Boundary Value Problems for the Classical and Mixed Integrodifferential Equations with Riemann-Liouville Operators

Abstract

By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded third-order integrodifferential equations with Riemann-Liouville operators.

1. Introduction

The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively because of both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations. The first fundamental results in this direction were obtained in 1920–1930 by Tricomi [1] and Gellerstedt [2]. The works of M. A. Lavrent’ev, A. V. Bitsadze, F. I. Frankl, M. Protter, and C. Morawetz have had a great impact on this theory, where outstanding theoretical results were obtained and pointed out important practical values. Bibliography of the main fundamental results on this direction can be found, among others, in the monographs of Bitsadze [3], Bers [4], Salakhitdinov and Urinov [5], and Nakhushev [6].

In the recent years, in connection with intensive research on problems of optimal control of the agroeconomical system, long-term forecasting, and regulating the level of ground waters and soil moisture, it has become necessary to investigate a new class of equations called “loaded equations.” Such equations were investigated for the first time in works of N. N. Nazarov and N. Kochin. However, they did not use the term “loaded equation.” For the first time, the term has been used in works of Nakhushev [7], where the most general definition of a loaded equation is given and various loaded equations are classified in detail, for example, loaded differential, integral, integrodifferential, functional equations and so forth, and numerous applications are described [6, 8].

Basic questions of the theory of boundary value problems for partial differential equations are the same for the boundary value problems for the loaded differential equations. However, existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems.

Works of Nakhushev, M. Kh. Shkhankov, A. B. Borodin, V. M. Kaziev, A. Kh. Attaev, C. C. Pomraning, E. W. Larsen, V. A. Eleev, M. T. Dzhenaliev, J. Wiener, B. Islomov and D. M. Kuriazov, K. U. Khubiev, and M. I. Ramazanov et al. are devoted to loaded second-order partial differential equations.

It should be noted that boundary value problems for loaded equations of a hyperbolic, parabolic-hyperbolic, and elliptic-hyperbolic types of the third order are not well understood. We indicate only the works of V. A. Eleev, Islomov, D. M. Kur’yazov, and A. V. Dzarakhokhov.

The present paper is devoted to formulation and investigation of the analogue of the Cauchy-Goursat problem for the loaded equation of a hyperbolic type 𝜕𝑢𝜕𝑥^𝑥𝑥−𝑢_𝑦𝑦−𝜆𝑢−𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑢(𝑥,0)=0 (1) and a boundary value problem for a loaded equation of a mixed parabolic hyperbolic type 𝜕()𝜕𝑥𝐿𝑢−𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑢(𝑥,0)=0, (2) where 𝐿𝑢≡𝑢_𝑥𝑥−1−sgn𝑦2𝑢_𝑦𝑦−1+sgn𝑦2𝑢_𝑦𝐷−𝜆𝑢,_{𝛼𝑖0𝑥}𝑓(𝑥)=1Γ−𝛼_𝑖∫^𝑥₀𝑓(𝑡)𝑑𝑡(𝑥−𝑡)^1+𝛼𝑖,𝛼_𝑖𝑑<0,𝐷𝑑𝑥^{𝛼𝑖−10𝑥}𝑓(𝑥),0<𝛼_𝑖<1. (3)𝐷^−𝛼𝑖_0𝜉𝐷^𝛼𝑖_0𝜉𝑓≡𝐷⁰_0𝜉𝑓≡𝑓(𝜉). Assume that 𝛼_𝑛<𝛼_𝑛−1<⋯<𝛼₁=𝛼<1 and coefficients 𝑎_𝑖=𝑎_𝑖(𝑥)∈𝐶¹[0,1]∩𝐶³(0,1), 𝜆, 𝜇 are given real parameters, and 𝜆>0.

2. Analogue of the Cauchy-Goursat Problem for a Loaded Equation of the Hyperbolic Type

Let 𝐷⊂𝑅² be a domain bounded at 𝑦<0 by the characteristics 𝐴𝐶∶𝑥+𝑦=0,𝐵𝐶∶𝑥−𝑦=1 (4) of (1) and the segment 𝐴𝐵 of the axis 𝑦=0.

Let us consider the following analogue of the Cauchy-Goursat problem for the loaded equation (1) in the domain 𝐷.

Problem A. Find a solution 𝑢(𝑥,𝑦) to (1), which is regular in the domain 𝐷 continuous in 𝐷, has continuous derivatives 𝑢_𝑥, 𝑢_𝑦, up to 𝐴𝐵∪𝐴𝐶, and satisfies the boundary value conditions 𝑢_𝑦𝑥,𝑦_𝐴𝐵(𝑥)𝑢=𝜈,0≤𝑥<1,𝑥,𝑦_𝐴𝐶=𝜓₁(𝑥),𝜕𝑢𝑥,𝑦𝜕𝑛_𝐴𝐶=𝜓₂(𝑥),10≤𝑥≤2, (5) where 𝑛 is an inner normal and 𝜈(𝑥), 𝜓₁(𝑥), 𝜓₂(𝑥) are real-valued functions.

Theorem 1. If 2𝜈(𝑥)=√2𝜓₂(0)−𝜓^′₁(0) and 𝜈(𝑥)∈𝐶[0,1]∩𝐶²(),𝜓0,1₁(𝑥)∈𝐶¹10,2∩𝐶³10,2,𝜓₂(𝑥)1∈𝐶0,2∩𝐶²10,2, (6) then there exists a unique solution to the problem 𝐴 in the domain 𝐷.

Proof of Theorem 1. An important role in proving Theorem 1 is played by the following.
Lemma  2.  Any regular solution to (1) is represented in the form𝑢𝑥,𝑦=𝑧𝑥,𝑦+𝑤(𝑥), (7)where 𝑧(𝑥,𝑦) is the solution of the equation𝜕𝑧𝜕𝑥^𝑥𝑥−𝑧_𝑦𝑦−𝜆𝑧=0 (8)and 𝑤(𝑥)—is the solution of the following ordinary differential equation:𝑤^′′′(𝑥)−𝜆𝑤^′(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑤(𝑥)=𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑧(𝑥,0). (9)
Proof of Lemma  2. Let 𝑢(𝑥,𝑦), represented by Formula (7), be the solution of (1). Then, substituting 𝜕𝑢𝜕𝑥^𝑥𝑥−𝑢_𝑦𝑦−𝜆𝑢−𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑢()=𝜕𝑥,0𝑧𝜕𝑥^𝑥𝑥−𝑧_𝑦𝑦−𝜆𝑧+𝑤^′′′(𝑥)−𝜆𝑤^′(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑤(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑧(𝑥,0)=0 (10) satisfies (1).
Then, vice versa, let 𝑢(𝑥,𝑦) be a regular solution to (1) and let 𝑤(𝑥) be a certain solution equation 𝑤^′′′(𝑥)−𝜆𝑤^′(𝑥)=𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑢(𝑥,0). (11)
Let us prove the validity of relation (7). Manifestly, the function 𝑢+𝜇𝑥,𝑦=𝑧𝑥,𝑦𝜆^𝑥₀ch√𝜆(𝑥−𝑡)−1𝑢(𝑡,0)𝑑𝑡 (12) is a solution to (1), where 𝑧(𝑥,𝑦) is a solution to (8) and the function 𝜔(𝑥)=𝜇𝜆^𝑥₀ch√𝜆(𝑥−𝑡)−1^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑢(𝑡,0)𝑑𝑡 (13) is a partial solution to (1). Hence, (1) entails the validity of representation (7); that is, 𝑢(𝑥,𝑦)=𝑧(𝑥,𝑦)+𝑤(𝑥).
It follows from the latter representation that 𝑢(𝑥,0)=𝑧(𝑥,0)+𝑤(𝑥). Then, (11) provides 𝑤^′′′(𝑥)−𝜆𝑤^′(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑤(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑧(𝑥,0)=0, (14) and the function 𝑧(𝑥,𝑦)=𝑢(𝑥,𝑦)−𝑤(𝑥) satisfies (8).
Lemma 2 is proved.
Invoking that the function 𝑎𝑒^√𝜆𝑥+𝑏𝑒^{−√𝜆𝑥}+𝑐 satisfies (8), we can assume without loss of generality that 𝑤(0)=𝑤^′(0)=𝑤^′′(0)=0, (15) when studying problem A.
Let us solve the Cauchy problem for (9) with conditions (15) with respect to 𝑤(𝑥).
The solution to the Cauchy problem for (9) with conditions (15) has the form 𝜔(𝑥)=1𝜆^𝑥₀ch√𝜆(𝑥−𝑡)−1𝑓(𝑡)𝑑𝑡, (16) where 𝑓(𝑥)=𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑤(𝑥)+𝜇^𝑛_𝑖=1𝑎_𝑖(𝑥)𝐷^𝛼𝑖_0𝑥𝑧(𝑥,0). (17)
The last equality with respect to designation and after some transformation becomes 𝜔(𝑥)−𝜇^𝑥₀𝐾(𝑥,𝑡)𝜔(𝑡)𝑑𝑡=𝐹(𝑥),0≤𝑥≤1, (18) where 𝐾()=1𝑥,𝑡Γ−𝛼_𝑖∫^𝑥₀𝐴_𝑖(𝑡)(𝑡−𝑠)^{−1−𝛼𝑖}×ch√𝜆(𝑥−𝑡)−1𝑑𝑡,𝛼_𝑖1<0,𝜆Γ1−𝛼_𝑖∫^𝑥_𝑠𝜕𝐴_𝑖(𝑡)(𝑡−𝑠)^−𝛼𝑖×ch√𝜆(𝑥−𝑡)−1𝜕𝑡𝑑𝑡,0<𝛼_𝑖𝐹(𝑥)=𝜇<1,𝜆^𝑥₀ch√𝜆(𝑥−𝑡)−1^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑥𝑧(𝑡,0)𝑑𝑡. (19)
And with recurring index 𝑖=1,2,…,𝑛 implied summation. Solving the next equation with respect to [7] and Dirichlet’s formula we have 𝜔(𝑥)=^𝑥₀𝑃(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑥𝑧(𝑡,0)𝑑𝑡, (20) where 𝑃()=𝜇𝑥,𝑡𝜆ch√𝜆(𝑥−𝑡)−1+𝜇^𝑥_𝑡ch√𝜆(𝑠−𝑡)−1𝑅𝑥,𝑠;𝜇𝑑𝑠. (21)
By virtue of representation (7), problem A is reduced to problem A* of finding a solution 𝑧(𝑥,𝑦) of (8), which is regular in the domain conditions 𝑧_𝑦𝑥,𝑦_𝐴𝐵=𝜈(𝑥),0<𝑥<1,𝑧(𝑥,𝑦)_𝐴𝐶=𝜓₁(𝑥)−𝑤(𝑥),𝜕𝑧𝑥,𝑦𝜕𝑛_𝐴𝐶=𝜓₂(𝑥)−1√2𝑤^′(𝑥)1,0≤𝑥≤2, (22) where 𝜔(𝑥) is defined by (20).
Similarly to [9, 10], we can write out the solution to (8) with conditions (22) by means of the general representation in view of (6) and [11]: 𝑧𝑥,𝑦=₀^𝑥+𝑦𝜈(𝑡)𝐼₀𝜆𝑥+𝑦−𝑡𝑥−𝑦−𝑡𝑑𝑡−𝜓^∗₁(0)𝐼₀𝜆𝑥²−𝑦²+𝜓^∗₁𝑥+𝑦2+𝜓^∗₁𝑥−𝑦2+1√𝜆^0𝑥+𝑦𝜆𝜓^∗₁(𝑡)−√2𝜓₂^∗′(𝑡)×sin√𝜆𝑡−𝑥+𝑦2+1𝑑𝑡√𝜆^{0(𝑥−𝑦)/2}𝜆𝜓^∗₁(𝑡)−√2𝜓₂^∗′(𝑡)×sin√𝜆𝑡−𝑥−𝑦2𝑑𝑡−2₀^{(𝑥−𝑦)/2}𝜓^∗₁(𝑡)×𝐵_𝑡+10,2𝑡;𝑥+𝑦,𝑥−𝑦𝑑𝑡√𝜆^𝑦0𝜆𝜓^∗₁(−𝑡)−√2𝜓₂^∗′()√−𝑡×sin𝜆−2𝑦−𝑡𝑑𝑡√𝜆^{0(𝑥−𝑦)/2}𝐵_𝑡×0,2𝑡;𝑥+𝑦,𝑥−𝑦𝑑𝑡_𝑡0𝜆𝜓^∗₁(𝑧)−√2𝜓₂^∗′(𝑧)sin√𝜆(−𝑡+𝑧)𝑑𝑧, (23) where 𝜓^∗₁(𝑥)=𝜓₁(𝑥)−𝑤(𝑥),𝜓^∗₂(𝑥)=𝜓₂(𝑥)−1√2𝑤^′(𝑥), (24)𝐵(𝑡,𝑧;𝑥+𝑦,𝑥−𝑦) is the Riemann-Hadamard function [12] and, 𝐼₀[𝑧] is the modified Bessel function [13].
Assuming that 𝑦=0 in (23) and invoking (20), we obtain the following functional correlation, transferred from the domain 𝐷 onto 𝐴𝐵: 𝜏(𝑥)+₀^𝑥/2𝐾(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)𝑑𝑡=Φ(𝑥),0≤𝑥≤1, (25) where 𝜏(𝑥)(),𝐾()𝑥=𝑧𝑥,0𝑥,𝑡=2𝑃2,𝑡+𝜆𝑥^𝑥_2𝑡𝑃𝑠2,𝑡𝐼₁(+1𝜆𝑥𝑥−𝑠)𝑑𝑠√𝜆^𝑥2𝑡𝐾^∗𝑥2×𝑠,𝑡𝜆𝑃2−1,𝑡4𝑃^′𝑠2Φ(𝑥),𝑡𝑑𝑠,=2𝜓₁𝑥2−𝜓₁(0)𝐼₀√+𝜆𝑥_𝑥0𝜈(𝑡)𝐼₀√𝜆(𝑥−𝑡)𝑑𝑡+𝜆𝑥^𝑥₀𝐼₁𝜆𝑥(𝑥−𝑡)𝜓₁𝑡2+1𝑑𝑡√𝜆^𝑥0𝐾^∗𝑥2,𝑡2×𝜆𝜓₁𝑡2−√22𝜓^′₂𝑡2𝐾𝑑𝑡,_∗(𝑥,𝑡)=sin√𝜆()+𝑡−𝑥_𝑥𝑡𝜆𝑥𝐼₁𝜆𝑥(𝑥−2𝑠)sin√𝜆(𝑡−𝑠)𝑑𝑠, (26)(27)𝐼₁(𝑥)=𝐼₁(𝑥)/𝑥, 𝐼₀(𝑥), are 𝐼₁(𝑥) are the modified Bessel functions [13].
We can assume without loss of generality that 𝛼_𝑖>0 when 𝑖=1,2,…,𝑚 and 𝛼_𝑖<0 when 𝑖=𝑚+1,…,𝑛.(1)Let 𝑖≥𝑚+1, and then for any function 𝜏(𝑥)∈𝐶[0,1] applying the Dirichlet permutation integration formula from (25) we have 𝜏(𝑥)+𝜇Γ−𝛼_𝑖0^𝑥/2𝜏(𝑡)𝑑𝑡_𝑡^𝑥/2𝐴_𝑖𝐾(𝑥,𝜉)𝑑𝜉(𝜉−𝑡)^1+𝛼𝑖=Φ(𝑥). (28)(2)Let 𝑖=1,2,…,𝑚, and then taking account of conversion [3] into (25) we get 𝜏(𝑥)−𝜇Γ1−𝛼_𝑖0^𝑥/2𝜏(𝑡)𝑑𝑡_𝑡^𝑥/2𝜕𝐴_𝑖𝐾(𝑥,𝜉)/𝜕𝜉(𝜉−𝑡)^𝛼𝑖=Φ(𝑥). (29)
Hence, we conclude that the integral equations (28), (29) with respect to (6), [7] always have a solution, which is unique [14].
Thus, it is proved that problem A is uniquely solvable. Theorem 1 is proved.

3. Investigation of Problem C for (2)

Formulation of Problem C for (2). Let Ω₁⊂𝑅² be a domain bounded by the segments 𝐴𝐵, 𝐵𝐵₀, 𝐴𝐴₀, and 𝐴₀𝐵₀ of the straight lines 𝑦=0, 𝑥=1, 𝑥=0, and 𝑦=ℎ, respectively, when 𝑦>0. Ω₂ is a characteristic triangle bounded by the segment 𝐴𝐵 of the axis 𝑂𝑋 and two characteristics 𝐴𝐶∶𝑥+𝑦=0,𝐵𝐶∶𝑥−𝑦=1 (30) of (2) for 𝑦<0.

Let us introduce the following notation: 𝐼=𝑥,𝑦∶0<𝑥<1,𝑦=0,Ω=Ω₁∪Ω₂∪𝐼. (31)

Let us term the function 𝑢(𝑥,𝑦)∈𝐶(Ω)∩𝐶¹(Ω)∩𝐶^3,1(Ω₁)∩𝐶^3,2(Ω₂) satisfying (2) in Ω₁ and Ω₂ as a regular solution of (2).

Problem C. Find the function 𝑢(𝑥,𝑦), possessing the following properties:(1)𝑢(𝑥,𝑦)∈𝐶(Ω); (2)𝑢_𝑥(𝑢_𝑦) is continuous up to𝐴𝐴₀∪𝐴𝐶(𝐴𝐵∪𝐴𝐶);(3)𝑢(𝑥,𝑦) is a regular solution of (2) in the domains Ω₁ and Ω₂;(4)the sewing conditions 𝑢_𝑦(𝑥,−0)=𝑢_𝑦(𝑥,+0),(𝑥,0)∈𝐼 (32) are satisfied on 𝐴𝐵;(5)𝑢(𝑥,𝑦) satisfies the boundary value conditions 𝑢(𝑥,𝑦)_𝐴𝐴0=𝜑₁𝑦,𝑢(𝑥,𝑦)_𝐵𝐵0=𝜑₂𝑦,𝑢_𝑥(𝑥,𝑦)_𝐴𝐴0=𝜑₃𝑦,0≤𝑦≤ℎ,𝑢(𝑥,𝑦)_𝐴𝐶=𝜓₁(𝑥),𝜕𝑢(𝑥,𝑦)𝜕𝑛_𝐴𝐶=𝜓₂(𝑥)1,0≤𝑥≤2, (33)where 𝑛 is the inner normal and 𝜑₁(𝑦), 𝜑₂(𝑦), 𝜑₃(𝑦), 𝜓₁(𝑥), and 𝜓₂(𝑥) are given functions.

We note the unique solvability of problem C and Gellerstedt problems for a loaded differential equation (2) when 𝛼_𝑖≡0 was proven by Islomov and Baltaeva [11].

Theorem 3. If 𝜆>0 and 𝜑₁(0)=𝜓₁(0),𝜑_𝑗𝑦∈𝐶¹[],,𝜑0,1𝑗=1,2₃𝑦∈𝐶[0,1]∩𝐶¹(),𝜓0,1₁(𝑥)∈𝐶¹10,2∩𝐶³10,2,𝜓₂(𝑥)1∈𝐶0,2∩𝐶²10,2, (34)(35) then there exists a unique solution to the problem C in the domain Ω.

Proof of Theorem 3. The following theorem holds.
Lemma  4.  Any regular solution of (2) (when 𝑦≠0) is represented in the form𝑢𝑥,𝑦=𝑧𝑥,𝑦+𝑤(𝑥), (36)where 𝑧(𝑥,𝑦) is a solution to the equation𝜕0=𝑧𝜕𝑥^𝑥𝑥−𝑧_𝑦𝑧−𝜆𝑧,𝑧>0,_𝑥𝑥−𝑧_𝑦𝑦−𝜆𝑧,𝑧<0. (37)𝑤(𝑥) is a solution of the following ordinary differential equation:𝑤^′′′(𝑥)−𝜆𝑤^′(𝑥)−𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑤(𝑥)=𝜇^𝑛_𝑖=1𝑎_𝑖𝐷^𝛼𝑖_0𝑥𝑧(𝑥,0). (38)
The lemma is proved similarly to Lemma  2.
Invoking that the function 𝑎𝑒^√𝜆𝑥+𝑏𝑒^{−√𝜆𝑥}+𝑐 satisfies (37), we can subordinate the function 𝑤(𝑥) to the conditions 𝑤(0)=𝑤^′(0)=𝑤^′′(0)=0. (39)
Solution of the Cauchy problem for (38) with the conditions (39) can be represented in the form (20).
By virtue of representation (36), (2) and the boundary value conditions (33), in view of (39), are reduced to the form (37): 𝑧𝑥,𝑦_𝐴𝐴0=𝜑₁𝑦,𝑧𝑥,𝑦_𝐵𝐵0=𝜑₂𝑦−𝑤(1),𝜕𝑧𝑥,𝑦𝜕𝑥_𝐴𝐴0=𝜑₃𝑦,0≤𝑦≤ℎ,𝑧(𝑥,𝑦)_𝐴𝐶=𝜓₁(𝑥)(𝑥)1−𝑤,0≤𝑥≤2,𝜕𝑧(𝑥,𝑦)𝜕𝑛_𝐴𝐶=𝜓₂(𝑥)−1√2𝑤^′(𝑥)1,0≤𝑥≤2. (40)(41)(42)
Derivation of Basic Functional Relations. As it is known from problem A, the solution to (37) with the boundary value conditions (41), (42), and 𝜕𝑧𝑥,𝑦𝜕𝑦_𝑦=0=𝜈(𝑥),0<𝑥<1, (43) is given by the formula (23).
Assuming that 𝑦=0 in (23), in view of (39), and 𝑧(𝑥,0)=𝜏(𝑥),0≤𝑥≤1, (44) we obtain the functional relation, transferred from the domain Ω₂ onto 𝐴𝐵: 𝜏(𝑥)+₀^𝑥/2𝐾(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)−𝑑𝑡_𝑥0𝐼₀√𝜆(𝑥−𝑡)𝜈(𝑡)𝑑𝑡=𝑓₁(𝑥), (45) where 𝑓₁(𝑥)=2𝜓₁𝑥2−𝜓₁(0)𝐽₀√𝜆𝑥+𝜆𝑥^𝑥₀𝐼₁𝜆𝑥(𝑥−𝑡)𝜓₁𝑡2+1𝑑𝑡√𝜆^𝑥0𝐾^∗𝑥2,𝑡2×𝜆𝜓₁𝑡2−√2𝜓^′₂𝑡2𝑑𝑡, (46) where 𝐾(𝑥,𝑡) can be represented in the form (26).
Denoting 𝑓₁(𝑥)=𝜏(𝑥)+₀^𝑥/2𝐾(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)𝑑𝑡−𝑓₁(𝑥) (47) from (44) and using the inversion formula for such equations [10] 𝜈(𝑥)=𝐶^√0,𝜆_𝑂𝑋𝑓₁(𝑥)≡𝑓^′₁(𝑥)−𝜆^𝑥₀𝑓₁(𝑡)𝐼₁√𝜆(𝑥−𝑡)𝑑𝑡, (48) in view of (35) and (45), we obtain 𝜈(𝑥) with respect to 𝜏(𝑥) in the form 𝜈(𝑥)=𝜏^′(𝑥)−𝜆^𝑥₀𝜏(𝑡)𝐼₁√𝜆(+𝑥−𝑡)𝑑𝑡_{0𝑛𝑥/2𝑖=1}𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)×𝐾^′(𝑥,𝑡)−𝜆^𝑥_2𝑡𝐾(𝑠,𝑡)𝐼₁√𝜆(𝑥−𝑠)𝑑𝑠𝑑𝑡−𝑓^′₁(𝑥)+𝜆^𝑥₀𝑓₁(𝑡)𝐼₁√𝜆(𝑥−𝑡)𝑑𝑡. (49)
Due to the property of Problem C and in view of (43), (44), we obtain [9] 𝜏^′′(𝑥)−𝜆𝜏(𝑥)=𝑘+𝜈(𝑥) (50) from (37) in Ω₁, tending 𝑦→−0. Here 𝑘 is an unknown constant to be defined.
The equality (50) is a second functional relation between 𝜏(𝑥) and 𝜈(𝑥), transferred from the domain Ω₁ to 𝐴𝐵.
Existence of Solution to Problem C. Solving (50) with respect to 𝜏(𝑥) with the conditions 𝜏(0)=𝜑₁(0),𝜏^′(0)=𝜑₃(0), (51) we have 𝜏(𝑥)=1√𝜆^𝑥0sh√𝜆()𝜈(𝑡)−𝑘𝑥−𝑡𝑑𝑡𝜆1−ch√𝜆𝑥+𝜑₁(0)ch√1𝜆𝑥+√𝜆𝜑₃(0)sh√𝜆𝑥. (52)
Omitting the function 𝜈(𝑥), in (49) and (52), in view of the sewing condition, we obtain an integral equation with a shift with respect to 𝜏(𝑥): 𝜏(𝑥)−^𝑥₀𝐾₁()𝜏(𝑡)−𝑥,𝑡𝑑𝑡_0𝑥/2𝐾₂(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)𝑘𝑑𝑡=−𝜆1−ch√𝜆𝑥+Φ₁(𝑥), (53) where 𝐾₁(𝑥,𝑡)=ch√𝜆()−√𝑥−𝑡𝜆^𝑥_𝑡sh√𝜆(𝑥−𝑠)𝐼₁√𝜆(𝐾𝑠−𝑡)𝑑𝑠,₂()1𝑥,𝑡=−√𝜆^𝑥2𝑡sh√𝜆()×𝐾𝑥−𝑠_′(𝑠,𝑡)−𝜆^𝑥_2𝑡𝐾(𝑧,𝑡)𝐼₁√𝜆(Φ𝑠−𝑧)𝑑𝑧𝑑𝑠,₁(𝑥)=𝜑₁(0)ch√+1𝜆𝑥√𝜆𝜑₃(0)sh√𝜆𝑥−^𝑥₀𝐾₁(𝑥,𝑡)𝑓₁(𝑡)𝑑𝑡. (54)
Assuming that 𝛼(𝑥)=Φ₁(𝑥)−𝑘𝜆1−ch√+𝜆𝑥_0𝑥/2𝐾₂(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)𝑑𝑡, (55) we write (53) in the form 𝜏(𝑥)−^𝑥₀𝐾₁(𝑥,𝑡)𝜏(𝑡)𝑑𝑡=𝛼(𝑥),0≤𝑥≤1. (56)
Hence, (56) is an integral Volterra equation of the second kind, which is unconditionally and uniquely solvable in class 𝐶(0≤𝑥≤1). Thus, the solution of (56) has the from 𝜏(𝑥)=𝛼(𝑥)+^𝑥₀𝑅₁(𝑥,𝑡)𝛼(𝑡)𝑑𝑡, (57) where 𝑅₁(𝑥,𝑡) is the resolvent of the kernel 𝐾₁(𝑥,𝑡).
In view of (55) and the Dirichlet formula, (57) has 𝜏(𝑥)−₀^𝑥/2𝐾^∗₂(𝑥,𝑡)^𝑛_𝑖=1𝑎_𝑖(𝑡)𝐷^𝛼𝑖_0𝑡𝜏(𝑡)𝑑𝑡=Φ₂(𝑥), (58) where 𝐾^∗₂(𝑥,𝑡)=𝐾₂(𝑥,𝑡)+^𝑥_2𝑡𝐾₂(𝑠,𝑡)𝑅₁()Φ𝑥,𝑠𝑑𝑠,₂(𝑥)=Φ₁(𝑥)+^𝑥₀𝑅₁(𝑥,𝑡)Φ₁(𝑡)−𝑘𝑑𝑡𝜆1−ch√𝜆𝑥+^𝑥₀1−ch√𝜆𝑡𝑅₁(𝑥,𝑡)𝑑𝑡. (59)
Hence, we conclude that (58) always has a solution that is unique and can be represented in the form [14] 𝜏(𝑥)=Φ₂(𝑥)+₀^𝑥/2𝑅₂(𝑥,𝑡)Φ₂(𝑡)𝑑𝑡,0≤𝑥≤1, (60) where 𝑅₂(𝑥,𝑡) is the resolvent of the kernel 𝐾^∗₂(𝑥,𝑡).
Hence, by virtue of the condition 𝜏(1)=𝜑₂(0)−𝑤(1), 𝑘 are determined uniquely. Upon determining 𝜏(𝑥), we find the functions 𝜈(𝑥) and 𝑤(𝑥) from (49) and (20).
Thus, the solution of problem C in the domain Ω₂ in view of (20) and (23) is determined uniquely according to the formula (36), and in the domain Ω₁ we arrive to the problem for an nonloaded equation of the third order [9]. Thus, the solution of problem B in the domains Ω₁ and Ω₂ can be constructed from (36) in view of (20), (23), and Problem Γ₁₁ [9].
Thus, problem C is uniquely solvable. Theorem 3 is proved.