Numerical method for the equation with fractional derivative on state and with functional delay on time Текст научной статьи по специальности «Математика»

ISSN 1810-0198. Вестник ТГУ, т. 20, вып. 5, 2015

С другой стороны,

œ œ

a f Aika (fk) = ^ Aikzk,

i=l

i=l

поэтому limi

a [ fi) - z

Ei

0.

Из непрерывности оператора а следует, что г = а(/), поэтому г € а о Рр(д), тогда применяя теорему 2, получаем истинность вспомогательного утверждения для а о Рр.

Из построенного доказательства видно, что мультиоператор Рр слабо замкнут. Теорема доказана.

ЛИТЕРАТУРА

1. Борисович Ю.Г., Гельман Б.Д., Мышкис А.Д., Обуховский В.В. Введение в теорию многозначных оторбажений и дифференциальных включений. Издание 2-е, испр. и доп.-М: Книжный дом «Либроком», 2011. 224 с.

2. Hale J.K., Kato J. Phase Space for Retarded Equations with Infinite Delay // Funkcial. Ekvac. 1978. № 1, 21. P. 11-41.

БЛАГОДАРНОСТИ: Работа поддержана грантами Российского Фонда Фундаментальных Исследований № 14-01-00468 и № 14-01-92004.

Поступила в редакцию 9 июня 2015 г.

Petrosyan G.G. A THEOREM ON THE WEAK CLOSURE OF SUPERPOSITION MULTIOPERATORS

We prove a theorem on the weak closure of superposition multioperators for multifunction corresponding multivalued mapping, which is subject to conditions such as upper Caratheodory conditions. This theorem is a generalization of Theorem 1.5.30 from [1].

Key words: multioperators; weak convergence; measurable sections; compact set.

Петросян Гарик Гагикович, Воронежский государственный педагогический университет, г. Воронеж, Российская Федерация, кандидат физико-математических наук, старший преподаватель кафедры высшей математики, e-mail: [email protected]

Petrosyan Garik Gagikovich, Voronezh State Pedagogical University, Voronezh, the Russian Federation, Candidate of Physics and Mathematics, Senior Lecturer of the Higher Mathematics Department, e-mail: [email protected]

УДК 519.633

NUMERICAL METHOD FOR THE EQUATION WITH FRACTIONAL DERIVATIVE ON STATE AND WITH FUNCTIONAL DELAY ON TIME

© V.G. Pimenov, A.S. Hendy

Key words: fractional equation; delay; grid schemes; interpolation; extrapolation; convergence order.

For fractional diffusion equation with functional after-effect on time, the implicit numerical method is constructed and the order of its convergence is obtained. The method is a fractional analogue of the Crank-Nicholson method, and also uses interpolation and extrapolation of the prehistory of model on time.

ISSN 1810-0198. Вестннк Try, t. 20, HUO. 5, 2015

Consider a one-dimensional fractional diffusion equation [1] with functional delay [2]

du d au

~dt = + f(x, t, u(x t), ut(x, ■)),

x € [0,X], t € [0, T], d > 0, ut(x, ■) = {u(x,t + s), —t ^ s < 0} is the prehistory function, 1 < a ^ 2, the Riemann fractional derivative is define by

x

da u 1 d2 f u({,x)

dxa r(2 - a) dx2 J (x - C)a-1 ^ 0

with initial conditions: u(x,t) = <^(x,t), x € [0,X], t € [—t, 0], and with boundary conditions: u(0, t) = 0, u(X, t) = b(t), t € [0, T].

Let h = X/N, introduce x» = ih, i = 0,..., N, and A = T/M, tj = jA, j = 0,..., M. t/A = K is positive integer. Denote by uj approximations of functions u(xi,tj) at the nodes.

For every fixed i = 0,..., N let's introduce discrete prehistory for the time points tj, j = 0,..., M : {uk}j = {uk, j — m ^ k ^ j}. The mapping I : (uk}j ^ v»(t), t € [tj — t, tj + A/2] will be called the interpolation-extrapolation operator of discrete prehistory. We will use piecewise-linear interpolation with extrapolation by continuation

( i(ui(t — ti-i)+ ui-1 (ti — t)), ti-1 < t < ti, 1 < l < j, v*(t) = ^ i(uj(t — tj-i) + uj-i(tj — t)), tj < t < tj + A/2, [ ^(x,,t), —t < t < 0.

For approximation of a fractional derivative, we will use the right-shifted formula of Grunwald-Letnikov [1]

dau 1 1 ^^ r(k_a)

lim V -y---f u(x — (k — 1)Ax, t),

dxa r(—a) K^œ (Ax)a k=0 r(k + 1)

were Ax = x/K. Also, we will use further designations = r(—a)r(fc+i) • Then

dau(xi,tj+1 ) 1 . . . 1 ^ i

dxa ~ ~ 2(^a,xuj+i + ^a'Xuj), ^«'Xuj = g«>fcuj + .

fc=0

Applying the usual approximation to a derivative on time and interpolation designs with extrapolation, as a result we receive analog of Crank-Nicholson method

uj .1 — uj d • •

j+iA j = 2 (5a.xuj+i + 5a,xuj) + j i , (1)

fj+1 = f (xi,tj + A, vi(ij + A), vj+a (■)), i = 1,•••,N,j = 0,.. •, M — 1,

with the corresponding initial conditions u0 = ^(x^to), i = 0, ...,N, vj (t) = ^(x^t), t < to, i = 0,..., N, and with boundary conditions u0 = 0, j = b(tj). j = 0,..., M. The computational work in solving this system of equations is equivalent to solving two triangular systems at each time step. If a = 2 , we receive the Crank-Nicholson scheme for the heat

conductivity equation with functional delay. This is because of ¿2>xuj = —-—1— •

ISSN 1810-0198. Вестник ТГУ, т. 20, вып. 5, 2015

Now, we will reduce the proposed scheme to the general scheme which appeared in [2]. Let's j ,Uj, ■■■ , Uj

introduce yj = (U ,u2, • • • € RN 1, then (1) rewrite in form

(E - A)yj+1 = (E + A)yj + AFj,

As the matrix of E — A is reversible, this equation can be written in the following form as clarified in [2]

yj+i = Syj + A$(ij ,1 ({yi}j), A).

L e m m a 1. The error of approximation of a method of (1) without interpolation has order Aj + h.

L e m m a 2. The error of approximation of a method of (1) with piecewise-linear interpolation and extrapolation by continuation has order Aj + h.

L e m m a 3. The eigenvalue X of matrix S = (E — A)-i(E + A) satisfy the condition |A| < 1. From this follows that ||Sn|| ^ S for any natural degree n.

From the statements of these lemmas and results of [2], we can state the following theorem.

Theorem 1. The proposed method (1) with piecewise-linear interpolation and extrapolation by continuation converges with order Aj + h.

REFERENCES

1. Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation // Journal of Computational Physics. 2006. V. 213. P. 205-214.

2. Pimenov V.G., Lozhnikov A.B. Difference Schemes for the Numerical Solution of the Heat Conduction Equation with Aftereffect // Proceedings of the Steklov Institute of Mathematics. 2011. V. 275. S. 1. P. 137-148.

ACKNOWLEDGEMENTS: This work was supported by Government of the Russian Federation program 02.A03.21.0006 on 27.08.2013 and by RFBR Grant 13-01-00089.

Received 25 May 2015.

Пименов В.Г., Хенди А.С. ЧИСЛЕННЫЙ МЕТОД ДЛЯ УРАВНЕНИЯ С ДРОБНОЙ ПРОИЗВОДНОЙ ПО ПРОСТРАНСТВУ И ФУНКЦИОНАЛЬНЫМ ЗАПАЗДЫВАНИЕМ ПО ВРЕМЕНИ

Для дробного уравнения диффузии с функциональным последействием по времени сконструирован неявный численный метод и получен порядок его сходимости. Метод является дробным аналогом метода Кранка-Никольсон с использованием интерполяции и экстраполяции предыстории модели по времени.

Ключевые слова: дробные уравнения; запаздывание; сеточные схемы; интерполяция; экстраполяция; порядок сходимости.

Pimenov Vladimir Germanovich, Ural Federal University named after the first President of Russia B.N. Yeltsin, Ekaterinburg, the Russian Federation, Doctor of Physics and Mathematics, Professor, the Head of the Computational Mathematics Department, e-mail: [email protected]

Пименов Владимир Германович, Уральский федеральный университет имени первого Президента России Б.Н. Ельцина, г. Екатеринбург, Российская Федерация, доктор физико-математических наук, профессор, заведующий кафедрой вычислительной математики, e-mail: [email protected]

Hendy Ahmed Said, Ural Federal University named after the first President of Russia B.N. Yeltsin, Ekaterinburg, the Russian Federation; University of Benha, Benha, Egypt, Post-graduate Student of the Mathematics Department, e-mail: [email protected]

Хенди Ахмед Саид, Уральский федеральный университет имени первого Президента России Б.Н.Ельцина, г. Екатеринбург, Российская Федерация; Университет г. Бенха, г. Бенха, Египет, аспирант департамента математики, e-mail: [email protected]