CC BY
0INTEGRAL CONDITION FOR BENNEY-LUKE TYPE EQUATION THE PROBLEM GIVEN WITH THE RIGHT AND
INVERSE PROBLEM
Dekhkanov Khusan Tursunovich
Namangan State University doctoral student (PhD) https://doi.org/10.5281/zenodo.13894825 Abstract. The article studies the fractional order equation of the form D"u(t) + A(D"u(t)) + A2(D"u(t)) + Au(t) = f, where 0 <a< 1, for the time interval 0 < t < T
rT
with the integral boundary condition £ u(t)dt = p. Here, A : H ^ H is a self-adjoint,
unbounded, positive operator defined on a separable Hilbert space H, and A-1 is a compact operator. The article demonstrates the existence and uniqueness of solutions to the problem using the Fourier method. Additionally, it addresses the inverse problem concerning the determination of the right-hand side of the equation.
Key words. Integral, boundary condition, Benney-Luke equation, Hilbert, orthogonal. Аннотация. В статье рассматривается задача нахождения решения дробно-ордinaries уравнения вида D"u(t ) + A(D"u (t )) + A2(D"u(t )) + Au(t ) = f, где 0 <a< 1, на
ct
временном интервале 0 < t < T с интегральным краевым условием I u(t)dt = p. Oператор
J 0
A : H ^ H является самосопрженным, неограниченным, положительным оператором, определенным в сепарабельном гильбертовом пространстве H, а A"1 является компактным оператором. В статье с использованием метода Фурье показана существование и единственность решения задачи. Кроме того, рассматривается обратная задача, связанная с определением правой части уравнения.
Ключевые слова: интеграл, граничное условие, уравнение типа Бенни-Лука, Гильберт, ортогональный.
Annotatsiya. Maqolada Dau(t ) + A(Dau(t )) + A2(Dau(t )) + Au (t ) = f, 0 <a < 1,
T
ko'rinishidagi kasr tartibli tenglama uchun vaqt bo'yicha 0 < t < T, Ju(t)dt = p integral
0
chegaraviy shartni qanotlantruvchi yechimni topish masalasi o'rganilgan, A : H ^ H o'z-o'ziga qo'shma, chegaralanmagan, musbat H separabel Hilbert fazosida aniqlangan operator va Akompakt operator. Maqolada Furye usulidan foydalanib masalaning yechimi mavjudligi va yagonaligi ko'rsatilgan. Bundan tashqari maqolada tenglamaning o'ng tomonini topish bo'yicha teskari masala ham o'rganilgan.
Kalit so'zlar: Integral, chegaraviy shart, Benney-Luke tipidagi tenglama, Hilbert,
ortoganal
Let , H be a separable Hilbert space with the scalar product (•,•) and the norm || • || . Let the operator A : H ^ H . H be a self-adjoint, positive, unbounded arbitrary operator with the domain of definition D(A). Suppose that operator A has a complete orthonormal system of
eigenfunctions [vk} in the space H and the corresponding set of positive eigenvalues [Xk}
We can renumber eigenvalues non-descendingly, that is, write 0 < \ <X1----> +œ
Let a vector-function (or simply function) h(t) be defined in the interval [0, +to) with values in H. We will consider the Caputo fractional derivative of order a e (0,1) which is defined as (see, e.g., [1]-[2])
Dah(t) =-1-Y dt > 0.
t W r(1 -a) Y(t-%)a b
Let a e (0,1) is a given number. The main problem in this paper is the following non-local problem
'Dau(t) + A(Dau(t)) + A2 (Dau(t)) + Au(t) = f (t), 0 < t < T,
(1)
I u(t)dt = p,
- 0
where elements j, f e H . We call this problem the forward problem.
We allow parameter a to take the value 1, i.e., the equation in (1) can go to the classical equation,
since the results obtained in this work are also new for such equations
Definition. A function u(t) e C((0, T]; H) is called the solution of the forward problem if it has properties
A2(Dfu(t)), A(Dtau(t)),Dfu(t), Au(t) e C((0,T];H) and satisfies the conditions of problem (1)
Denote by E (z) the two-parameter Mittag-Leffler function
n
Er/( z) "Yrz , V
n=0 r(rn + /)
where r > 0 and / is an arbitrary complex number. If / = 1 then we have the classical Mittag-Leffler function
to _k
z
E/ z) = = Er (z)
n=0
r(rk +1)
The main result of this paper reads as follows
Theorem. Let j e D(A) and f e H. The problem (1) has a unique solution and the solution can be represented as a series converging uniformly in t
to ^
u(t) = -TO^ (jkEa1( /kt<X )
k=1 Ea,2(-/k1 )T
+ fk [taEa !(-/kta)Eaa(-/kT a)T - Ea>(-/kta)Eaa+2(-/T a)T ^H 1 + Ak +Ak
4k
whert fk, jk are the Fourier coefficients of the functions /=■ 2
1 + Ak
In addition to the forward problem, the article also studies the inverse problem of finding the right-hand side of the equation. Let us consider the inverse problem u(r) = ¥, 0 <r< T (2)
in which the unknown element f e H, characterizing the action of heat sources, does not depend on u(t) and ¥,jeH are given elements.
Definition. A pair {u(t), f} of function u(t) e C((0,T];H) and f e H with the properties A2(Dfu(t)), A(Dau(t)),Dfu(t) and Au(t) e C((0,T];H) and satisfying conditions (1),
(2) is called the solution of the inverse problem (1), (2).
Theorem . Let ф, YeD(A) . Then the inverse problem (1), (2) has a unique solution {u (t ), f} and this solution has the following form Аннотация
n 1
u(t ) = -^ (фкЕа1(-^кП
к=1 Ea,2(-VkT )T
+ fk 2 [taEa,a+¿-Kta)E^-^Ta)T-ЕаЛ(-ъta )Ta)Ta>,
where and
f=£ fk
kvk
k=0
* кЕаЛ(-икТ a)T-фкЕаЛ(-^кта)
f, =-k rk y—rk rk y--(1+ K + K).
REFERENCES
1. C.Lizama, Abstract linear fractional evolution equations, Handbook of Fractional Calculus with Applications V. 2, J.A.T. Marchado Ed. DeGruyter, 465-497 (2019)
2. R.Gorenflo, A.A.Kilbas, F.Mainardi and S.V.Rogozin, Mittag-Leffler Functions, Related Topics and Applications; Springer: Berlin/Heidelberg, Germany, (2014), doi: 10.1007/978 3662-61550-8. [CrossRef]
