1. Introduction and Formulation of a Problem Development of the theory of the equations with fractional derivatives is stimulated with development of the theory of the integer order differential equations. About applications to physics, biology, mathematical modeling, and so forth, one can find works [1–3]. Notice works [4–7], devoted to the studying of BVPs for parabolic-hyperbolic equations, involving fractional derivatives. BVPs for the mixed type equations involving the Caputo and the Riemann-Liouville fractional differential operators were investigated in works [8, 9].

Note that with intensive research on problem of optimal control of the agroeconomic system, regulating the label of ground waters and soil moisture, it has become necessary to investigate a new class of equations called “loaded equations.” It was given the most general definition of “loaded equations” and various loaded equations are classified in detail by Nakhushev [10]. After this work very interesting results on the theory of boundary value problems for the loaded equations parabolic, parabolic-hyperbolic, and elliptic-hyperbolic types were published; for example, see [11–13].

In this direction, some local and nonlocal problems for the loaded elliptic-hyperbolic type equations of the second and third order in double-connected domains were investigated (see [14–17]).

BVPs for the loaded mixed type equations with fractional derivative have not been investigated yet.

In the given work, we consider the following equation:(1)0=uxx-DCoyαu+px∫x1t-xβ-1ut,0dt,at y>0uxx-uyy+qx+y∫x+y1t-x-yγ-1ut,0dt,at y<0,with the following operation [18]:(2)DCoyαf=1Γ1-α∫0yy-t-αf′tdt,where 0<α,β,γ<1.

Let Ω be domain, bounded with segments A1A2¯=x,y: x=1, 0≤y≤h, B1B2¯=x,y: x=0, 0≤y≤h, and B2A2¯=x,y: y=h,0≤x≤1 at y>0 and characteristics A1C: x-y=1; B1C: x+y=0 of (1) at y<0, where A11;0, A21;h, B10;0, B20;h, C1/2;-1/2.

Let us enter designations(3)Dxa-βfx=1Γβ∫xat-xβ-1ftdt, 0<β<1,(4)Ω+=Ω∩y>0,Ω-=Ω∩y<0,I1=x: 12<x<1,I2=y: 0<y<h.In the domain of Ω the following problem is investigated.

Problem 1. To find a solution u(x,y) of (1) from the class of functions, (5)W=ux,y: ux,y∈CΩ-∩C2Ω-, uxx∈CΩ+,DCoyαu∈CΩ+,satisfying boundary conditions(6)ux,yA1A2=φy, y∈I¯2,(7)ux,yB1B2=ψy, y∈I¯2,(8)ux,yA1C=ωx, x∈I1¯,and gluing condition (9)limy→+0 y1-αuyx,y=λuyx,-0, x,0∈A1B1,where φ(y), ψ(y), and ω(x) are given functions, λ=const λ≠0, and besides ω(1)=φ(0).

2. The Uniqueness of Solution of Problem <xref ref-type="statement" rid="problem1">1</xref> It is known that (1) at y≤0 on the characteristics coordinates ξ=x+y and η=x-y totally looks like(10)uξη=-qξ4∫ξ1t-ξγ-1ut,0dt.Let us enter designations ux,0=τx, 0≤x≤1; uy(x,-0)=ν-x, 0<x<1; (11)limy→+0 y1-αuyx,y=ν+x, 0<x<1.It is known that solution of the Cauchy problem for (1) in the domain of Ω- can be represented as follows:(12)ux,y=τx+y+τx-y2-12∫x+yx-yν-tdt-14∫x+yx-yqξdξ∫ξx-ydη∫ξ1t-ξγ-1τtdt.After using condition (8) and taking (3) into account from (12) we will get(13)ν-x=x-12ΓγqxDx1-γτx-τ′x+ω′x+12.Considering designations and gluing condition (9) we have (14)ν+x=λν-x.Further from (1) at y→+0 taking into account (2), (14), and(15)limy→0 D0yα-1fy=Γαlimy→0 y1-αfy,we get [8](16)τ′′x-λΓαν-x+ΓβpxDx1-βτx=0.

Theorem 2. Satisfying conditions (17)p0≤0,p′x≤0;λq0≤0,λx-1qx′≥0,the solution u(x,y) of Problem 1 is unique.

Proof. It is known that if homogeneous problem has only trivial solution, then we can state that original problem has unique solution. For this aim we assume that Problem 1 has two solutions; then denoting difference of these solutions as u(x,y) we will get appropriate homogenous problem.

We multiply (16) to τ(x) and integrated it from 0 to 1:(18)∫01τ′′xτxdx-λΓα∫01τxν-xdx+Γβ∫01τxpxDx1-βτxdx=0.We will investigate the integral (19)I=λΓα∫01τxν-xdx-Γβ∫01τxpxDx1-βτxdx.Taking (13) into account, ω(x)=0, we get(20)I=λΓαΓγ2∫01τxx-1qxDx1-γτxdx-λΓα∫01τxτ′xdx-Γβ∫01τxpxDx1-βτxdx=λΓα2∫01qxτx1-xdx∫x1t-xγ-1τtdt-λΓα2∫01dτ2x-∫01τxpxdx∫x1t-xβ-1τtdt.Considering τ(1)=0, τ0=0 (deduced from conditions (6) and (7) in homogeneous case) and on a base of the formula (see [19, p. 188]),(21)x-t-γ=1Γγcosπγ/2∫0∞zγ-1coszx-tdz, 0<γ<1.After some simplifications from (20) we will get (22)I=λΓαq04Γ1-γsinπγ/2∫0∞z-γ∫01τtcoszt dt2+∫01τtsinzt dt2dz+λΓα4Γ1-γsinπγ/2∫0∞z-γdz∫01∂∂xx-1qx∫x1τtcoszt dt2+∫x1τtsinzt dt2dx-p02Γ1-βsinπβ/2∫0∞z-β∫01τtcoszt dt2+∫01τtsinzt dt2dz-12Γ1-βsinπβ/2∫0∞z-βdz∫01∂∂xpx∫x1τtcoszt dt2+∫x1τtsinzt dt2dx.

Thus, owing to (17) from (22) it is concluded that τ(x)≡0. Hence, based on the solution of the first boundary problem for (1) [9, 20] taking into account (6) and (7) we will get ux,y≡0 in Ω¯+. Further, from functional relations (13), taking into account τ(x)≡0, we get that ν-(x)≡0. Consequently, based on the solution (12) we obtain u(x,y)≡0 in closed domain Ω¯-.

3. The Existence of Solution of Problem <xref ref-type="statement" rid="problem1">1</xref> Theorem 3. Satisfying conditions (17) and(23)φy,ψy∈CI2¯∩C1I2,ωx∈C1I1¯∩C3I1,px∈CA1B1¯∩C2A1B1,qx+y∈CΩ-¯∩C2Ω-,the solution of the investigating problem exists.

Taking (13) into account from (16) we will obtain(24)τ′′x+λ2ΓαΓγ1-xqxDx1-γτx+λΓατ′x=λΓαω′x+12-ΓβpxDx1-βτx.From here, (25)τ′′x+λΓατ′x=fx,where (26)fx=λ2ΓαΓγx-1qxDx1-γτx+λΓαω′x+12-ΓβpxDx1-βτx.Solution of (25) together with conditions (27)τ0=ψ0,τ1=φ0has a form(28)τx=1-xψ0+xφ0+∫01Gx,tf1tdt,where f1x=fx+λΓαψ0-φ0. One has(29)Gx,t=eλΓαx-1eλΓαt-eλΓαeλΓαxeλΓα-1λΓα;0≤t≤xeλΓαt-1eλΓαt-eλΓαeλΓαxeλΓα-1λΓα;t≤x≤1.G(x,t) is Green’s functions of the boundary value problem for (25) with conditions (27). Further, considering (26) and using (3) from (28) we will get (30)τx=λΓα2∫0xτtdt∫0ts-1t-sγ-1K1x,sqsds-∫0xτtdt∫0tt-sβ-1K1x,spsds+λΓα2∫x1τtdt∫0xs-1t-sγ-1K1x,sqsds-∫x1τtdt∫0xt-sβ-1K1x,spsds+λΓα2∫x1τtdt∫txs-1t-sγ-1K2x,sqsds-∫x1τtdt∫xtt-sβ-1K2x,spsds+Fx,where (31)Fx=λΓα∫01Gx,tω′t+x2dt-λΓαψ0-φ0∫01Gx,tdt,K1x,t=eλΓαt-1eλΓαx-eλΓαeλΓαxeλΓα-1λΓα;K2x,t=eλΓαx-1eλΓαt-eλΓαeλΓαxeλΓα-1λΓα.Folding separately the integrals with limits ∫0x and ∫x1 we rewrite integral equation (30) as follows:(32)τx=∫01Kx,tτtdt+Fx.Here (33)Kx,t=∫0tK1x,sλΓα21-st-sγ-1qs-t-sβ-1psds;0≤t≤x∫0xK1x,sλΓα2s-1t-sγ-1qs-t-sβ-1psds+∫xtK2x,sλΓα2s-1t-sγ-1qs-t-sβ-1psds;x≤t≤1.For better understanding, we present the statement of equivalence.

Remark 4. Satisfying all conditions of Theorem 3, the existence and the uniqueness of solutions of Problem 1 in the class of W are equivalent to the unique solvability of the Fredholm type integral equation (32) in the class of C10,1∩C2(0,1).

Now we need to investigate the integral equation (32); for this aim we will estimate the kernel and the right side.

Owing to class (23) of the given functions and after some evaluations from (33) we will conclude that(34)K1x,t=eλΓαt-1eλΓα-1λΓα1-eλΓα1-x≤1-eλΓαλΓα=const<∞ at 0≤t≤x,Kx,t=∫0tK1x,sλΓα21-st-sγ-1qs-t-sβ-1psds≤const∫0tt-sσ-1ds≤const·tσ≤const at 0≤t≤x, where σ=minγ,β.

Accordingly we can get K(x,t)≤const at x≤t≤1.

Taking into account ω(x)∈C1I1¯∩C3I1 and G(x,t)≤const and Gx(x,t)≤const from F(x)=λΓ(α)∫01Gx,tω′(t+x)/2dt-λΓ(α)ψ(0)-φ0∫01Gx,tdt we will get Fx≤const and F′x≤const. Hence on the base of theory of Fredholm integral equations of the second kind the integral equation (32) is solvability and we can write a solution of this equation via resolvent-kernel:(35)τx=Fx-∫01Rx,tFtdt,where R(x,t) is the resolvent-kernel of K(x,t).

We will find unknown functions ν-(x) and ν+(x) accordingly from (13) and (14):(36)ν-x=1-x2qx∫x1t-x1-γdt∫01Rt,sFsds-1-x2qx∫x1t-x1-γFtdt-F′x+∫01∂Rx,t∂xFtdt+ω′x+12,and ν+(x)=λν-(x).

We write solution of Problem 1 in the domain Ω+ as follows [18, 21]:(37)ux,y=∫0yGξx,y,0,ηψηdη-∫0yGξx,y,1,ηφηdη+∫01G0x-ξ,yτξdξ-∫0y∫01Gx,y,0,ηpξdξ dη∫ξ1t-ξβ-1τtdt.Here G0(x-ξ,y)=1/Γ1-α∫0yη-αG(x,y,ξ,η)dη, (38)Gx,y,ξ,η=y-ηα/2-12∑n=-∞∞e1,α/21,α/2-x-ξ+2ny-ηα/2-e1,α/21,α/2-x+ξ+2ny-ηα/2is Green’s function of the first boundary problem for (1) with the Riemann-Liouville fractional differential operator (note that it is true for the Caputo operator too) (see [20]), and (39)e1,δ1,δz=∑n=0∞znn!Γδ-δnis the Wright type function [18].

Solution of Problem 1 in the domain Ω- will be found by formula (12). Hence, Theorem 3 is proved.