FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

12.07.2015 Views
62 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangProof. Studying the spectrum of the problem is reduced to studying thespectrum of the linear operator bunchClearly,L(λ) = J + M −1{ n ∑M −1( n ∑i=1i=1}a i (x)D αi0x ω i − λM −1 .)a i (x)D α i0x ω i(x)u ∈ G 1 .M −1 is a positive operator. If for the function of distribution rn(r) it ispossible choose a nondecreasing function ϕ(r) (0 ≤ r ≤ ∞) possessing theproperties1. lim ϕ(r) = ∞, ϕ(r) ↑ (r → ∞);r→∞[ ] ′ ϕ ′ (r)2. lim ln(ϕ(r)) = limr→∞r→∞ ϕ(r) < ∞;n(r)3. limr→∞ ϕ(r) = 1,then by theorem of Keldysh,limr→∞n(r)n(r, M −1 ) = 1.As ϕ(r), in our case, obviously it is possible to take the function ϕ(r) =r 1/2 . This proves Theorem 2.9.□Consider the differential expressionLy = −y ′′ + D α 0xyon the finite interval [0, 1].Let T be an operator defined in the Hilbert space H = L 2 (0, 1) bythe operator L with boundary conditions y(0) = y(1) = 0. Certainly, theoperator T is a weak disturbance of the operator{u ′′ ,Au =u(0) = 0, u(1) = 0.Formulate a theorem which is quite important in studying the operatorsof the type T.Theorem 2.10. The operator T is dissipative.Proof.( )(T y, y) = − d2dx 2 y, y + (D0xy, α y), (D0xy, α y) ≥ 0.Thus the operator T is dissipative.□

Boundary Value Problems for Differential Equations of Fractional Order 637. Construction of a Biorthogonal SystemConsider the Sturm–Liouville problem for a fractional differential equation⎧⎪⎨ u ′′ + a m (x)u ′ + a i (x)D α i0x ω i(x)u + λu = 0,u(0) cos(α) + u⎪ ′ (0) sin(α) = 0,(2.47)⎩u(1) cos(β) + u ′ (1) sin(β) = 0and the problem⎧⎪⎨ z − [a m (x)z(x)] ′ − ω i (x)D αix1 a i(t)z(t) + λz = 0,z(0) cos(α) + z⎪⎩′ (0) sin(α) = 0,z(1) cos(β) + z ′ (1) sin(β) = 0.We will call the problems (2.47) and (2.48) mutually adjoint.Following M. M. Dzhrbashyan [10], we introduce the functionsω(λ) = u(1, λ) + u ′ (1, λ) sin(β),˜ω(λ ∗ ) = z(0, λ ∗ ) + z ′ (0, λ ∗ ) sin(α),where u(x, λ) is a solution of the following Cauchy problem(2.48)u ′′ + a m (x)u ′ + a i (x)D α i0x ω i(x)u + λu = 0, (2.49)u(0) = sin(α), u ′ (0) = − cos(α) (2.50)and z(x, λ ∗ ) is a solution of the problemz ′′ (x) + a m (x)z ′ (x) + ω i (x)D α ix1 a i(x)z + λ ∗ z = 0, (2.51)z(1) = sin(β), z ′ = − cos(β). (2.52)The existence and uniqueness of solutions of mutually adjoint problems(2.47) and (2.48) are already proved [3].Clearly if u(x, λ) is a solution of the problem (2.49)–(2.50), then anecessary and sufficient condition for it to be also a solution of the problem(2.47) isω(λ) = u(1, λ) cos(β) + u ′ (1, λ) sin(β).There is a similar statement for the solutions of the problems (2.51)–(2.52).For construction of an orthogonal system of eigenfunctions and associatedfunctions of mutually adjoint problems by method of M. M. Dzhrbashyanwe need the followingTheorem 2.11. Let a m (0) = a m (1) = 0. Then for any values of theparameters λ, λ ∗ the identityholds.(λ − λ ∗ )∫ 10u(x, λ)z(x, λ ∗ ) dx = ω(λ) − ˜ω(λ ∗ ) (2.53)

Page 4 and 5: 24 T. S. Aleroev, H. T. Aleroeva, N







Page 20 and 21: CHAPTER 2The Sturm-Liouville Proble












Page 46 and 47: CHAPTER 3Solving Two-Point Boundary






Page 58 and 59: Since∣∣f(t, u 2 ) − f(t, u 1

Page 60 and 61: agreement with the fact that (3.33)

Page 62: 82 T. S. Aleroev, H. T. Aleroeva, N

operator

fractional

differential

boundary

equations

equation

aleroev

theorem

eigenvalues

aleroeva

FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER ... View more FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Delete template?

Save as template ?

FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER