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

Authors: B. Islomov; U. I. Baltaeva
Journal: International Journal of Partial Differential Equations
 (2013)

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## 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.


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## Body

## 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(1)∂∂x(uxx-uyy-λu)-μ∑i=1n‍ai(x)D0xαiu(x,0)=0
and a boundary value problem for a loaded equation of a mixed parabolic hyperbolic type
(2)∂∂x(Lu)-μ∑i=1n‍ai(x)D0xαiu(x,0)=0,
where
(3)Lu≡uxx-1-sgny2uyy-1+sgny2uy-λu,D0xαif(x)={1Γ(-αi)∫0xf(t)dt(x-t)1+αi,αi<0,ddxD0xαi-1f(x),0<αi<1.D0ξ-αiD0ξαif≡D0ξ0f≡f(ξ). Assume that αn<αn-1<⋯<α1=α<1 and coefficients ai=ai(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

LetD⊂R2 be a domain bounded at y<0 by the characteristics
(4)AC:x+y=0,BC:x-y=1
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 ux, uy, up to AB∪AC, and satisfies the boundary value conditions
(5)uy(x,y)|AB=ν(x),0≤x<1,u(x,y)|AC=ψ1(x),∂u(x,y)∂n|AC=ψ2(x),0≤x≤12,
where n is an inner normal and ν(x), ψ1(x), ψ2(x) are real-valued functions.Theorem 1.
If2ν(x)=2ψ2(0)-ψ1′(0) and
(6)ν(x)∈C[0,1]∩C2(0,1),ψ1(x)∈C1[0,12]∩C3(0,12),ψ2(x)∈C[0,12]∩C2(0,12),
then there exists a unique solution to the problem A in the domain D.Proof of Theorem1.
An important role in proving Theorem1 is played by the following.
Lemma  2.Any regular solution to (1) is represented in the form(7)u(x,y)=z(x,y)+w(x),wherez(x,y)is  the solution of the equation(8)∂∂x(zxx-zyy-λz)=0andw(x)—is the solution of the following ordinary differential equation:(9)w′′′(x)-λw′(x)-μ∑i=1n‍aiD0xαiw(x)=μ∑i=1n‍aiD0xαiz(x,0).
Proof of Lemma  2. Let u(x,y), represented by Formula (7), be the solution of (1). Then, substituting
(10)∂∂x(uxx-uyy-λu)-μ∑i=1n‍aiD0xαiu(x,0)=∂∂x(zxx-zyy-λz)+w′′′(x)-λw′(x)-μ∑i=1n‍aiD0xαiw(x)-μ∑i=1n‍aiD0xαiz(x,0)=0
satisfies (1).
Then, vice versa, letu(x,y) be a regular solution to (1) and let w(x) be a certain solution equation
(11)w′′′(x)-λw′(x)=μ∑i=1n‍aiD0xαiu(x,0).
Let us prove the validity of relation (7). Manifestly, the function
(12)u(x,y)=z(x,y)+μλ∫0x(chλ(x-t)-1)u(t,0)dt
is a solution to (1), where z(x,y) is a solution to (8) and the function
(13)ω(x)=μλ∫0x(chλ(x-t)-1)∑i=1n‍ai(x)D0xαiu(t,0)dt
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 thatu(x,0)=z(x,0)+w(x). Then, (11) provides
(14)w′′′(x)-λw′(x)-μ∑i=1n‍ai(x)D0xαiw(x)-μ∑i=1n‍ai(x)D0xαiz(x,0)=0,
and the function z(x,y)=u(x,y)-w(x) satisfies (8).
Lemma 2 is proved.
Invoking that the functionaeλx+be-λx+c satisfies (8), we can assume without loss of generality that
(15)w(0)=w′(0)=w′′(0)=0,
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
(16)ω(x)=1λ∫0x(chλ(x-t)-1)f~(t)dt,
where
(17)f~(x)=μ∑i=1n‍ai(x)D0xαiw(x)+μ∑i=1n‍ai(x)D0xαiz(x,0).
The last equality with respect to designation and after some transformation becomes(18)ω(x)-μ∫0xK(x,t)ω(t)dt=F(x),0≤x≤1,
where
(19)K(x,t)={1Γ(-αi)∫0xAi(t)(t-s)-1-αi×(chλ(x-t)-1)dt,αi<0,1λΓ(1-αi)∫sx∂Ai(t)(t-s)-αi×(chλ(x-t)-1)∂tdt,0<αi<1,F(x)=μλ∫0x(chλ(x-t)-1)∑i=1n‍ai(t)D0xαiz(t,0)dt.
And with recurring indexi=1,2,…,n implied summation. Solving the next equation with respect to [7] and Dirichlet’s formula we have
(20)ω(x)=∫0xP(x,t)∑i=1n‍ai(t)D0xαiz(t,0)dt,
where
(21)P(x,t)=μλ{μ∫tx(chλ(s-t)-1)R(x,s;μ)dschλ(x-t)-1+μ∫tx(chλ(s-t)-1)R(x,s;μ)ds}.
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
(22)zy(x,y)|AB=ν(x),0<x<1,z(x,y)|AC=ψ1(x)-w(x),∂z(x,y)∂n|AC=ψ2(x)-12w′(x),0≤x≤12,
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]:
(23)z(x,y)=∫0x+yν(t)I0[λ(x+y-t)(x-y-t)]dt-ψ1*(0)I0[λ(x2-y2)]+ψ1*(x+y2)+ψ1*(x-y2)+1λ∫0x+y(λψ1*(t)-2ψ2*′(t))×sinλ(t-x+y2)dt+1λ∫0(x-y)/2(λψ1*(t)-2ψ2*′(t))×sinλ(t-x-y2)dt-2∫0(x-y)/2ψ1*(t)×Bt(0,2t;x+y,x-y)dt+1λ∫0y(λψ1*(-t)-2ψ2*′(-t))×sinλ(y-t)dt-2λ∫0(x-y)/2Bt(0,2t;x+y,x-y)dt×∫0t(λψ1*(z)-2ψ2*′(z))sinλ(-t+z)dz,
where
(24)ψ1*(x)=ψ1(x)-w(x),ψ2*(x)=ψ2(x)-12w′(x),B(t,z;x+y,x-y) is the Riemann-Hadamard function [12] and, I0[z] is the modified Bessel function [13].
Assuming thaty=0 in (23) and invoking (20), we obtain the following functional correlation, transferred from the domain D onto AB:
(25)τ(x)+∫0x/2K(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt=Φ(x),0≤x≤1,
where
(26)τ(x)=z(x,0),K(x,t)=2P(x2,t)+λx∫2txP(s2,t)I-1[λx(x-s)]ds+1λ∫2txK*(x2,t)×(λP(s2,t)-14P′(s2,t))ds,(27)Φ(x)=2ψ1(x2)-ψ1(0)I0[λx]+∫0xν(t)I0[λ(x-t)]dt+λx∫0xI-1[λx(x-t)]ψ1(t2)dt+1λ∫0xK*(x2,t2)×(λψ1(t2)-22ψ2′(t2))dt,K*(x,t)=sinλ(t-x)+∫txλxI-1[λx(x-2s)]sinλ(t-s)ds,I-1(x)=I1(x)/x, I0(x), are I1(x) are the modified Bessel functions [13].
We can assume without loss of generality thatαi>0 when i=1,2,…,m and αi<0 when i=m+1,…,n.(1)
Leti≥m+1, and then for any function τ(x)∈C[0,1] applying the Dirichlet permutation integration formula from (25) we have
(28)τ(x)+μΓ(-αi)∫0x/2τ(t)dt∫tx/2AiK(x,ξ)dξ(ξ-t)1+αi=Φ(x).(2)
Leti=1,2,…,m, and then taking account of conversion [3] into (25) we get
(29)τ(x)-μΓ(1-αi)∫0x/2τ(t)dt∫tx/2∂AiK(x,ξ)/∂ξ(ξ-t)αi=Φ(x).
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. Theorem1 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
(30)AC:x+y=0,BC:x-y=1
of (2) for y<0.Let us introduce the following notation:(31)I={(x,y):0<x<1,y=0},Ω=Ω1∪Ω2∪I.Let us term the functionu(x,y)∈C(Ω-)∩C1(Ω)∩C3,1(Ω1)∩C3,2(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)
ux(uy) 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(32)uy(x,-0)=uy(x,+0),(x,0)∈I
are satisfied on AB;(5)
u(x,y) satisfies the boundary value conditions
(33)u(x,y)|AA0=φ1(y),u(x,y)|BB0=φ2(y),ux(x,y)|AA0=φ3(y),0≤y≤h,u(x,y)|AC=ψ1(x),∂u(x,y)∂n|AC=ψ2(x),0≤x≤12,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 αi≡0 was proven by Islomov and Baltaeva [11].Theorem 3.
Ifλ>0 and
(34)φ1(0)=ψ1(0),φj(y)∈C1[0,1],(j=1,2),φ3(y)∈C[0,1]∩C1(0,1),(35)ψ1(x)∈C1[0,12]∩C3(0,12),ψ2(x)∈C[0,12]∩C2(0,12),
then there exists a unique solution to the problem C in the domain Ω.Proof of Theorem3.
The following theorem holds.
Lemma  4.Any regular solution of (2) (wheny≠0) is represented in the form(36)u(x,y)=z(x,y)+w(x),wherez(x,y)is a solution to the equation(37)0=∂∂x{zxx-zy-λz,z>0,zxx-zyy-λz,z<0.w(x)is a solution of the following ordinary differential equation:(38)w′′′(x)-λw′(x)-μ∑i=1n‍aiD0xαiw(x)=μ∑i=1n‍aiD0xαiz(x,0).
The lemma is proved similarly to Lemma  2.
Invoking that the functionaeλx+be-λx+c satisfies (37), we can subordinate the function w(x) to the conditions
(39)w(0)=w′(0)=w′′(0)=0.
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):
(40)z(x,y)|AA0=φ1(y),z(x,y)|BB0=φ2(y)-w(1),∂z(x,y)∂x|AA0=φ3(y),0≤y≤h,(41)z(x,y)|AC=ψ1(x)-w(x),0≤x≤12,(42)∂z(x,y)∂n|AC=ψ2(x)-12w′(x),0≤x≤12.
Derivation of Basic Functional Relations.As it is known from problem A, the solution to (37) with the boundary value conditions (41), (42), and
(43)∂z(x,y)∂y|y=0=ν(x),0<x<1,
is given by the formula (23).
Assuming thaty=0 in (23), in view of (39), and
(44)z(x,0)=τ(x),0≤x≤1,
we obtain the functional relation, transferred from the domain Ω2 onto AB:
(45)τ(x)+∫0x/2K(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt-∫0xI0[λ(x-t)]ν(t)dt=f1(x),
where
(46)f1(x)=2ψ1(x2)-ψ1(0)J0[λx]+λx∫0xI-1[λx(x-t)]ψ1(t2)dt+1λ∫0xK*(x2,t2)×(λψ1(t2)-2ψ2′(t2))dt,
where K(x,t) can be represented in the form (26).
Denoting(47)f~1(x)=τ(x)+∫0x/2K(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt-f1(x)
from (44) and using the inversion formula for such equations [10]
(48)ν(x)=COX0,λ[f~1(x)]≡f~1′(x)-λ∫0xf~1(t)I-1[λ(x-t)]dt,
in view of (35) and (45), we obtain ν(x) with respect to τ(x) in the form
(49)ν(x)=τ′(x)-λ∫0xτ(t)I-1[λ(x-t)]dt+∫0x/2∑i=1n‍ai(t)D0tαiτ(t)×(K′(x,t)-λ∫2txK(s,t)I-1[λ(x-s)]ds)dt-f1′(x)+λ∫0xf1(t)I-1[λ(x-t)]dt.
Due to the property of Problem C and in view of (43), (44), we obtain [9]
(50)τ′′(x)-λτ(x)=k+ν(x)
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
(51)τ(0)=φ1(0),τ′(0)=φ3(0),
we have
(52)τ(x)=1λ∫0xshλ(x-t)ν(t)dt-kλ(1-chλx)+φ1(0)chλx+1λφ3(0)shλx.
Omitting the functionν(x), in (49) and (52), in view of the sewing condition, we obtain an integral equation with a shift with respect to τ(x):
(53)τ(x)-∫0xK1(x,t)τ(t)dt-∫0x/2K2(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt=-kλ(1-chλx)+Φ1(x),
where
(54)K1(x,t)=chλ(x-t)-λ∫txshλ(x-s)I-1[λ(s-t)]ds,K2(x,t)=-1λ∫2txshλ(x-s)×(λ∫2txK(z,t)I-1[λ(s-z)]dzK′(s,t)-λ∫2txK(z,t)I-1[λ(s-z)]dz)ds,Φ1(x)=φ1(0)chλx+1λφ3(0)shλx-∫0xK1(x,t)f1(t)dt.
Assuming that(55)α(x)=Φ1(x)-kλ(1-chλx)+∫0x/2K2(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt,
we write (53) in the form
(56)τ(x)-∫0xK1(x,t)τ(t)dt=α(x),0≤x≤1.
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
(57)τ(x)=α(x)+∫0xR1(x,t)α(t)dt,
where R1(x,t) is the resolvent of the kernel K1(x,t).
In view of (55) and the Dirichlet formula, (57) has
(58)τ(x)-∫0x/2K2*(x,t)∑i=1n‍ai(t)D0tαiτ(t)dt=Φ2(x),
where
(59)K2*(x,t)=K2(x,t)+∫2txK2(s,t)R1(x,s)ds,Φ2(x)=Φ1(x)+∫0xR1(x,t)Φ1(t)dt-kλ[1-chλx+∫0x(1-chλt)R1(x,t)dt].
Hence, we conclude that (58) always has a solution that is unique and can be represented in the form [14]
(60)τ(x)=Φ2(x)+∫0x/2R2(x,t)Φ2(t)dt,0≤x≤1,
where R2(x,t) is the resolvent of the kernel K2*(x,t).
Hence, by virtue of the conditionτ(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 domainΩ2 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. Theorem3 is proved.



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