key: cord-0064229-jc8m56j2
authors: Yuldashev, T. K.; Abdullaev, O. Kh.
title: Unique Solvability of a Boundary Value Problem for a Loaded Fractional Parabolic-Hyperbolic Equation with Nonlinear Terms
date: 2021-06-09
journal: Lobachevskii J Math
DOI: 10.1134/s1995080221050218
sha: 55098c5c2a0fa9d9f67443ff9eee560c5cafe3ea
doc_id: 64229
cord_uid: jc8m56j2

This work is devoted to study the existence and uniqueness of solution of an analogue of the Gellerstedt problem with nonlocal assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov–Caputo operator of differentiation. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of successive approximations of factorial law for Volterra type nonlinear integral equations.

One of the most important areas of mathematical analysis is the theory of fractional order integrodifferential operators. Today, the theory and applications of operators of fractional differentiation and integration have become a powerful direction of theoretical and applied research in different sciences and technologies. In particular, the operators of fractional differentiation and integration are used in the study of problems associated with the study of the coronavirus COVID-19 (see, for example [1] [2] [3] ). In [4] , the mathematical problems of an Ebola epidemic model by fractional order equations are considered. In [5] , fractional models of the dynamics of tuberculosis infection are considered.

On May 29, 1947 A. N. Gerasimov made report at the Institute of Mechanics of the USSR Academy of Sciences (see [6] ). There, he has introduced a concept of fractional derivative, which today we know as a Gerasimov-Caputo fractional derivative. This Gerasimov's report was published in [7] . Fractional order differentiation operators of Riemann-Liouville and Gerasimov-Caputo describe diffusion processes [3, vol. 1, 47-85] . A physical and engineering interpretations of the generalized fractional operators are given in [3, vol. 4-8] , [8] [9] [10] [11] [12] [13] . In [9] , in particular, by the aid of operational calculus of Mikusinski type were studied the problems of existence and representation of solution of initial value problem for the general ordinary linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients.

In [14, 15] , the fractional integro-differentiation operators are applied in studying the dielectric relaxation in glass-forming liquids with different chemical compositions. For this a classical Debyetype model was used. This model describes exponential relaxation and it was determined by a first-order differential equation. The ubiquitous feature of the dynamics of supercooled liquids and amorphous polymers is just non-exponential relaxation, which is the result of slow relaxation. To successfully describe the relaxation dynamics of glassy materials, the author in [15] proposed a new model of dielectric relaxation containing derivatives and integrals of the noninteger order, which are a natural generalization of the Debye equation.

The theory of boundary value problems for equations of mixed type of fractional order is also one of the intensively developing directions in the general theory of partial differential equations. It should be noted that local and nonlocal boundary value problems for equations of parabolic-hyperbolic and elliptichyperbolic types, including various integro-differential operators of fractional order, have been studied by many authors (see works [16] [17] [18] [19] [20] [21] ).

There are works [22, 23] , in which were investigated local and nonlocal boundary value problems for parabolic-hyperbolic equations with Gerasimov-Caputo operator without a loaded part. Similar problems were considered in [24] [25] [26] for loaded equations of parabolic type, whose solutions trace was included in various fractional integro-differential operators of Riemann-Liouville, Erdaley-Kober and others types.

When modelling the problems of optimal control of the agroeconomic system for regulating groundwater marks and soil moisture, it became necessary to study boundary value problems for loaded partial differential equations [27, 28] . A large number of publications, in particular, [29] [30] [31] [32] [33] , have been devoted to study of boundary value problems of various kinds for loaded equations of parabolic, parabolichyperbolic and elliptic-hyperbolic and other types.

In the present paper we study the existence and uniqueness of solution of the Gellerstedt type problem with nonlocal boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov-Caputo operator of differentiation.

We consider a loaded fractional order parabolic-hyperbolic equation with nonlinear terms:

where

a i (x, y), f i (x, y; u(x, 0)) are given functions, p i , α are constants and p i > 0, 0 < α < 1, i = 1, 2.

The main goal of this work is to study the unique solvability of a boundary value problem with an integral gluing condition for the equation (1) . Let Ω be region, bounded by intervals:

for y > 0, and by characteristics:

We introduce the designations (see Fig. 1 ):

Problem BV BV BV . It is required to find a solution u(x, y) of the equation (1) from the class of functions:

satisfying boundary value conditions:

and integral gluing condition:

where ψ(x), ϕ i (y)(i = 1, 2) and λ k (x)(k = 1, 5) are given continuous functions, φ 2 (0) = ψ (0) and 4 k=1 λ 2 k (x) = 0.

As we know, the same above problems for the equation (1) for the integer order α = 1 have not been investigated, too. On the another hand, we would like to note, that fundamental solution of equation (1) for y > 0, α = 1, completely coincides with the fundamental solution of the heat equation u xx − u t = 0. Therefore, all results in this work remain valid in case of integer order α = 1, too.

We note that solution of the Cauchy-Goursat problem with condition (5) and u(x, 0) = τ (x) for equation (1) in Ω 2 has the form

Further, from the equation (1) as y → +0 taking into account (2), (6) and

Hence, taking into account (see the equation (7)) the following relation

By virtue of properties of functions from the class (3) for solution of the problem BV , we have

By integration the equation (8) with initial value conditions (9) we derive

where

Theorem 1. We suppose that p i = const > 1 and the following conditions are fulfilled:

where L i = const > 0(i = 1, 2). Then the problem has a unique solution.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 42 No. 5 2021

Proof. By virtue of properties (13) and (14), taking designations (11) and (12) into account we have estimates:

where M 1 , g 10 , f i0 = const > 0 (i = 1, 2). The equations (10) and (7) we consider as a system of nonlinear second kind integral equations of Volterra type with respect to unknown functions τ (x) and u(x, y) for y ≤ 0:

We define a sequence of functions τ n (x) and u n (x, y)(n = 0, 1, . . .) from the following system of recurrent equations:

By virtue of properties (13) and estimates (16), we have following estimates:

where m i , λ 10 , c i1 = const > 0(i = 1, 2). Further, taking these estimates into account, from the iteration process (18) we derive

where β = max Γ(α)M 1 c 11 , m 1 c p 1 11 + f 10 ,

To continue the proof of the Theorem 1 we need on following lemma.

are true, then there holds

where β * = const > 0, k is a fixed natural number.

Proof. By virtue of Lipschitz condition (15) and estimate (20) , we have

So, the Lemma is proved. P Further, we take into account that for a functions F (τ ) = τ p (x) and G(u) = u p (x, y) in the case p > 1 Lipschitz conditions hold:

Therefore, taking estimates (19) into account, from successive approximations (18) we obtain

where c 12 , c 22 = const > 0. We put L 2 λ 10 Γ(α) = max{m 2 c 22 ; m 1 c 12 + L 1 } and β = Γ(α)λ 10 (f 20 + m 2 c p 2 21 ). Then assuming that M 1 < L 2 λ 10 (you can always get it by imposing a condition on the function λ 1 (x)) from last estimate, we have

Taking the estimate (21) into account, from successive approximations (18) yields

Taking also the estimate (22) and last inequality into account, from iteration process (18) we obtain

Continuing the above reasoning for arbitrary n, we have

By virtue of the obtained estimates, we conclude that the functional sequences of functions {τ n (x)} ∞ n=1 and {u n (x, y)} ∞ n=1 has a unique limit functions τ (x) and u(x, y): Now we prove the uniqueness of the solution of the system (17) . We suppose that this system (17) have two solutions u 1 (x, y); τ 1 (x) and u 2 (x, y); τ 2 (x) . Then introducing the following designations u(x, y) = u 1 (x, y) − u 2 (x, y) and τ (x) = τ 1 (x) − τ 2 (x), from this system (17) 

Taking into account the class of given functions of the system (23), we have

We assume that 0 < max m 2 c 22 4 , L 2

= δ is small number. Then from the last inequalities we obtain estimates

where

Strengthening the first estimate of (24), taking into account the second one of (24), we have

where

Introducing the designation |v(x, y)| = |u(x, y)| + |τ (x)|, from (25) we obtain the following estimate

We take into account that I 3 (v) ≤ I 2 (v) and

From the last estimate we have

where δ 0 = max{δ, 4δλ 10 Γ(α)}. Taking into account that I(v) is a linear operator with respect to function v(x, y), from the estimate (26) we obtain

For the estimation I 2 (v) we have

It is easy to check that

where σ n = 2σ n−1 + n and σ 0 = 0.

As n → ∞ from (27) implies that |v(x, y)| = 0. Consequently, from the designation |v(x, y)| = |u(x, y)| + |τ (x)| = 0, we come to identities |u(x, y)| ≡ 0 and |τ (x)| ≡ 0. Hence, u 1 (x, y) = u 2 (x, y) and τ 1 (x) = τ 2 (x). Consequently, the solution of the system (17) is unique.

After determination τ (x) we restore the unique solution of the considering problem BV in the domain Ω 2 as a solution of the Cauchy-Goursat problem (see the equation (7)). Further, we take the existence of function τ (x) into account and use the solution of second boundary value problem for the equation (1) in domain Ω 1 [34] :

u(x, y) = y 0 G ξ (x, y, 0, η)ϕ 2 (η)dη − y 0 G ξ (x, y, 1, η)ϕ 1 (η)dη + is Wright type function.

Notice that the Wright type function e 1,δ 1,δ (z) is partial case of the generalized (Fox-)

⎦ , when p = 0, q = 2, b 1 = 0, β 1 = 1 (see [34] ). G(x, y, ξ, η)f 1 (ξ, η; τ (ξ))dξdη.

The Theorem 1 is proved. P

Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model

Forecast analysis of the epidemic trend of COVID-19 in the United States by a generalized fractional-order SEIR model

Handbook of Fractional Calculus with Applications

On a fractional order Ebola epidemic model

A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative

Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union

Generalization of laws of the linear deformation and their application to problems of the internal friction

Fractional calculus and its applications in physics

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

Applications of variable-order fractional operators: A review

Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids

Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives

A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications

Experimental evidence for fractional time evolution in glass forming materials

On fractional relaxation

Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative

An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative

Fundamental solution of the fractional-order diffusion-wave equation

Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator

Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator

Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator

On a nonlocal problem with gluing condition of integral form for parabolic-hyperbolic equation with Caputo operator

Analog of the Gellerstedt problem for the mixed type equation with integral-differential operators of fractional order

About a problem for the degenerate mixed type equation involving Caputo and Erdelyi-Kober operators fractional order

A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative

Fractional Calculus and Their Applications

Loaded Equations and Their Applications

Non-local problem for the loaded mixed type equations with integral operator

The Dirichlet problem for the loaded Lavrent'ev-Bitsadze equation

On a problem of heat equation with fractional load

Initial-boundary problem for parabolic-hyperbolic equation with loaded summands

Boundary-value problems for loaded third-order parabolichyperbolic equations in infinite three-dimensional domains

Fractional Order Partial Equations

A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities

The authors are grateful to the reviewer for helpful comments and advice.