SOLVABILITY OF A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS

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1 Electronic Journal of Differential Equations, Vol , No. 7, pp. 2. ISSN: URL: or SOLVABILITY OF A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS ANAR T. ASSANOVA Communicated by Ludmila S. Pulkina Abstract. We study a nonlocal problem with integral conditions for a hyperbolic equation two independent variables. By introducing additional functional parameters, we investigated the solvability and construction of approximate solutions. The original problem is reduced to an equivalent problem consisting of the Goursat problems for a hyperbolic equation with parameters and the boundary value problem with integral condition for the ordinary differential equations with respect to the parameters. Based on the algorithms for finding solutions to the equivalent problem, we propose algorithms for finding the approximate solutions, and prove their convergence. Coefficient criteria for the unique solvability of nonlocal problem with integral conditions for hyperbolic equation with mixed derivative are also established.. Introduction On the domain Ω =, T ], ω], we consider the nonlocal problem for hyperbolic equation with integral conditions 2 u t x = At, x u x + Bt, x u + Ct, xu + ft, x,. t Kt, ξut, ξdξ = ψt, t, T ],.2 Mτ, xuτ, xdτ = ϕx, x, ω],.3 where ut, x is unknown function, the functions At, x, Bt, x, Ct, x, and ft, x are continuous on Ω, the functions Kt, x and ψt are continuously differentiable by t on Ω and, T ], respectively, the functions Mt, x and ϕx are continuously differentiable by x on Ω and, ω], respectively, < a ω, < b T. The compatibility condition is given below. Let CΩ, R be a space of functions u : Ω R continuous on Ω with norm u = t,x Ω ut, x. 2 Mathematics Subject Classification. 35L5, 35L53, 35R3, 34B. Key words and phrases. Hyperbolic equation; nonlocal problem; integral condition; algorithm; approximate solution. c 27 Texas State University. Submitted March 7, 27. Published July 6, 27.

2 2 A. T. ASSANOVA EJDE-27/7 A function ut, x CΩ, R, having the partial derivatives ut,x x CΩ, R, ut,x t CΩ, R, and 2 ut,x t x CΩ, R is called a classical solution to problem..3, if it satisfies equation. and integral conditions.2,.3. Mathematical modelling of various physical processes often leads to the nonlocal problems for hyperbolic equations. Problems with integral conditions arise while researching the processes of heat distribution, plasma physics, clean technology of silicon ores, moisture transfer in capillary-porous media, etc. 8, 9,,, 2, 3, 5, 6, 7, 8, 24, 25, 26, 27]. Some classes of nonlocal boundary value problems with integral conditions for hyperbolic equations are studied in 6, 8-28]. Solvability conditions for the considered problems are obtained in the different terms. Problem..3 for Kt, x = Mt, x and Kt, x = Mt, x = are studied in,, 9, 2, 2, 22, 23]. Under the assumptions of continuous differentiability of the equation coefficients, the conditions for the unique solvability of that problem have been obtained. For Kt, x = Kx, Mt, x = Mt, the problem..3 for the system of hyperbolic equations is studied in 28] by contractive mapping principle. For Kt, x = Mt, x, problem..3 for the system of hyperbolic equations is studied in 7]. The unique solvability conditions for this problem are established in the terms of initial data. Nonlocal problems with general integral conditions for hyperbolic equations belong to the field of little studied problems of mathematical physics. This formulation of problem is considered for the first time. The aim of this work is to construct algorithms for finding a solution to problem..3 and establish conditions for the existence and uniqueness of classical solution to problem..3. In Section 2, a scheme of method used 4, 5, 6] is provided. By introduction of new unknown functions being a linear combination of solutions values on characteristics, the problem..3 is reduced to an equivalent problem consisting of the Goursat problem for hyperbolic equations with functional parameters and the boundary value problems with integral conditions for ordinary differential equations with respect to the parameters entered. Algorithm for finding the approximate solution to the investigated problem is constructed. The algorithm consists of two parts: in the first part, we solve two boundary value problems with integral condition for ordinary differential equations, and in the second part, we solve the Goursat problem for hyperbolic equation with parameters. Boundary value problems with integral condition for ordinary differential equations are intensively studied in recent years, and they find numerous applications in the applied problems, 2, 3]. In Section 3, the conditions for the existence of unique solution to the boundary value problems with integral condition for ordinary differential equations are presented. In Section 4, the convergence of algorithm is proved, and the conditions for the unique solvability of problem..3 are given in the terms of initial data. 2. Method s scheme and algorithm Notation: µt = ut, 2 u,, λx = u, x 2 u,, ũt, x, where ũt, x is a new unknown function. We make a following replacement of desired function ut, x in problem..3: ut, x = ũt, x + µt + λx and transit

3 EJDE-27/7 SOLVABILITY OF A NONLOCAL PROBLEM 3 to the problem 2 ũ = At, x ũ + Bt, x ũ + Ct, xũ + At, x λx t x x t + Bt, x µt + Ct, xλx + Ct, xµt + ft, x, Kt, ξdξµt + Mτ, xdτλx + 2. ũt, =, t, T ], 2.2 ũ, x =, x, ω], 2.3 Kt, ξũt, ξdξ + t, T ], Mτ, xũτ, xdτ + x, ω]. Kt, ξλξdξ = ψt, Mτ, xµτdτ = ϕx, A triplet of functions ũt, x, µt, λx, satisfying the hyperbolic equation 2., the conditions on characteristics 2.2, 2.3, and the functional relations 2.4 and 2.5 at µ = λ, will be called a solution to problem if the function ũt, x CΩ, R has the partial derivatives ũt, x x CΩ, R, ũt, x t CΩ, R, 2 ũt, x t x CΩ, R, the functions µt and λx are continuously differentiable on, T ] and, ω], respectively. Relation µ = λ is a compatibility condition of data. Problem is equivalent to problem..3. If the function u t, x is a solution to problem..3, then the triplet of functions ũ t, x, µ t, λ x, where ũ t, x = u t, x µ t λ x, µ t = u t, 2 u,, λ x = u, x 2 u,, is a solution to problem The converse is also true. If the triplet of functions ũ t, x, µ t, λ x is a solution to problem , then the function u t, x defined by the equality u t, x = ũ t, x + µ t + λ x, where u t, 2 u, = µ t, u, x 2 u, = λ x is a solution to problem..3. At fixed µt, λx, problem is the Goursat problem with respect to the function ũt, x on the domain Ω. Relations 2.4 and 2.5 allow us to determine the unknown parameters µt, λx, where the functions µt, λx satisfy condition µ = λ. By conditions 2.2, 2.3, relations 2.4 at t = and 2.5 at x = yield K, ξdξµ + Mτ, dτλ + Taking into account µ = λ, we obtain K, ξdξλ + K, ξλξdξ = ψ, 2.6 Mτ, µτdτ = ϕ. 2.7 K, ξλξdξ = ψ, 2.8

4 4 A. T. ASSANOVA EJDE-27/7 Mτ, dτµ + Let us define the following condition. Condition i. Assume B t = B 2 x = Mτ, µτdτ = ϕ. 2.9 Kt, ξdξ for all t, T ], Mτ, x dτ for all x, ω]. From relation 2.4 we determine the parameter µt = { } Kt, ξũt, ξdξ + Kt, ξλξdξ ψt, 2. B t for t, T ]. Similarly, from relation 2.5 we determine parameter λx = { B 2 x Mτ, xũτ, xdτ + } Mτ, xµτdτ ϕx, 2. for x, ω]. Assumptions on the data of problem..3 allow us to differentiate 2. and 2. by t and x, respectively. Then we obtain µt = Ḃt B 2t Kt, ξũt, ξdξ { Kt, ξ ũt, ξdξ B t t ũt, ξ } + Kt, ξ dξ + Ḃt t B 2t Kt, ξλξdξ B t λx = Ḃ2x B2 2x + B 2 x Mτ, x Kt, ξ λξdξ Ḃt t B 2tψt + B t ψt, t, T ], Mτ, xũτ, xdτ { Mτ, x ũτ, xdτ B 2 x x ũτ, x } dτ + Ḃ2x x B2 2x Mτ, xµτdτ Mτ, x µτdτ Ḃ2x x B2 2xϕx + B 2 x ϕx, for x, ω]. We introduce the new unknown functions ṽt, x = and the following notation ũt, x, wt, x = x G t, ũ, w = Ḃt B 2t Kt, ξũt, ξdξ { Kt, ξ ũt, ξdξ + B t t L t, λ = Ḃt B 2 t Kt, ξλξdξ B t ũt, x, t } Kt, ξ wt, ξdξ, Kt, ξ λξdξ, t

5 EJDE-27/7 SOLVABILITY OF A NONLOCAL PROBLEM 5 G 2 x, ũ, ṽ = Ḃ2x B 2 2 x B 2 x Mτ, xũτ, xdτ { Mτ, x ũτ, xdτ + x Then equations 2.2 and 2.3 can be written in the form } Mτ, xṽτ, xdτ. µt = G t, ũ, w + L t, λ + B t ψt Ḃt t, T ], 2.4 tψt, λx = G 2 x, ũ, ṽ + L 2 x, µ + B 2 x B 2 Ḃ2x ϕx B2 2 x, ω]. 2.5 xϕx, Thus, we have a closed system of equations , , for determining the unknown functions ṽt, x, wt, x, ũt, x, λx, λx, µt, µt. Relation 2.4 in conjunction with 2.9 present a boundary value problem with integral condition for a differential equation with respect to µt, and the relation 2.5 in conjunction with 2.8 present a boundary value problem with integral condition for a differential equation with respect to λx. Boundary value problem with integral condition 2.4, 2.9 is equivalent to relation 2.4, and boundary value problem with integral condition 2.5, 2.8 is equivalent to relation 2.5 at µ = λ. If µt, λx, µt, λx are known, then we find the functions ṽt, x, wt, x, ũt, x from Conversely, if we know the functions ṽt, x, wt, x, ũt, x, then we can find µt, µt, λx, λx from boundary value problems 2.4, 2.9 and 2.5, 2.8. The unknowns are both ṽt, x, wt, x, ũt, x, and µt, µt, λx, λx. Therefore, to find solution of problem , we use an iterative method: determine the triplet ũ t, x, µ t, λ x as a limit of sequence ũ m t, x, µ m t, λ m x, m =,, 2,..., according to the following algorithm: Step. Assuming ũt, x =, wt, x =, λx =, in the right-hand side of equation 2.4, we find initial approximations µ t, µ t, t, T ] from the boundary value problem with integral condition 2.4, 2.9. Assuming ũt, x =, ṽt, x =, µt = in the right-hand side of equation 2.5, we find initial approximations λ x, λ x, x, ω] from the boundary value problem with integral condition 2.5, Find ṽ t, x, w t, x, ũ t, x, t, x Ω from the Goursat problem for λx = λ x, µt = µ t, λx = λ x, µt = µ t. Step. Assuming ũt, x = ũ t, x, wt, x = w t, x, λx = λ x in the right-hand side of equation 2.4, we find µ t, µ t, t, T ] from the boundary value problem with integral condition 2.4, 2.9. Assuming ũt, x = ũ t, x, ṽt, x = ṽ t, x, µt = µ t in the right-hand side of equation 2.5, we find λ x, λ x, x, ω] from the boundary value problem with integral condition 2.5, Find ṽ t, x, w t, x, ũ t, x, t, x Ω from the Goursat problem for λx = λ x, µt = µ t, λx = λ x, µt = µ t. And so on. Step m. Assuming ũt, x = ũ m t, x, wt, x = w m t, x, λx = λ m x in the right-hand side of equation 2.4, we find µ m t, µ m t, t

6 6 A. T. ASSANOVA EJDE-27/7, T ] from the boundary value problem with integral condition 2.4, 2.9. Assuming ũt, x = ũ m t, x, ṽt, x = ṽ m t, x, µt = µ m t in the righthand side of equation 2.5, we find λ m x, λ m x, x, ω] from the boundary value problem with integral condition 2.5, Find ṽ m t, x, w m t, x, ũ m t, x, t, x Ω, from the Goursat problem for λx = λ m x, µt = µ m t, λx = λ m x, µt = µ m t, m =, 2,.... The constructed algorithm consists of two parts: we solve the boundary value problems with integral condition for the ordinary differential equations 2.4, 2.9 and 2.5, 2.8 in the first part, and we solve the Goursat problem for hyperbolic equations with functional parameters in the second part. 3. Boundary value problems with integral condition for the differential equations Consider the boundary value problem with integral condition for the ordinary differential equations µt = ġ t, t, T ], 3. Mτ, dτµ + Mτ, µτdτ = ϕ, 3.2 where the function g t is continuously differentiable on, T ], the function Mt, x is continuous on Ω, ϕ is a constant, < b T. The function µt C, T ], R having the derivative µt C, T ], R, is called a solution to problem 3., 3.2, if it satisfies ordinary differential equation 3. and boundary condition 3.2. We also consider the boundary value problem with integral condition for the ordinary differential equation of the type λx = ġ 2 x, x, ω], 3.3 K, ξdξλ + K, ξλξdξ = ψ, 3.4 where the function g 2 x is continuously differentiable on, ω], the function Kt, x is continuous on Ω, ψ is a constant, < a ω. The function λx C, ω], R having the derivative λx C, ω], R, is called a solution to problem 3.3, 3.4, if it satisfies ordinary differential equation 3.3 and boundary condition 3.4. General solution to equation 3. has the form µt = g t + C, t, T ], where C is a constant. Since Condition i holds, the constant C is uniquely determined by 3.2: C = 2B 2 ϕ 2B 2 Mτ, { g τ + g } dτ. Then the unique solution to problem 3., 3.2 has the form µt = g t + 2B 2 ϕ 2B 2 Mτ, { g τ + g } dτ, 3.5

7 EJDE-27/7 SOLVABILITY OF A NONLOCAL PROBLEM 7 for t, T ]. Analogously, the general solution to equation 3.3 has the form λx = g 2 x + C 2, x, ω], where C 2 is a constant. Since Condition i holds, the constant C 2 is uniquely determined by 3.4: C 2 = 2B ψ 2B K, ξ { g 2 ξ + g 2 } dξ. Then unique solution to problem 3.3, 3.4 has the form λx = g 2 x + 2B ψ 2B K, ξ { g 2 ξ + g 2 } dξ, 3.6 for x, ω]. Below we give conditions for the unique solvability of boundary value problems with integral condition 3., 3.2 and 3.3, 3.4. Theorem 3.. Suppose Condition i is holds. Then problem 3., 3.2 has a unique solution µ t C, T ], R representable in the form 3.5, and t,t ] µ t K g t, ϕ, 3.7 t,t ] where K = + ] + 2b 2 B 2 Mt,. t,b] Theorem 3.2. Suppose Condition i holds. Then problem 3.3, 3.4 has a unique solution λ x C, ω], R representable in the form 3.6, and x,ω] λ x K 2 g 2x, ψ, 3.8 x,ω] where K 2 = + ] + 2a 2 B K, x. x,a] 4. Algorithm s convergence and main result In Section 2, an algorithm for finding a solution to problem , which is equivalent to problem..3, is constructed. To formulate the main result, we let us give few assumptions and notation. Let Condition i hold, and introduce the notation: α = At, x, β t,x Ω H = α + β + γ, = Bt, x, γ t,x Ω κ = Kt, x, κ t,x Ω σ = Mt, x, σ t,x Ω β = t,t ] B t], δ = t,t ] Ḃt, = Ct, x, t,x Ω Kt, x 2 =, t,x Ω t Mt, x 2 =, t,x Ω x β 2 = x,ω] B 2x], δ 2 = x,ω] Ḃ2x, l a = aβ κ { + T, ωhe HT +ω}, l 2 b = bβ 2 σ { + T, ωhe HT +ω},

8 8 A. T. ASSANOVA EJDE-27/7 l 2 a = aβ {δ β κ + κ 2 + δ β κ + κ + κ 2 T, ωhe HT +ω}, l 22 b = bβ 2 {δ 2 β 2 σ + σ 2 + δ 2 β 2 σ + σ + σ 2 T, ωhe HT +ω}. In Section 3, the conditions for unique solvability of boundary value problems with integral condition 3., 3.2 and 3.3, 3.4 are established. At fixed ṽt, x, wt, x, ũt, x on each step of the algorithm we solve the boundary value problems with integral condition 2.2, 2.9 and 2.3, 2.8. Besides, in problem 2.2, 2.9, we consider λx is also known. Same for µt in problem 2.3, 2.8. At fixed λx, µt, λx, µt, we solved the Goursat problem The following statement gives the conditions for the convergence of proposed algorithm and the existence of unique solution to problem Theorem 4.. Let: the functions At, x, Bt, x, Ct, x, and ft, x be continuous on Ω; 2 the functions Kt, x and ψt be continuously differentiable by t on Ω and, T ], respectively; the functions Mt, x and ϕx be continuously differentiable by x on Ω and, ω], respectively; 3 Condition i hold; 4 the inequality q = K l a + K 2 l 2 b, l 2 a, l 22 b < be fulfilled. Then problem has a unique solution. Proof. Let conditions 3 be fulfilled. We use step of algorithm and consider the boundary value problem with integral condition µt = B t ψt Ḃt t, T ], 4. tψt, Mτ, dτµ + B 2 Mτ, µτdτ = ϕ. 4.2 λx = Ḃ2x ϕx x, ω], 4.3 B 2 x xϕx, K, ξdξλ + B2 2 K, ξλξdξ = ψ. 4.4 Condition 3 and the conditions of Theorems 3. and 3.2 yield the unique solvability of problems 4., 4.2 and 4.3, 4.4. We find initial approximations µ t and λ x from the boundary value problems 4., 4.2 and 4.3, 4.4. Then, similar to the estimates 3.7 and 3.8, for the functions µ t, λ x and their derivatives µ t, λ x the estimates hold: t,t ] µ t K β ψt, ϕ, 4.5 t,t ] t,t ] µ t β ψt + δ β 2 ψt. 4.6 t,t ] t,t ] x,ω] λ x K 2 β 2 ϕx, ψ, 4.7 x,ω]

9 EJDE-27/7 SOLVABILITY OF A NONLOCAL PROBLEM 9 λ x β 2 ϕx + δ 2β2 2 ϕx. 4.8 x,ω] x,ω] x,ω] Solving the Goursat problem for the found values of parameters, we find ṽ t, x, w t, x, ũ t, x for all t, x Ω. The following inequalities are valid: ṽ t, x T, ωe HT +ω t,x Ω ft, x, w t, x T, ωe HT +ω t,x Ω ft, x, ũ t, x T, ωe HT +ω t,x Ω ft, x, where ft, x = At, x λ ] x + Bt, x µ t + Ct, x λ x + µ t + ft, x. Then, we determine the functions µ m t, λ m x, µ m t, λm x, ṽ m t, x, w m t, x, ũ m t, x from the mth step of the algorithm, and we obtain µ m+ t, λ m+ x, µ m+ t, λ m+ x, ṽ m+ t, x, w m+ t, x, ũ m+ t, x, from step m +, m =, 2,.... Evaluating the corresponding differences of successive approximations, we obtain t,t ] µm+ t µ m t K β Kt, ξũ m t, ξ ũ m t, ξ dξ + t,t ] ] Kt, ξλ m ξ λ m ξ dξ, x,ω] λm+ x λ m x K 2 β 2 + x,ω] Mτ, xũ m τ, x ũ m τ, x dτ Mτ, xµ m τ µ m τ dτ ], t,t ] µm+ t µ m t L t, λ m λ m + G t, ũ m ũ m, w m w m ], 4. t,t ] λ m+ x λ m x x,ω] ] L 2 x, µ m µ m + G 2 x, ũ m ũ m, ṽ m ṽ m, x,ω] 4.2

10 A. T. ASSANOVA EJDE-27/7 ṽ m+ t, x ṽ m t, x T, ωe HT +ω{ α λ m+ x λ m x x,ω] + β t,t ] µm+ t µ m t + γ x,ω] λm+ x λ m x ]} + t,t ] µm+ t µ m t, w m+ t, x w m t, x T, ωe HT +ω{ α λ m+ x λ m x x,ω] + β t,t ] µm+ t µ m t + γ x,ω] λm+ x λ m x ]} + t,t ] µm+ t µ m t, ũ m+ t, x ũ m t, x T, ωe HT +ω{ α λ m+ x λ m x x,ω] + β t,t ] µm+ t µ m t + γ x,ω] λm+ x λ m x ]} + t,t ] µm+ t µ m t. Suppose that m+ = x,ω] λm+ x λ m x + t,t ] µm+ t µ m t, λ m+ x λ m x, x,ω] t,t ] µm+ t µ m t. 4.5 Then, from relations , taking into account the notation introduced and estimations , we obtain the main inequality m+ q m. 4.6 Condition 4 of the theorem leads to the convergence of sequence m as m, i.e., =. This gives the uniform convergence of sequences λ m x, λ m x, µ m t, µ m t, to λ x, λ x, µ t, µ t,respectively, as m. Functions λ x and µ t are continuous and continuously differentiable on, ω] and, T ], respectively. Based on estimates , we establish the uniform convergence of sequences ṽ m t, x, w m t, x, ũ m t, x to the functions ṽ t, x, w t, x, ũ t, x, respectively, with respect to t, x Ω. Obviously, the functions ũ t, x, ṽ t, x, and w t, x are continuous on Ω. Solving the problems on the m + th step of the algorithm and passing to the limit as m, we obtain that the functions ũ t, x, λ x, µ t together with their derivatives satisfy the Goursat problem and boundary value problems with integral condition 2.4, 2.9 and 2.5, 2.8. We carry out the inverse transition from problem 2.4, 2.9 to relation 2.4, and pass from problem 2.5, 2.8 to relation 2.5. Then the triplet of functions ũ t, x, λ x, µ t is solution to problem

11 EJDE-27/7 SOLVABILITY OF A NONLOCAL PROBLEM Next we prove the uniqueness of solution to Let the triplet of functions ũ t, x, λ x, µ t and another triplet of functions ũ t, x, λ x, µ t be two solutions to the problem. We introduce the notation = x,ω] λ x λ x + t,t ] µ t µ t,. λ x λ x, x,ω] t,t ] µ t µ t, After calculation, analogous to , we obtain q. 4.7 By condition 4 of the theorem, we have q <. Then inequality 4.7 takes place only for. This gives us λ x = λ x, µ t = µ t and ũ t, x = ũ t, x. Therefore, the solution to problem is unique. The next assertion follows from the equivalence of problem..3 and problem Theorem 4.2. Let conditions 4 of Theorem 4. be fulfilled. Then problem..3 has a unique classical solution. Proof. Conditions 4 of Theorem 4. imply the existence of a unique solution to , the triplet of functions ũ t, x, λ x, µ t. According to the algorithm presented above, for each m =,, 2,..., this triplet is determined as a limit of sequence triplets ũ m t, x, µ m t, λ m x as m. Then solution to problem..3, the function u t, x, exists and is determined by the equality u t, x = ũ t, x + λ x + µ t. Acknowledgements. The author thanks the anonymous referee for the careful reading of this article and the useful suggestions. This research was partially supported by Grant of Ministry of Education and Science of the Republic of Kazakhstan, No 822/ΓΦ4. References ] A. A. Abramov, L. F. Yukhno; Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition, Computational Mathematics and Mathematical Physics, 52 22, No 2, pp ] A. A. Abramov, L. F. Yukhno; Solving a system of linear ordinary differential equations with redundant conditions, Computational Mathematics and Mathematical Physics, 54 24, No 4, pp ] A. A. Abramov, L. F. Yukhno; A solution method for a nonlocal problem for a system of linear differential equations, Computational Mathematics and Mathematical Physics, 54 24, No, pp ] A. T. Asanova, D. S. Dzhumabaev; Unique Solvability of the Boundary Value Problem for Systems of Hyperbolic Equations with Data on the Characteristics, Computational Mathematics and Mathematical Physics, 42 22, No, pp ] A. T. Asanova, D. S. Dzhumabaev; Unique solvability ofnonlocal boundary value problems for systems of hyperbolic equations, Differential Equations, 39 23, No, pp ] A. T. Asanova, D. S. Dzhumabaev; Well-posedness of nonlocalboundary value problems with integral condition for the system of hyperbolic equations,journal of Mathematical Analysis and Applications, 42 23, No, pp ] A. T. Assanova; Nonlocal problem with integral conditions for the system of hyperbolic equations in the characteristic rectangle, Russian Math. Iz. VUZ, 6 27, No 5, pp. 7 2.

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