INVERSE COEFFICIENT PROBLEM FOR THE SEMI-LINEAR FRACTIONAL TELEGRAPH EQUATION

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1 Electronic Journal of Differential Equations, Vol. 25 (25), No. 53, pp. 3. ISSN: URL: or ftp ejde.math.txstate.edu INVERSE COEFFICIENT PROBLEM FOR THE SEMI-LINEAR FRACTIONAL TELEGRAPH EQUATION HALYNA LOPUSHANSKA, VITALIA RAPITA Abstract. We establish the unique solvability for an inverse problem for semilinear fractional telegraph equation D α t u + r(t)dβ t u u = F (x, t, u, D β t u), (x, t) (, T with regularized fractional derivatives Dt αu, Dβ t u of orders α (, 2), β (, ) with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution u of the first boundary-value problem for such equation, and an unknown continuous coefficient r(t) under the over-determination condition Z u(x, t)ϕ(x)dx = F (t), t [, T with given functions ϕ and F.. Introduction The existence and uniqueness theorems for fractional Cauchy problems were proved in [2, 4, 5, 7,, 2, 3, 4, 5, 6, 24 and other works. The conditions of classical solvability of the first boundary-value problem for equation D β t u(x, t) A(x, D)u(x, t) = F (x, t) with regularized fractional derivative (see, for example, [4) and some elliptic differential second order operator A(x, D) were obtained in [8 and [9. Equations with fractional derivatives are applied in studying of anomalous diffusion and various processes in physics, mechanics, chemistry and engineering. The telegraph fractional equations in theory of thermal stresses is considered, for example, in [2. Inverse problems to such equations arise in many branches of science and engineering. Some inverse boundary-value problems to diffusion-wave equation with different unknown functions or parameters were investigated, for example, in [, 3, 6, 9, 7, 2, 22, 25. In particular, the article [ was devoted to determination of a source term for a time fractional diffusion equation with an integral type over-determination condition. 2 Mathematics Subject Classification. 35S5. Key words and phrases. Fractional derivative; inverse boundary value problem; over-determination integral condition; Green s function; integral equation. c 25 Texas State University - San Marcos. Submitted April 22, 25. Published June, 25.

2 2 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 In this note we prove the existence and uniqueness of a classical solution (u, r) of the inverse boundary-value problem D α t u + r(t)d β t u u = F (x, t, u, D β t u), (x, t) (, T, (.) u(x, t) =, (x, t) [, T, (.2) u(x, ) = F (x), x, (.3) u t (x, ) = F 2 (x), x, (.4) u(x, t)ϕ (x)dx = F (t), t [, T (.5) for regularized telegraph equation, where α (, 2), β (, ), is a bounded domain in R N, N 3, with a boundary of class C +s, s (, ), F, F, F 2, F, ϕ given functions. We shall use Green s functions to prove the solvability of this problem. 2. Definitions and auxiliary results We shall use the notation: Ω =, Q i = Ω i (, T, i =,, Q 2 =, D(R m ) is a space of indefinitely differentiable functions with compact supports in R m, m =, 2,..., D(Q ) = {v C (Q ) : ( t )k v t=t =, k =,,... }, D (R m ) and D (Q ) are spaces of linear continuous functionals (generalized functions [23, p. 3-5) over D(R m ) and D(Q ), respectively, (f, ϕ) stands for the value of f D (R m ) on the test function ϕ D(R m ) and also the value of f D (Q ) on ϕ D(Q ). We denote by f g the convolution of generalized functions f and g, use the function { θ(t)t λ f λ (t) = Γ(λ) for λ > (t) for λ, f +λ where Γ(z) is the Gamma-function, and θ(t) is the Heaviside function. Note that f λ f µ = f λ+µ. Also note that the Riemann-Liouville derivative v (α) t (x, t) of order α > is defined by v (α) t (x, t) = f α (t) v(x, t) and while D α t v(x, t) = t Γ( α)[ t v(x, τ) v(x, ) dτ (t τ) α t α = v (α) t (x, t) f α (t)v(x, ), for α (, ), Dt α t v τ (x, τ) v(x, t) = Γ(2 α)[ t (t τ) α dτ v t(x, ) (t τ) α = v (α) t (x, t) f α (t)v(x, ) f 2 α (t)v t (x, ) for α (, 2). We denote D t v = v t. Let C(Q ), C(Q ), C[, T be spaces of continuous functions on Q, Q and [, T, respectively, C γ ( ) (C γ ( )) be a space of bounded continuous functions on ( ) satisfying Hölder continuity condition, C γ (Q i ) (C γ ( Q i )), i =,, be a

3 EJDE-25/53 INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 3 space of bounded continuous functions on Q i ( Q i ) which for all t (, T satisfies Hölder continuity condition with respect to space variables, C γ (Q R 2 ) be a space of bounded continuous functions f(x, t, v, z) on Q R 2 which for all t (, T, v, z R satisfies Hölder continuity condition with respect to space variables x, C β (Q ) = {v C(Q ) : D β t v C(Q )}, C γ β (Q ) = {v C γ (Q ) : D β t v C γ (Q )}, C 2,α (Q ) = {v C(Q ) : v, D α t v C(Q )}, C 2,α ( Q ) = {v C 2,α (Q ) : v, v t C( Q )}. We use the following assumptions: (A) F C γ (Q R 2 ), γ (, ), F (x, t, v, w) A + B [ v q + w p, F (x, t, v, w ) F (x, t, v 2, w 2 ) D [ v v 2 q + w w 2 p for all v, w, v, v 2, w, w 2 R where p, q [, 2, A, B, D are some positive constants, (A2) F C γ ( ), F Ω =, (A3) F 2 C γ ( ), (A4) F, D β F, D α F C[, T, there exists f := inf t [,T D β F (t) >, (A5) ϕ C 2 ( ), ϕ Ω =. Definition 2.. A pair of functions (u, r) C 2,α ( Q ) C[, T satisfying (.) on Q and the conditions (.2)-(.5) is called a solution of the problem (.)-(.5). From (.3), (.4) and (.5), it follows the necessary agreement conditions F (x)ϕ (x)dx = F (), F 2 (x)ϕ (x)dx = F (). (2.) We introduce the operators (Lv)(x, t) v (α) t (x, t) v(x, t), (x, t) Q, v D (Q ), (L reg v)(x, t) D α t v(x, t) v(x, t), (x, t) Q, v C 2,α ( Q ). Definition 2.2. A vector-function (G (x, t, y, τ), G (x, t, y), G 2 (x, t, y)) is called a Green s vector-function of the problem (L reg u)(x, t) = g (x, t), (x, t) Q, (2.2) u(x, t) =, (x, t) Q, (2.3) u(x, ) = g (x), u t (x, ) = g 2 (x), x, (2.4) if for rather regular g, g, g 2 the function t u(x, t) = dτ G (x, t, y, τ)g (y, τ)dy + 2 j= G j (x, t, y)g j (y)dy, (2.5) for (x, t) Q, is the classical solution (in C 2,α ( Q )) of the first boundary-value problem (2.2) (2.4). It follows from Definition 2.2 that (LG )(x, t, y, τ) = δ(x y, t τ), (x, t), (y, τ) Q, (L reg G j )(x, t, y) =, (x, t) Q, y, j =, 2,

4 4 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 G (x,, y) = δ(x y), t G (x,, y) =, G 2 (x,, y) =, t G 2(x,, y) = δ(x y), x, y where δ is Dirac delta-function. Lemma 2.3 ([5). The following relations hold: G j (x, t, y) = t f j α (τ)g (x, t, y, τ)dτ, (x, t) Q, y, j =, 2. Lemma 2.4. A Green s vector-function of the first boundary-value problem (2.2) (2.4) exists. The above lemma is proved following the strategy in [7, Lemma 2. Theorem 2.5. If g C γ (Q ), γ (, ), g j C γ ( ), j =, 2, g Ω = then there exists a unique solution u C 2,α ( Q ) of (2.2) (2.4). It is defined by where u(x, t) = ( G g ) (x, t) + ( G g )(x, t) + ( G 2 g 2 )(x, t), (x, t) Q (2.6) t (G g )(x, t) = dτ G (x, t, y, τ)g (y, τ)dy, (G j g j )(x, t) = G j (x, t, y)g j (y)dy, j =, 2. Proof. Taking Lemmas 2.3 and 2.4 into account, as in [7, 8,, 24 for the Cauchy problems, we show that the function (2.6) belongs to C 2,α ( Q ) and satisfies the problem (2.2) (2.4). We use the estimates founded in [7,, 5, 24: G (x, t, y, τ) C (t τ) x y N 2, x y 2 < 4(t τ) α, G j (x, t, y) C j tj α x y N 2, j =, 2, x y 2 < 4t α, G (x, t, y, τ) Ĉ(t τ) α x y N ( x y 2 4(t τ) α ) + N where c, C j, C j, Ĉj G j (x, t, y) C (t τ) α x y N, x y 2 > 4(t τ) α 2(2 α) e c( x y 2 4a (t τ) α ) 2 α j Ĉjt ( x y 2 ) N 2(2 α) x y 2 c( x y N 4t α e 4t α ) 2 α C jt j x y N, j =, 2, x y 2 > 4t α (j =,, 2) are positive constants; G j (x + x, t + t, y, τ) G j (x, t, y, τ) A j (x, t, y, τ)[ x + t α/2 γ, D β t G j (x + x, t + t, y, τ) D β t G j (x, t, y, τ) A β,j (x, t, y, τ)[ x + t α/2 γ (x, t), (x + x, t + t) Q, (y, τ) Q j, j =,, 2

5 EJDE-25/53 INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 5 with some < γ < where non-negative functions A j (x, t, y, τ), A β,j (x, t, y, τ) have the same kind of estimates as G j (x, t, y, τ), D β t G j (x, t, y, τ), j =,, 2, have, respectively, and G j (x, t, y, τ) = G j (x, t, y), A j (x, t, y, τ) = A j (x, t, y), A β,j (x, t, y, τ) = A β,j (x, t, y) for j =, 2. Note that for the general boundary-value problem to a parabolic equation with partial derivatives the last properties of a Green s vectorfunction were obtained in [. 3. Existence and uniqueness theorems for the inverse problem Now we prove the existence of a solution for the inverse problem (.) (.5). It follows from the theorem 2.5 that under assumptions (A), (A2), (A3) for a given r C[, T the solution u C 2,α ( Q ) of the first boundary-value problem (.) (.4) satisfies u(x, t) = t r(τ)dτ G (x, t, y, τ)dτ β u(y, τ)dy (3.) + h (x, t, u, D β t u) + h(x, t), (x, t) where h (x, t, u, D β t u) = h(x, t) = t dτ G (x, t, y, τ)f (y, τ, u(y, τ), Dτ β u(y, τ))dy, 2 G j (x, t, y)f j (y)dy, (x, t). j= (3.2) Conversely, any solution u C γ β (Q ) of (3.) belongs to C 2,α ( Q ) and is the solution of (.) (.5). It follows from the equation (.) and assumption (A5) that Dt α u(x, t)ϕ (x)dx + r(t) D β t u(x, t)ϕ (x)dx = u(x, t) ϕ (x)dx + F (x, t, u(x, t), D β t u(x, t))ϕ (x)dx, t (, T. Using (.5) we obtain D α F (t) + r(t)d β F (t) = u(x, t) ϕ (x)dx + F (x, t, u(x, t), D β t u(x, t))ϕ (x)dx (here D α F (t) = Dt α F (t) and so on); that is, by using (A4), [ r(t) = u(x, t) ϕ (x)dx + F (x, t, u(x, t), D β t u(x, t))ϕ (x)dx Ω D α F (t) [D β F (t), t (, T. (3.3)

6 6 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 Note that, for u C γ β (Q ), the function r(t), defined by (3.3), belongs to C[, T. By substituting the right-hand side of (3.3) into (3.) in place of r(t) we obtain t u(x, t) = [D β F (τ) dτ G (x, t, y, τ) [ F (z, τ, u(z, τ), Dτ β u(z, τ))ϕ (z)dz (3.4) + u(z, τ) ϕ (z)dz D α F (τ) Dτ β u(y, τ)dy + h (x, t, u(x, t), D β t u(x, t)) + h(x, t), (x, t) Q, where the functions h, h are defined by (3.2). We have reduced the problem (.) (.5) to system (3.3), (3.4). Conversely, a pair (u, r) C γ β (Q ) C[, T satisfying (3.3) and (3.4) is a solution of the problem (.)-(.5). Thus, under assumptions (A) (A5) and (2.), a pair of functions (u, r) C 2,α ( Q ) C[, T is the solution of (.)-(.5) if and only if the function u C γ β (Q ) is the solution of (3.4), with r C[, T defined by (3.3). Theorem 3.. Under the assumptions (A) (A5) and the condition (2.), there exists T (, T (Q = (, T ) and a solution (u, r) C 2,α ( Q ) C[, T of (.)-(.5). The function u is the solution of (3.4), and r(t) is defined by (3.3). Proof. From the previous conclusion, it is sufficient to prove the solvability of the equation (3.4) in C γ β (Q ). We shall use the Schauder principle. Let r C[,T = max t [,T r(t), and v C γ β (Q) = max Let R be some positive number, and { sup v(x, t), sup D β t v(x, t), (x,t) Q (x,t) Q v(x + x, t) v(x, t) sup (x,t) Q, x < x γ, D β t v(x + x, t) D β t v(x, t) } sup (x,t) Q, x < x γ, M R = M R (Q ) = {v C γ β (Q ) : v C γ β (Q) R}. On M R we consider the operator t (P v)(x, t) := [D β F (τ) dτ G (x, t, y, τ) [ v(z, τ) ϕ (z)dz + F (z, τ, v(z, τ), D β t v(z, τ))ϕ (z)dz D α t F (τ) D β τ v(y, τ)dy + h (x, t, v, D β t v) + h(x, t), (x, t), v M R with the functions h, h defined by (3.2). At the beginning we show the existence of R >, T (, T and therefore M R = M R(Q ) such that P : M R M R. For v M R, (x, t) Q we find the estimates h (x, t, v, D β t v)

7 EJDE-25/53 INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 7 t = dτ G (x, t, y, τ)f (y, τ, v(y, τ), D β t v(y, τ))dy t [ dτ G (y,τ) Ω: (x, t, y, τ) F (y, τ, v(y, τ), D β t v(y, τ)) dy y x <2(t τ) α/2 + G (y,τ) (x, t, y, τ) F (y, τ, v(y, τ), D β : t v(y, τ)) dy y x >2(t τ) α/2 t [ C [ dτ A (y,τ) Ω: (t τ) y x N 2 + v(y, τ q + D β t v(y, τ)) p dy y x <2(t τ) α/2 + C Ĉ (y,τ) : y x >2(t τ) α/2 t t C (t τ) α y x N [ A + v(y, τ q + D β t v(y, τ)) p dy 2(t τ) α/2 diam Ω (t τ) α 2(t τ) α/2 [ rdr + (t τ) [ (t τ) α + (t τ) α ln diam Ω dτ (t τ) α/2 k t α[ A + B (R q + R p ) dr [ dτ A + B (R q + R p ) r [ A + B (R q + R p ) where C, C, Ĉ, ˆk, k are positive constants, and α = α ϱ, with ϱ an arbitrary number in (, ). Also we have G j (x, t, y)f j (y)dy [ G (y,τ) Ω: j (x, t, y)dy + G (y,τ) Ω j (x, t, y)dy F j : C( Ω) y x <2t α/2 y x >2t α/2 c [ (y,τ) : + y x <2t α/2 (y,τ) : y x >2t α/2 t j α dy x y N 2 ˆk j t j [ diam Ω + r N 2t α/2 k j t j F j C( Ω), j =, 2 t j ( y x 2 ) N y x 2 2(2 α) c( y x N 4t α e 4t α ) 2 α dy 2 α t αn ĉ 2(2 α) e ( r 2 ) 2 α t α where c, ĉ, ˆk j, k j (j =, 2) are positive constants. Therefore, h(x, t) k j t j F j C( Ω), (x, t) Q. Similarly, j=,2 F (y, τ, v(y, τ), D β t v(y, τ))ϕ (z)dy F j C( Ω) dr F j C( Ω) [ A + B (R q + R p ) ϕ (z) dz (y, τ) Q

8 8 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 and therefore t [D β F (τ) dτ G (x, t, y, τ)f (z, τ, v(z, τ), D β t v(z, τ))ϕ (z)dz k f tα[ A + B (R q + R p ) ϕ (z) dz (y, τ) Q. Then, given R, we obtain the estimate (P v)(x, t) k f tα (c R + c 2 R 2 ) + H (t) q t α R 2 + H, (x, t) Q where c = dx D α F C[,T, Ω ) c 2 = 2B (f + ϕ (z) dz + ϕ (z) dz, ( H (t) = A k t α + ) ϕ (z) dz + k F f C( Ω) + k 2 t F 2 C( Ω) H, q = k (c + c 2 ). f In the same way for v M R, (x, t) Q, x < we obtain (P v)(x + x, t) (P v)(x, t) x γ q t α R 2 + H, (x, t) Q. Here and later q j, c j, H j (j=,2,... ) are positive numbers. For every g C γ (Q ) we have D β ( t G g ) t (t τ) β dτ (x, t) = G (x, t, y, τ)g(y, τ)dy, (x, t) Q Γ( β) and, as previously, we find that ( G g ) (x, t) q t α β g C(Q), (x, t) Q, q = const >. D β t Furthermore, D β t D β t ( ) G F (x, t) = t = t ( ) G2 F 2 (x, t) = t = t t t (t τ) β dτ Γ( β) f β (t τ) ( G F (x, τ)dτ, )τ (t τ) β dτ Γ( β) f β (t τ) ( G 2 F 2 (x, τ)dτ. )τ G (x, τ, y)f (y)dy f β (t)f (x) G 2 (x, τ, y)f 2 (y)dy f 2 β (t)f 2 (x) Since ( G j F j )τ (j =, 2) are continuous functions on Q, we have D β( ) t Gj F j (x, t) c2+j t β, (x, t) Q, j =, 2. So, D β t (P v)(x, t) q 2 t α2 R 2 + H 2, (x, t) Q

9 EJDE-25/53 INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 9 and similarly D β t (P v)(x + x, t) D β t (P v)(x, t) x γ q 3 t α2 R 2 + H 3, (x, t) Q, x < for all v M R with α 2 = α β and As a result we obtain P v C γ β (Q) H 2 = c 5 + c 6 F C( Ω) + c 7 t F 2 C( Ω), H 3 = c 8 + c 9 F C( Ω) + c F 2 C( Ω). max t [,T max{q t α R 2 + H, q t α R 2 + H, q 2 t α2 R 2 + H 2, q 3 t α2 R 2 + H 3 } max t [,T [qr2 t α2 + H v M R, R with q = max{q T β, q T β, q 2, q 3 }, H = max{h, H, H 2, H 3 }. To show the inequality max{qr 2 t α2 + H, } R t [, T, v M R (3.5) consider the function w(s) = qs 2 t α2 s, s. We find w (s) = 2qt α2 s and prove that s = s (t) = [2qt α2 is the point of the minimum of f(s). Then the inequality P v C γ β (Q ) R v MR is satisfied for some R max{, H}, T = min{t, T } if qt α2 s 2 s H and 2qt α2 max{, H} for all t [, t. We have qt α2 s s = 4qt. There exists α 2 t > such that 4qt α2 H and 2qt α2 max{, H} for all t [, t. Then t = min{[, [ }. 4qH /α2 2q max{, H} /α2 Note that (2.) and (A4) imply F C( Ω) >. We have proven the existence of R, T > such that P : M R M R. The operator P is continuous on M R = {v C β (Q ) : v Cβ (Q ) = max{ v C(Q ), D β t v C(Q )} R}; thus, on M R. Namely, for v, v 2 M R, (x, t) Q, (P v )(x, t) (P v 2 )(x, t) t [ = [D β F (τ) dτ G (x, t, y, τ) v 2 (z, τ) ϕ (z)dz[dτ β v 2 (y, τ) Ω Dτ β v (z, τ) [v (z, τ) v 2 (z, τ) ϕ (z)dz Dτ β v (y, τ) Ω + [F (z, τ, v (z, τ), Dτ β v (z, τ) F (z, τ, v 2 (z, τ), Dτ β v 2 (z, τ)ϕ (z)dz dy + h (z, τ, v (z, τ), Dτ β v (z, τ) h (z, τ, v 2 (z, τ), Dτ β v 2 (z, τ) t c sup [D β F (τ) dτ G (x, t, y, τ) dy (x,t) Q

10 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 [ v 2 C(Q ) D β v D β v 2 C(Q ) + D β v C(Q ) v v 2 C(Q ) + 2D v v 2 q C(Q ) + 2D D β v D β v 2 p C(Q ) t + 2D sup dτ G (x, t, y, τ) [ dy v v 2 q C(Q (x,t) Q Ω ) + 2D D β v D β v 2 p C(Q ) c 2 (T ) α R v v 2 C γ β (Q ). Similarly, by using previous estimates, D β t (P v )(x, t) D β t (P v 2 )(x, t) c 3 (T ) α2 R v v 2 C γ β (Q ) for all v, v 2 M R, (x, t) Q. The operator P is compact on M R (and thus on M R ): it was established earlier the uniform boundedness of the set P M R := {(P v)(x, t), (x, t) Q : v M R} in addition, it follows from the properties of Green s operators that for all ε > there exists δ = δ(ε) such that for all (x, t) Q, x < δ, t < δ and for all v M R sup (P v)(x + x, t + t) (P v)(x, t) < ε, (x,t) Q sup D β t (P v)(x + x, t + t) D β t (P v)(x, t) < ε. (x,t) Q As a result, the operator P is equicontinuous on MR. According to Schauder principle there exists a solution u MR of the equation (3.4). Theorem 3.2. Assume that F C (Q R 2 ) and is bounded, D β F C[, T and D β F (t), t [, T, ϕ satisfies the assumption (A5) and ϕ(x), x. Then a solution (u, r) C 2,α ( Q ) C[, T of the problem (.)-(.5) is unique. Proof. Take two solutions (u, r ), (u 2, r 2 ) C 2,α ( Q ) C[, T of (.)-(.5) and substitute them into equation (.). For u = u u 2, r = r r 2 we obtain the equation D α t u = u r (t)d β t u r(t)d β t u 2 + F (x, t, u, D β t u ) F (x, t, u 2, D β t u 2 ). By Hadamard lemma F (x, t, u, D β t u ) F (x, t, u 2, D β t u 2 ) = F (x, t)u + F 2 (x, t)d β t u with some known functions F j, j =, 2, which are continuous and bounded on Q, depend on u, u 2, D β t u, D β t u 2. Then the previous equation becomes D α t u + ( r (t) F 2 (x, t) ) D β t u = u + F (x, t)u r(t)d β t u 2, (x, t) Q. It follows from the boundary condition that and from the initial conditions u(x, t) =, x Ω, t [, T, u(x, ) =, u t (x, ) =, x.

11 EJDE-25/53 INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION Let G (x, t, y, τ) be the main Green s function of the first boundary-value problem for the equation D α t u + ( r (t) F 2 (x, t) ) D β t u = u + F (x, t)u. Then the function u(x, t) satisfies the equation t u(x, t) = dτ G (x, t, y, τ)dτ β u 2 (y, τ)r(τ)dy, (x, t) Q. It follows from the over-determination condition (.5) that [ r(t)d β F (t) = u(x, t) ϕ (x) ( ) + F (x, t)u(x, t) + F 2 (x, t)d β t u(x, t) ϕ (x) dx, (3.6) and then u(x, t) satisfies the equation t dτ [ (F u(x, t) = D β K (x, t, τ) (z, τ)ϕ (z) + ϕ (z) ) u(z, τ)dz F (τ) (3.7) + F 2 (z, τ)ϕ (z)dτ β u(z, τ) dz, (x, t) Q where K (x, t, τ) = G (x, t, y, τ)dτ β u 2 (y, τ)dy is continuous with respect to x and integrable in time function. If F (z, τ) = F 2 (z, τ) =, (z, τ) Q then by the uniqueness of the solution of this linear second type Volterra integral equation we obtain u(x, t) =, (x, t) Q. Then it follows from (3.6) that r(t)d β F (t) =, t [, T. Since D β F (t) on [, T (under the assumptions of this theorem), it follows that r(t) for t [, T. Assume that F (z, τ) + F 2 (z, τ), (z, τ) Q. (3.8) If F 2 (z, τ) =, (z, τ) Q then by the uniqueness of the solution of the linear second type Volterra integral equation (3.7) with integrable kernel K (x, t, τ) ( F (z, τ)ϕ (z) + ϕ (z) ) we obtain, as in previous case, that u(x, t) =, (x, t) Q and then, from (3.6), that r(t), t [, T. In the general case denote V (x, t) = ( F (x, t)ϕ (x) + ϕ (x) ) u(x, t) + F 2 (x, t)ϕ (x)d β t u(x, t). Then (3.7) implies t dτ ( V (x, t) = F D β (x, t)ϕ (x) + ϕ (x) ) K (x, t, τ)v (z, τ)dz F (τ) t + F 2 (x, t)ϕ (x)d β dτ t D β K (x, t, τ)v (z, τ)dz; F (τ) that is, V (x, t) = t dτ K(x, t, z, τ)v (z, τ)dz, (x, t) Q,

12 2 H. LOPUSHANSKA, V. RAPITA EJDE-25/53 where K(x, t, z, τ) = [F D β (x, t)ϕ (x) + ϕ (x) + F 2(x, t)ϕ (x)(t τ) β F (τ) Γ( β) K (x, t, τ). By the uniqueness of the solution of this linear second type Volterra integral equation with integrable kernel K(x, t, z, τ) we obtain V (x, t) =, (x, t) Q. Note that (3.8) implies Note also that F (x, t)ϕ (x) + ϕ (x) + F 2 (x, t)ϕ (x), (x, t) Q. D β t u(x, t) = f β (t) u(x, t) = f β (t) f β (t) u(x, t) = u(x, t) =, (x, t) Q, for the function u(x, t) satisfying zero initial conditions. Then it follows from the previous results that u(x, t) =, (x, t) Q and from (3.6), by the assumptions of this theorem, we obtain r(t), t [, T. In separate case F (x, t)ϕ (x) + ϕ (x) = for all (x, t) Q we may put V (x, t) = F 2 (x, t)ϕ (x)d β t u(x, t) and, as before, obtain the linear second type Volterra integral equation t V (x, t) = dτ K (x, t, z, τ)v (z, τ)dz, (x, t) Q with integrable kernel K (x, t, z, τ) = F 2 (x, t)ϕ (x)k (x, t, τ)(t τ) β/ Γ( β)d β F (τ). As before, from here we obtain V (x, t) =, (x, t) Q and, as in the general case, since F 2 (x, t)ϕ (x) on Q, conclude that u(x, t) =, (x, t) Q and r(t), t [, T. The similar result holds for the inverse problem on determination of a pair of functions (u, b): a solution u of the first (or second) boundary-value problem for the equation D α t u = u + b(t)u = F (x, t), (x, t) Q and an unknown coefficient b(t) under the same over-determinating condition (.2). We may study the cases N =, 2 in the same way. Acknowledgments. The authors are grateful to Prof. Mokhtar Kirane for useful discussions. References [ T. S. Aleroev, M. Kirane, S. A. Malik; Determination of a source term for a time fractional diffusion equation with an integral type over-determination condition, Electronic J. of Differential Equations, 23 (23), no. 27, 6. [2 V. V. Anh, N. N. Leonenko; Spectral analysis of fractional kinetic equations with random datas, 4 (2), no. 5/6, [3 J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki; Uniqueness in an inverse problem for a one-dimentional fractional diffusion equation, Inverse problems, 25 (2), 6. [4 M. M. Djrbashian, A. B. Nersessyan; Fractional derivatives and Cauchy problem for differentials of fractional order, Izv. AN Arm. SSR. Matamatika, 3 (968), 3-29.

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