MIXED TYPE INTEGRO-DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER CAPUTO OPERATORS AND SPECTRAL PARAMETERS Текст научной статьи по специальности «Математика»

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta

2021. Volume 57. Pp. 190-205

MSC2020: 35M12, 35J25, 35L20, 30E20, 45E05 © T.K. Yuldashev, E. T. Karimov

MIXED TYPE INTEGRO-DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER CAPUTO OPERATORS AND SPECTRAL PARAMETERS

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

Keywords: integro-differential equation, mixed type equation, small parameter, spectral parameters, fractional Caputo operators, unique solvability.

DOI: 10.35634/2226-3594-2021-57-10 Introduction

Fractional calculus plays an important role for the mathematical modeling in many scientific and engineering disciplines (see more detailed information in [1]). In [2], some basic problems in continuum and statistical mechanics are considered. In [3], the mathematical problems of Ebola epidemic model are studied. In [4,5], the fractional model for the dynamics of tuberculosis infection and novel coronavirus (nCov-2019), respectively, is studied. The construction of various models of theoretical physics problems by the aid of fractional calculus is described in [6, vol. 4, 5], [7, 8]. A specific physical interpretation of the Hilfer and Caputo fractional derivatives, describing the random motion of a particle moving on a real line at Poisson paced times with finite velocity is given in [9]. A detailed review on the application of fractional calculus in solving applied problems is given in [6, vol. 6-8], [10]. More detailed information as well as a bibliography related to the theory of fractional integro-differentiation, including the Hilfer and Caputo fractional derivatives can also be found in [11].

Applications for the equations of mixed type were studied in [12-14]. In particular, in the work [12] I. M. Gel'fand considered an example of gas motion in a channel surrounded by a porous medium and at that, gas motion in the channel was described by a wave equation, while outside the channel a diffusion equation was posed. Ya. S. Uflyand in [13] considered a problem on propagation of electric oscillations in compound lines, when the losses on a semiinfinite line were neglected and the rest of the line was treated as a cable with no leaks. He reduced this problem to a mixed parabolic-hyperbolic type equation. In [14], a hyperbolic-parabolic system arising in pulse combustion is investigated. Mixed type fractional differential equations are studied in many works of scientists, in particular in [15-24].

One of the important sections of the theory of integral and differential equations is the theory of integro-differential equations. The presence of an integral term in differential equations of the first and second orders has an important role in the theory of dynamical systems with automatic control [25,26]. Mixed type integer order integro-differential equations with degenerate kernels and spectral parameters are studied in [27,28].

In the present paper we study the issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters in multidimensional rectangular domain. We note that boundary value problems for integro-differential equations with spectral parameters have singularities in studying the questions of one value solvability [29,30].

§ 1. Statement of the problem

In multidimensional domain Q = {—T < t < T, 0 < x1;... , xm < 1} a mixed integro-differential equation of the following form is considered

(u f K1(t,s)U(s,x) ds, t> 0,

(1.1)

u J K2(t,s)U(s,x) ds, t< 0,

where

A£(U )

+

1 + sgn(t)

c - e

Ô2

i=1

C D0t

1 - sgn(t) 2

d 2

i=1

U (t,x) + U (t,x),

m

2

m

< m

B (U) = ^

E

i=1 m

i=1

d2 U dxidxi '

d 2U dxidxi '

t > 0, t < 0,

T and 1 are given positive real numbers, u is positive spectral parameter, e is positive small parameter, x e Q^, u is real non-zero spectral parameter, 0 = Kj(t,s) = a(t)bj(s), aj(t) e C2[-T; T], bj(s) e C[-T; T], Q^ = [0; /]m, 0 < ft < a ^ 1, 1 < ft < «2 ^ 2. Problem. Find in the domain Q an unknown function

U (t,x) g C(O) n Cai'2(O+) n Ca2'2(O-) n ci,;X1+2(n+) n Cax+2(O-) n

n C«i+2+0+...+0(O ) n C«2+2+0+...+0(Q ) n Cai +0+2+0+...+0(Q ) n (12)

(O+) n ,Xi,X2,...,Xm (O-) n Ci,Xi,X2,X3,...,Xm (O+) n (1-2)

n C«2+0+2+0+...+0(O ) n n Cai+0+...+0+2 (O ) n Ca2+0+...+0+2 (O )

n Ci,Xi,X2,X3 ,...,xm (O-) n • • • n Ct,xi,...,xm_i,xm (O + ) n Ci,xi,...,xm_i,xm (O-),

satisfying the mixed integro-differential equation (1.1) and the following boundary conditions

U(—T,x) = pi(x), cDq4U(—T,x) = <^(x), x G Om, (1.3)

U(t, 0,X2,X3, . . . = U(t,/,X2,X3, . . . , Xm) =

= U(t, xi, 0, x3,..., x) = U(t, x1, l, x3,..., xm) = ... = = U(t, x1;..., xm-1, 0) = U(t, x1;..., xm-1, l) =

Uxixi (tj 0 j x2 j x3 j ... j xm) Uxixi (tj lj x2 j x3 j . . . j xm)

= Uxx xi (t,x1, 0,x3, . . . ,xm) = Ur1xi (t,x1jljx3j... ,xm) = ... = (1.4)

Uxixi (tj x1j . . . j xm-1j 0) Uxixi (tj x1j . . . j xm-1j l) ... = Uxmxm (tj 0j x2j x3j . . . j xm) = Uxmxm (tj lj x2j x3j . . . j xm) = Uxmxm (tj x1j 0 j x3j . . . j xm) Uxmxm (tj x1j lj x3j . . . j xm) ...

= Uxmxm (t,x1, . . . ,xm-1, 0) = Uxmxm (t,x1, . . . , xm-1, l) = 0, -T < t < T,

where 0 < 0 < 1, ^¿(x) are given smooth functions, <£¿(0) = ^j(l) = 0, i = 1,2, Cr (Q) is a class of functions U(t, x1,..., xm) with continuous derivatives drr, fx!,..., fp^ in Q,

^'.^(n) is a class of functions U(t,x1,... ,xm) with continuous derivatives ITr, ,..., dp^-

in Q, C[+ir+0+l.;'xmo(Q) is a class of functions U(t,x1 ,...,xm) with continuous derivative

d2rU Q /^r+0+iii+0+r

dtr dx

in Q, ..., C['+i'+illx"m_+rxm (Q) is a class of functions U (t, x1,..., xm) with continuous derivative df,2^ in Q, r, s are positive real numbers, Q = {—T ^ t ^ T, x G Qmj,

Q_ = {-T < t < 0,m0 < x1,.. .,xm < lj, Q + = {0 < t < T, 0 < x1,.. .,xm < lj.

§ 2. Expansion of the solution of the problem (1.1)—(1.4) into Fourier series

The solution of the mixed integro-differential equation (1.1) in domain Q is sought in the form of a Fourier series

U(t,x)= £ <'|||'„m(t) Viii'nm (x), (2.1)

ni.....nm = 1

where

U+i'iii'nm (t) = U(tj x) ^ni^nm (x) dxj t > 0j

,± f + \ — )

U+i'iii'nm (t) = U(tj x) tfni'iii'nm (x) dxj t < 0j

u±i'iii'nm (t)H T (2.2)

U (t, x) ^rai'iii'„m (x) dx = ... U (t, x) $ ni.....„„ (x) dx 1 ■ ... ■ dx„

jo JO

m

Q i \ I I2 \ • • ^^m 0

»ni^nm (x) = ( \/yl sin—x 1 ■ ... ■ Sin l xm, n1,...,Km =1, 2,.

Substituting series (2.1) into equation (1.1), we obtain two fractional countable systems of ordinary integro-differential equations

C U+i 'm'nm (t) + £^ni'iii'nm C D0t U+i'iii'nm (t) + ^ni 'iii'nm .....nm (t) =

fT

v/ a1(t)b1(s)u+i' i'„m (s) dSj t> 0j Jo

(2.3)

CD0iu_i'iii'nm (t) + £^ni'iii'ramCD0tu_i.....nm (t) + .....nm^ u-_x,...,nm (t)

v J «2(t)b2(s)u_i'iii,nm (s) ds, t< 0,

(2.4)

where ^ni'iii,nm = fv+...+nm.

nm

nm

By the aid of notations

Tr+,..,nm =/ b1(s)u+i,...,nm (s) dS (2.5)

Tni,...,nm = J b2(sKi,...,„m (s) ds (2.6)

we present the countable systems of equations (2.3) and (2.4) as follows

CD0i u+i,...,nm (t) + e^ni,...,nmCD0tu+i,...,nm (t) + ^ni.....nm u+i,...,nm (t)

= ua1(t)<,..,nm > t> 0

CDa2uni,...,„m (t) + e,ni,...,nmCD0tUni,...,nm (t) + ^ni,...,nm W Uni,...,nm (t)

= 2W^,...,«™ > t<0.

(2.7)

(2.8)

The solutions of the countable systems of differential equations (2.7) and (2.8), satisfying conditions

d

Urai,...,nm (0) = C+li,...,ram , uni.....nm (0) = C1ni.....nm , dtuni,...,nm (0) = C2ni,...,;

have the form:

U+i ,...,nm (t) = VTni,...,nm X

ft

X I fl1(t - s)sai-1E(ai-^i;ai),ai (-£^ni)...,nm S^Ai, -^i,...,«™ S^) ds + (2.9)

0

.2

I C+ E \ tai — Ai — ,,2 ta^ t > 0

+ C1ni,...,nmE(ai—Ai,ai),1 ^ c,ni,...,nm b , ,ni,...,nm b j , b >

u—i,...,nm (t) VTni,...,nm X

^(a2— A2,a2),a2 '

-"ni,...,nm W " 'ni,...,n„

,0

X j «2(S - t)(-s)a2 —1E(a2—A ■ 2 (-S)a2—A2 -u2 w2(

~ni,...,nm*. ") 1 f^ni.....n^

■ H,«,..,«™ (-s)a2—A2, - A,...,nmw2(-s)a^ ds +

+ C2ni,...,nmE(a2—A2,a2),1 (-e,ni,...,nm (-1) 2 , -,ni,...,nm w (-1) 2) -

(2.10)

- C2ni,...,nm tE(a2—A2,a2),2 {-£,ni,...,nm (-t)a2 A2 , -,ni,...,nm w 2(-t)a2) , t < 0.

> HJni,...,nm

where C^ nm, Ci~i)... nm, (i = 1, 2) are for us unknown constants to be uniquely determined and

E(a,A),Y(z1 ,Z2)= V w , 1 2 —T, G C, Re (a) > 0, Re (ft > 0

^ n r(Y + am1 + pm2)

mi,m2=0

is Mittag-Leffler function of two variables.

From the nature of the statement of the problem (properties in (1.2)) it follows that the continuous conjugation condition is fulfilled: U (0 + 0, x) = U(0 - 0, x). So, taking the formula (2.2) into account, we have

<,...,nm (0 + 0)= / U(0 + 0,x)tfni,...,nm (x) dx =

, ^ (2.11) = / U(0 - 0,x)^ni)...)nm (x) dx = «ni,...«™ (0 - 0).

/nr

0

0

m

Analogously, taking (2.2) into account from the conditions (1.3) we obtain

uni,...,nm ( —T) = / U( —T (x) dx =

JqT

= (x) dx = ^lni,...,nm ,

Jap

CD0iUrai,...,ram ( —T) = / CD0tU( —T, x)^ni,...,nm (x) dx

(2.12)

(2.13)

^2(x)^ni,...,nm (x) dx = ^2ni,...;i

/np

where

^tni>...,nm = ^i(x) ^ni,...,nm (x) dx, i = 1, 2.

By the aid of continuous conjugation condition (2.11) from (2.9) and (2.10) we have that

Cini„..,nm = Cfni,...,nm. To find unknown coefficients Cr„i,...,„m and C2ni,...,nm in (2.10) we use the conditions (2.12) and (2.13) and come to the system of linear algebraic equations:

C1rai;...,ramE(«2- /2,<22),! ( —£^ni,...,ramT ^ , — ^ni,...,nmw T ) +

/1ni,...,nm ^ («2—P2,a2),1 V r^ni,...,nm± 1 P'ni,...,n„

2-^2 — » 2 w 2r

C — t—0

C1ni ,...,nm T

+ C2ni,...,nmTE(«2-^2,«2),2 ( e»ni;...,nmT 2 , »ni,...,nmw T 2) =

^1ni,...,nm ,

1

r(1- 0)J

+ C2ni,... ,nm T E(a2-/2, «2),2-0 (—e»ni;...,nm T 2 , — »ni,...,nm w T 2) =

^2ni,...,nm ,

+

where

^ini,...,nm = ^ini,...,nm vrni;...,nm ^ini,...,nm , i = 1, 2,

r0

^1n i,...,nm = J a2(s + T )( — s)a2-1 X

X E(a2- /2,a2),a2! — e»ni;...,nm ( — s) 2 ^ »ni,...,nm w ( — s) 2) ds, ^2ni,...,nm = a2(0) Ta2-2E(a2-^2,a2),a2-1 (—^i,...^™T<*2-/2 »ni,..,nm +

T E(a2-^2,a2),a2-1 (— e»ni;...,nmT ^ , — »ni,...,nmw T ) +

+ da 2(C — s)

ds

0

+ J ( — 0a2-1E(a2-A,a2),a2 ( — ^...n™ (—^2-/* ,— <,..,nm— ^^ d£ X

/-p 1 d2a2(Ç — s) d XX (s — C)a 2-1 ds2

If we assume that

X

E(a2-/2,a2), 1-0 ( e»ni;...,nmT ^ , »ni,...,nmw T )

r(1 — 0)

= 0,

(2.14)

(2.15)

(2.16)

*ni ,...,n™ (w) = T1-0 E(a2-/?2,a2),1 (—e»ni,...n™ T^/2 ,— »h ...^T^) X

X E(a2 /2,a2),2-0 (—e»ni,...,n™T<*2 ^ ^...n™w2T^) -

— T1-0 E(a2-/2,a2 ),2 ( — e»ni,...,n™ Ta 2-/2 , — »ni,...,n™ W2Ta 2 ) X (2.17)

" 1 "

nm

m

nm

then the system with respect to C1ni «m and C2ni,...«m is uniquely solvable. Solving this system (2.14) we arrive at the following presentations for these unknown coefficients

C1ni;...,nm = 7 7" [01ni,...,nmT1 dE(a2 — A2,a2),2—Q (-e,n1,...,nmTa2 ^ , -,ni,...,nmw2Ta2) -

^ni,...,nm (w)

- ^2ni,...,nmT ■ E(a2—A2,a2),2 (-e,«i ,...,«mTa2 > -,«i,...,nm W2^2)] , C2ni,...,nm = 7 7 [02«i,...,n™E(a2—A2,a2),1 (-e,«i,...,nmT 2 , -,ni,...,nm w T 2) -

^«1 ,...,nm (w)

"01«l,...,nm T

—Q

E(a2—A2,a2),1—Q ( e,ni;...,nmT A , ,ni,...,nmw T )

r(1 - 0)

Substituting these results into (2.10) and taking into account C+ni)...,nm = C1ni ..>nm in (2.9), we obtain the following representations:

u+i,...,nm (t> e,-,u) = ^ini.....nm N11ni,...,nm (t,e,-) - ^2ni,...,nm N12ni,...,nm (t,e,-) + (2 18)

+ urr+i,...)nm N13ni,...nm (t,e,u), t> 0,

,...,nm (t) = ^1ni.....nm N21ni,...,nm (t,e,-) - ^2ni,...,nm N22ni,...,nm (t, e,-) + (2 19)

+ urn;)...)nm N23ni,...,nm (t,e,-), t <

where

T1 - o

N11ni,...,nm (t,e,-) = -7 7"E(«2- 02,a2),2 -0 (-e,ni,...,nmT"2 °2 > -,ni,...,nm^^2) X

X E /'-en2 tai- 0i — ,,2

X E(ai- 0i,ai),1 V e,ni,...,nm^ , ,ni,...,nm^ j ,

N12ni;...;nm (t,e,—) =-E(a2-02,a2),2 (-e^i,..,^Ta2-02, -,ni;...;nm-Ta2) X

'«l,...,«™

X E /_e,2 tai— Ai _,2 ta^

X E(ai—Ai,ai),1 ^ fc,ni,...,nm 1 , ,ni,...,nm 1 J ,

N13ni,...,nm (t,e,w) = Mu«!,...,«™ (t,e) - M^m,...,«™ (t,e,w) + M^,...,«™ (t,e,w),

M11«l,...,«m (t,e)= / a1(t - s)sa 1 1 E(a i—Al,ai),a 1 (-e,ni)...)nm sai Al, -,2a1...,am ds,

0

M12ni,...,nm (t, w) = ^1«i.....«™ N11«1,...,«m (t, w),

M13ni,...,nm (t, w) = ^2«!,...,«™ N12«1,...,«m (t, W),

N21«!,...,«™ (t,e,w) = M21«!,...,«™ (t,e,w) + M22«i,...,«m (t,e,w),

N22«!,...,«™ (t,e,w) = M23«i,...,«m (t,e,w) + M24«i,...,«m (t,e,w),

N23«l,... ,«m (t,e,w) = M25«l,...,«m (t,e,w) - ^1«1.....n™ [M21«i,...,nm (t,e,w) - M22«l,... ,«m (t,e,w)] +

+ ^2«!,...,«™ [M23«i,...,nm (t,e,w) - M24«i,...,nm (t,e,w)] , T1—Q

M21«i,...,«m (t,e,w) = -— E(a2—A2,a2),2—Q H^i,...,«™T^A2 , -,«1,...,«™W^2) X

^«1,...,«m(w)

X E(a2 —A2,a2),1 (-e,«i,...,nm (-t)a2 2 , -,«i,...,nmw2(-t)a^ ,

T

X

M22«i,...,n™ (t,e,W) = --- X

^«l,...,nm (w)

E(a2—A2,a2),1—Q (-e,n1,...,nmT ^ , -,«1,...,«m w T ) -

1

r(1 - 0)

X

X t • E(a2 /2,a2),2 (—e»ni,...,nm ( —1)a 2 /2 J —»ni,...,nm^2( —1),

T 2 2 2

M23ni,...,nm (t,e,w) = -7 ^E(a2-/2,a2),2 (—e»ni,...,nmTa2 2 , —»ni,...,nmw X

/ \ J-'K<X2—P2,<X2),4\ ^»ni,...,nm

X E(a2-/2,a2),1 ( —e»ni,...,nm ( —t)a2-/2 , —»ni,...,nm ^ ( — t)^)

M24ni,...,nm (t,e,w) = -7 rE(a2-/2,a2),1 (—e»ni,...,nm^ ^ , — »ni,..,nm^^) X

.....n™ (w)

X t • E(a2 /2,a2),2 (—e»ni,...,nm ( —1)a 2 /2 , —»ni,...,nm^2( —1)a2)

M25ni,...,nm(t,e,w) = I* a2(s — t)(—s)a2-1 X

,2 (—s)a2-ft — ,,2 w2( — s)a2^ .,nml > 1 ni,...,n™

X E(a2-/2,a2),a2 (—e»ni,...,nm ( —s)^ , — ^2( —s)^) ds,

the quantity <£1ni'iii,nm and ^1ni'iii,nm are defined ^om (2.15) and (2.16), respectively. Substituting these presentations (2.18) and (2.19) into (2.5) and (2.6), we obtain

t+ ni.....nm

1 — V/ &1(s)N13ni,...,nm (s,e,w) ds

r-T /«T

(2.20)

/ &1(s) Nnni,...,nm (s,e,w) ds / &1(s)N12ni,...,nm (s,e,w) ds,

■,nm I ^W^'Uni w ^2ni.....n™

00

Tni ,...,nm

1 — V J b2(s)N23ni,...,nm(s,e,w) ds =

= ^1ni,...,nm J b2(s)N21ni,...,nm (s,e,w) ds — (2.21)

/ 62(s)N22ni,...,nm (s,e,w) ds. If the following conditions are fulfilled

v/ 61 (s) N13ni,...,nm(s,e,w) ds = 1, V/ &2(s) N23ni,...,nm(s,e,w) ds = 1, (2.22)

Jo J-T

then, by virtue of (2.20) and (2.21), from the presentations (2.18) and (2.19) we derive

,...,nm (t,e,w,V) =

Q11ni,...,nm (t, e, w, v)

Q12ni,...,nm(t,e, w,v), t> 0,

u-i,...,nm (t,e,w,V) =

Q21ni,...,nm(t,e,w,V) — ^2ni,...,nmQ22ni,...,nm(t, e, W, V), t < 0,

where

Q11ni,...,nm (t, e, w, V) = Nnni,...,nm (t, e, w) +

(2.23)

(2.24)

N13ni,...,nm (t,e, w)

T

+ V"-„T 7 ■ " ' ' ,-TV / 61(s)N11ni,...,nm (s,e,w) ds,

1 — V J0 61 (s) N13ni,...,nm (s,e, w) ds J0

Q12ni,...,nm (t, e, w, V) = N12ni,...,nm (t,e,w) +

N13ni,...,nm (t,e, w)

T

+ V"-„T7 3 iT' " ' ' ,-rT~ / 61(s)N12ni,...,nm (s,e,w) ds,

1 — Vj0 61(s)N13ni,...,nm (s,e, w) dW0

Q21ni,...,nm (t, e, w, V) = N21ni,...,nm (t,e,w) +

0

N23«!,...,«™ (t,e, w)

0

+ v-0 ^m,...,- 62(s)N21«1,...,«m(s,e,w) ds

1 - v/—T &2(s)N23«i,...,«m (s,e,w) ds J—t

Q22«!,...,«™ (t,e,w,v ) = N22«!,...,«™ (t, e, w) +

0

+ u"- n ---— b2(s)N22ni,...,nm (s,e,-) ds.

1 - uj- T b2(s)N23ni,...,nm (s, e, —) ds V- T

Now we substitute presentations (2.23) and (2.24) into the Fourier series (2.1) and obtain the following formal solution of the problem (1.1)—(1.4)

U(t,x,e,w,v) = ^ ^«i,...,«m (x) X

«1,... ,«m = 1

X [^1«i,...,nm Q11«i,...,nm (t,e,w,v ) - ^2«1.....«m Q12«l,... ,«m (t,£,W,V )] , t > 0,

œ

U(t,x,e,w,v)= ^ ^«i,...,«m (x) X

«1,... ,«m = 1

X [^1ni.....n™Q21«i,...,«m (t,e,w,v) + ^2«1.....n™Q22«i,...,«m (t,e,w,v)] , t < 0.

§3. The uniqueness of the solution of the problem (1.1)—(1.4)

Let condition (2.17) be violated. Then we suppose that

(2.25)

(2.26)

0"«l,...,«m (w) = T1 QE(a2—A2,a2),1 (-e,«i,...,nmTa2 A2 , -,«i,...,nmw2Ta2) X

.2 T a 2—A2 .2 . .2

,«m T , rni,...,«™

—Qp A „..2 T a2 —A2 ,.2 ,.2

X

X E(a2 — A2,a2),2—Q e,«i,...,«mTa2 , -^i,...,«™w2Ta2) -- T1 —Q E(a2— A2,a2 ),2 (-e,«2i ,...,«m Ta2 —A2 , -,«1,...,«™ W2Ta2 ) X (3.1)

E(a2 —A2,a2),1—Q (-e ^«1,...,«™T 2 A2 , -^„1.....n™ w T 2) - r(1 - 0)

for some values of w, where ,ni,...,«m = f n1 + ... + nm.

The set of positive solutions of this equation (3.1) with respect to the spectral parameter w is denoted by . We call the values w 6S1 irregular because the condition (2.17) is violated for them. The set A1 = (0; œ)\^ is called the set of regular values of the spectral parameter w, for which condition (2.17) is fulfilled.

If conditions in (2.22) are violated, then the kernels of the mixed integro-differential equation (1.1) have for each value of n1,..., nm two values of v1 and v2

11

V1 — —t-, V2 — —0-,

/0 &1(s)N13„i,...,„m (s,e,w) ds /— T &2(s)N23ni,...,nm (s,e,w) ds

where t 0

/ 61 (s) N13«!,...,«™ (s,e,w) ds = 0, / 62(s)N23«!,...,«™ (s,e,w) ds = 0.

./0 ./— T

We regard these real nonzero numbers as irregular kernel numbers of the mixed integro-differential equation (1.1) and denote their set {v1, v2} by S2. We take away the values v1 and v2 of the spectral parameter v from the set of nonzero real numbers (-œ; 0) U (0; œ). The resulting set A2 = (-œ; 0) U (0; œ)\S2 is called a set of regular values of the parameter v. For all values of v G A2 condition (2.22) is satisfied. We use the following notation

H = {n1,. . . , nm G N; w G A1; v G A2} ,

œ

where N is the set of natural numbers. This is the case when all values of the spectral parameters w and V are regular. Therefore, in this case, the solution of the problem (1. 1)—(1.4) in the domain Q is represented in the form of series (2.25) and (2.26).

To establish the uniqueness of the function U(t, x, e, w, v) we show that, under the zero conditions ^¿(x) = 0, x G Qm, i = 1, 2, the problem (1. 1)—(1.4) has only a trivial solution. We suppose that (x) = 0, i = 1, 2. Then ^ini,...,nm = 0, i = 1, 2, and from formulas (2.25) and (2.26) it implies that

U (t, x, e, w, v )tfni,...,nm (x) dx = 0.

/np

Hence, by virtue of completeness of systems of the eigenfunctions

2 nn1 t sin —i—x1

2 nn2 t sin —i—x2

2 . nnm

1 Sin—x

in the space L2 (Q™) we deduce that

U(t, x, e, w, V) = 0

for all x G [0, /]m and t G [—T; T]. Therefore, for the (n1;..., w, v) G K the solution of the problem (1.1)—(1.4) is unique, if this solution exists in the domain Q.

§ 4. Convergence of series (2.25) and (2.26)

We show that under certain conditions with respect to the functions ^(x) (i = 1, 2) the series (2.25) and (2.26) converge absolutely and uniformly in the domain Q. In this order we use the following well known properties of the Mittag-Leffler function:

1) for all k > 0, a, p, 7 G (0; 2], p ^ a ^ 7, t ^ 0 the function r-1£(Aa);7 (-kt^, -kta) is completely monotonous and there holds

(-1)s [ta-1E(^;«))7 (-kt^, -k ta)](s) ^ 0, s = 0,1, 2,...; 2) for all a, p G (0,2), 7 G R and argz1 = n there takes place the following estimate

C1

E(/,a),7 (z1, z2) J ^

1 + N

(4.1)

(4.2)

E(/,a),7 (e1Z1, Z2) — E(/,a),7 (e2Z1, Z2) I ^ |e1 — e21

C2

1 + |Z1 r

(4.3)

where 0 < C = const do not depend from z, e» G (0; e0) , 0 < e0 = const, i = 1, 2.

Indeed, according to the properties of the Mittag-Leffler function (formulas (4.1) and (4.2)) the functions Qyni,iii,nm (t, e,w, v) (i, j = 1, 2) are uniformly bounded on the segment [—T; T]. So for any positive integers n1,..., there exist finite constant numbers C0i (i =1, 2), that the following estimates take place

max < max |Qnni,.„,nm(t,e,w,v)| ; max IQ^i,...,nm(t,e,w,v)| ^ ^ C01,

(4.4)

max < max |Q21ni,...,nm(t,e,w,v)| ; max |Q22ni,...,nm(t,e,w,v)|^ ^ C02,

ni,...,nmeN I te[-T;0] te[-T;0]

where C0i = const, i = 1, 2.

Condition A. We suppose that the functions ^(x) G C2[0; /]m, i = 1, 2, on the domain [0; /]m have piecewise continuous third order derivatives. Then integrating by parts the following integrals three times with respect to each of variables xi; x2,..., xm we derive [28]

|(ini,...,nm 1 ^ I I 3 3

1 n n3, . . . , n3

3m

(3m) (ini,..

(4.5)

where

(3m) f d3m ( i(x)

(ini,...,nm = dx3dx3 dx3

jQm dxidx2 . . . dxm

Here the Bessel inequalities are true

■^ni.....nm (x) dx, i = 1, 2.

E

(3m) ( irai,..

rai,...,ram = 1

22 S< 2

d3m ( i(x)

3 3 3

i 2 m

dx, i = 1, 2.

(4.6)

Taking formulas (4.4)-(4.6) into account and applying the Cauchy-Schwarz inequality and Bessel inequality, for series (2.25) and (2.26) we obtain

|U(t,x,£,W,v)| ^ Y^

lu

±

'rai,...,rar

)| ■ |^rai,...ram (x)| ^

rai,...,ram = 1

^ Yii

E

rai,...,ram = 1 m

Yii

C0i £ [| (ini,.. . ,nm | + | (2ni

1

n 1 3 . . . nm ii,...,nm = i |o (3m) . V"^ (ini,...,nm + / y ni....."m

o E ni,...,nm = 1 V/ .V ^m

i ...^m

V/ ■ d3m ( 2 (x) I2

3 3 3 i 2 m

... nm

(3m) ( 2ni,..

(4.7)

d3m ( i(x)

dx3dx2... dxm

dx +

dx

< œ,

where ju = (^f) Coi (^)3m, i = 1, 2.

It follows from estimate (4.7) that the series (2.25) and (2.26) are convergent absolutely and uniformly in the domain Q for the (n1;..., nm, w, v) G K.

§ 5. Possibility of term differentiation of series (2.25) and (2.26)

For the (n1;..., nm, w, v) G K functions (2.25) and (2.26) formally differentiate in Q the required number of times

cDaiU(t,x,e,^,v) = ^ni,...,nm(x) x

ni,...,nm = i

x [(ini.....nmCDaiQiini,...,nm v) + (2ni.....nmCD01Qi2ni,...,nm v)]

(5.1)

t > 0,

cD0a2U(t,x,e,^,v) = ^ni,...,nm(x) x

ni,...,nm = i

x [(ini.....nm cD0i Q2ini,...,nm v) + (2ni,...,nmCD0i Q22ni,...,nm v)]

(5.2)

t < 0,

2

m

"i"

oo

m

o

m

o

2

2

l

o

o

Uxixi (t, x, e, w, v) = — ^ ^ni,...,nm (x) X

ni,...nm = 1

X [^1ni,...,nm Qi1ni,...,nm (t, e, w, V ) + ^2ni,...,nm Qi2ni,...,nm (t, e, w, V )] , —T < t < T,

(5.3)

Ux2x2 (t,x,e,w,V) = — jr (^P) ^ni>...,nm (x) X

ni,...,nm = 1

X [^1ni,...,nmQi1ni>...,nm (t, e, w, V) + + P2ni,...,nmQi2ni,...,nm (t, e, w, V)], —T < t < T, i = 1, 2.

(5.4)

The expansions of the following functions into Fourier series are defined in the domain Q in a similar way

Ux3X3 (t, x, e, w, V),..., UxmXm (t,x,e, w, v), cD^1 U^i (t,x,e, w, v), C Da Uxixi (t,x,e, w, v ), C D^l U^ (t,x,e,w, v ),..., c D£2 U^2^2 (t,x,e,w, v ),...,

CDai UXmXp (t, X, e, W, V) , CDa2 UXmXp (t, X, e, W, V) .

The convergence of series (5.1) and (5.2) is proved similarly to the proof of the convergence of series (2.25) and (2.26). So, we show the convergence of series (5.3) and (5.4). Taking into account formulas (4.4)-(4.6) and estimate (4.7) and applying the Cauchy-Schwarz inequality and Bessel inequality, we obtain

|UXlXi (t,x,e,w,v)| ^ Y^ (^T1) Ki,..,nm (t,e,w,v)| ■ |^ni,...,nm (x)| ^ ni,...nm = 1

m

n (f

ni,...,nm = 1

^ 72i

E

('J) C0i n2 [|^1ni,...,nm 1 + |^2ni,...

œ

+ E

1

_ni,...,n„

=1 n1n3... nm

(3m)

.,nm

ni ,...,nm

=1 n1n3... nm

(3m)

,,n„

Y2i

\

E

n2n6 n^

nl,...,nm = 1 1 2 ' ' m

dx +

+

d3 mP2(x)

3 3 3

1 2 m

d 3mp1(x)

5x3dx2... ôxm

dx < œ,

where Y2i = (yfy C0i (n1)

C0i (i)3 m-2 ,i =1, 2;

|UX2X2 (t,x,e,w,V)| ^ Y (^t) |U±l--nm (t,e,w,V) 1 " 1 ^ni>...,nm (x)| ^

ni,...,nm = 1

^ Y2i

E

m

2 \ /n

l I W 1

ni,...,nm = 1

33

,l,.,„m = 1 n3n2n3 . . . nm

(j) C0i X] n2 [| ^1ni,-,nm | + | P2ni,...

œ

+ E

(3m)

.,nm

33

=1 n3n2n3... nm

(3m) P 2ni,.. . nm

œ

œ

n

m

m

2

2

1

l

2

nm

œ

œ

n

m

œ

ni ,...,n

Y2i

\

E

626

Hin^no ...

ni,...,nm = i i 2 3 m

d3m (( i(x)

3 3 3

dxid^2 . . . d^m

dx +

+

d3m ( 2 (x)

dx

< œ.

3 3 3

i 2 m

The convergence of Fourier series for functions

Ux3X3 (t,x,e, w, V),..., UxmXm (t,x,e, w, v), CD° Uxixi (t,x,e, w, v), C D02 Uxi xi (t,x,e,w, v ), C D01 Ux2x2 (t,x,e,w, v ),..., c D^2 Ux2x2 (t,x,e,w, v ),...

CD°t UXmXm ^ ^ v) , CD0i UXmXm (t^ e,w,v) .

is proved in a similar way in the domain Q. It follows from these last estimates that the functions (2.25) and (2.26) possess properties (1.2) for regular values of spectral parameters w and v.

§ 6. Continuous dependence of the solution on a small parameter

We consider the continuous dependence of the solution of the problem (1.1)—(1.4) on a small parameter e for regular values of spectral parameters w and v. Let ei and e2 be two different values of a small positive parameter e. It is easy to check from (4.3), that the following estimates hold

max max |Qiini,...,nm(t,ei,w,v) - Qiini,...,nm(t,e2,w,v)| ^ Cu |ei - e2| , (6.1)

ni,...,nm€№e[0;T ]

max max IQfrni,...,«™ (t,£i,w,v) - Qfmi,...«™ (i,£f,w,v)| ^ Cfi |ei - £f| , (6.2) rai,...rameNte[ - T ;0]

where Cj = const, j, i =1, 2, G (0; e0), 0 < e0 = const, i = 1, 2.

Then, taking formulas (4.5), (4.6) and estimates (6.1), (6.2) into account and applying the Cauchy-Schwarz inequality and Bessel inequality, for series (2.25) and (2.26) we obtain

|U(t,x,e1,w,v) — U(t,x,e2,w,v| ^

E

lu

±

ni ,...,nm

(t,ei,w,v) - «J.....„m (t,e2,w,v)| ■ |^ni,...,nm (x)| ^

ni,...,nm = i

(Cii + Ci2) |ei - e2| [|(ini,...,nm | + |( 2ni,...,'

ni,...,nm = i

^ Y3i |ei - e21

E

.ni,...,nm = i

=i n3... nm

(3m)

(ini,.. . nm

+ E

ni,...,nm = i

=i n3 ...^m

(3m) ( 2ni,..

|ei - e2| Y3i

\

E

ni.....nm=i

=i .. .^m

+

d3m (( 2 (x)

3 3 3

i 2 m

dx

d3m (i(x)

dx3dx2... dxm

|ei - e2| ■ A,

dx +

(6.3)

(/—\ m 3m,

yfj (Cii + Cif) (£) m , i = 1, 2, A = const < to.

It follows from estimate (6.3) that |U(t, x, e1, w, v) — U(t, x, e2, w, v) | is small, if |e1 — e2| is small in the domain Q for the (n1,..., nm, w, v) G K.

m

2

1

"m

2

o

2

i

m

1

1

m

2

1

"m

2

"m

§ 7. Statement of the theorem

As a conclusion, we will formulate a theorem that we have already proved in this article. Thus, the following theorem is true.

Theorem 7.1. Let conditions of A be fulfilled. Then for the possible numbers ni;..., and regular values of spectral parameters u and v from the set N the problem (1.1)-(1.4) is uniquely solvable in the domain Q and this solution is represented in the form of series (2.25) and (2.26). Moreover, it is true that

limU(t,x,e,w,v) = U(t,x, 0,w,v),

£—^0

where U (t, x, 0, u, v) is the solution of a mixed fractional integro-differential equation of the form

rT

v/ K1(t,s)U(s,x) ds, t> 0,

A)(U ) — B (U)=<

v / K2(t,s)U(s,x) ds, t < 0,

-T

A0(U )

1 + sgn (t) Dai + 1 — Sgn(t) _ n«2

"C n01

D -

C n0t

U (t, x), B (U )=<

Vux.x., t> o,

i=1

w

!Vux.x., t < o

V i=1

with boundary value conditions (1.3) and (1.4) under consideration.

m

2

2

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Received 09.08.2020

Yuldashev Tursun Kamaldinovich, Candidate of Physics and Mathematics, Associate Professor, Uzbek-Israel Joint Faculty of High Technology and Engineering Mathematics, National University of Uzbekistan, ul. University, 4, Tashkent, 100174, Uzbekistan. ORCID: https://orcid.org/0000-0002-9346-5362 E-mail: tursun.k.yuldashev@gmail.com

Karimov Erkinjon Tulkinovich, Doctor of Physics and Mathematics, Senior Researcher, Department of Differential Equations, V. I. Romanovskiy Institute of Mathematics of Academy of Sciences of Uzbekistan, ul. University, 4a, Tashkent, 100174, Uzbekistan. ORCID: https://orcid.org/0000-0003-4443-6300 E-mail: erkinjon@gmail.com

Citation: T. K. Yuldashev, E. T. Karimov. Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2021, vol. 57, pp. 190-205.

Т. К. Юлдашев, Э. Т. Каримов

Смешанные интегро-дифференциальные уравнения с операторами Капуто дробного порядка и спектральными параметрами

Ключевые слова: интегро-дифференциальное уравнение, уравнение смешанного типа, малый параметр, спектральные параметры, дробные операторы Капуто, однозначная разрешимость.

УДК: 517.956

DOI: 10.35634/2226-3594-2021-57-10

Рассматриваются вопросы однозначной разрешимости краевой задачи для интегро-дифференци-ального уравнения смешанного типа с двумя операторами Капуто дробного порядка и спектральными параметрами. Интегро-дифференциальное уравнение смешанного типа является интегро-дифференциальным уравнением с частными производными дробного порядка как в положительной, так и в отрицательной частях рассматриваемой многомерной прямоугольной области. Порядок дробного оператора Капуто меньше в положительной части области, чем порядок соответствующего оператора в отрицательной части области. Используя метод рядов Фурье, получены две системы счетных систем обыкновенных дробных интегро-дифференциальных уравнений с вырожденными ядрами. Далее используется метод вырожденных ядер. Для определения произвольных постоянных интегрирования получена система алгебраических уравнений. Из этой системы были вычислены регулярные и нерегулярные значения спектральных параметров. Решение рассматриваемой задачи получено в виде рядов Фурье. Доказана однозначная разрешимость задачи для регулярных значений спектральных параметров. При доказательстве сходимости рядов Фурье используются свойства функции Миттаг-Леффлера, неравенство Коши-Шварца и неравенство Бесселя. Также изучена непрерывная зависимость решения задачи от малого параметра при регулярных значениях спектральных параметров. Результаты сформулированы в виде теоремы.

Поступила в редакцию 09.08.2020

Юлдашев Турсун Камалдинович, к. ф.-м. н., доцент, Узбекско-Израильский совместный факультет высокой технологии и инженерной математики, Национальный университет Узбекистана, 100174, Узбекистан, г. Ташкент, ул. Университетская, 4. ORCID: https://orcid.org/0000-0002-9346-5362 E-mail: tursun.k.yuldashev@gmail.com

Каримов Эркинжон Тулкинович, д. ф.-м. н., старший научный сотрудник, Институт математики им. В. И. Романовского АН Узбекистана, Узбекистан, 100174, г. Ташкент, ул. Университетская, 4а. ORCID: https://orcid.org/0000-0003-4443-6300 E-mail: erkinjon@gmail.com

Цитирование: Т. К. Юлдашев, Э. Т. Каримов. Смешанные интегро-дифференциальные уравнения с операторами Капуто дробного порядка и спектральными параметрами // Известия Института математики и информатики Удмуртского государственного университета. 2021. Т. 57. С. 190-205.