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerce, Roma, Edizioni Cremonese, 1962.

[4] Yu.Ya.Belov, K.V.Korshun, On some inverse problem for Burgers-type equation, Siberian journal of industrial mathematics, 16(2013), no. 3, 28-40.

[5] N.N.Yanenko, G.V.Demidov, Investigation of a Cauchy problem by weak approximation method, Doklady AN SSSR, 167(1966), no. 6, 1242-1244.

[6] A.A.Samarsky, On convergence of fractional steps method for heat transfer equations, Journal of computational mathematics and mathematical physics , 2(1962), no. 6, 1347-1354.

[7] Yu.Ya.Belov, On Estimates of Solutions of the Split Problems for Some Multidimensional Partial Differential Equations, Journal of Siberian Federal University. Mathematics & Physics, 2(2009), no. 3, 258-270.

О скорости сходимости метода Тонелли и метода слабой аппроксимации в задачах Коши для нагруженных уравнений

Юрий Я. Белов Кирилл В. Коршун

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Рассматривается задача Коши для нагруженного уравнения в частных производных, возникающая при решении коэффициентных обратных задач. Ранее доказана сходимость метода Тонелли и метода слабой аппроксимации для 'рассматриваемой задачи. В работе доказывается первый порядок сходимости данных методов.

Ключевые слова: дифференциальные уравнения, обратная задача, задача Коши, метод Тонелли, метод слабой аппроксимации.