## Boundary value problem for mixed-compound equation with fractional derivative, functional delay and advance.(Russian. English summary)Zbl 1438.35420

Summary: We study the Tricomi problem for the functional-differential mixed-compound equation $$LQu(x,y)=0$$ in the class of twice continuously differentiable solutions. Here $$L$$ is a differential-difference operator of mixed parabolic-elliptic type with Riemann-Liouville fractional derivative and linear shift by $$y$$. The $$Q$$ operator includes multiple functional delays and advances $$a_1(x)$$ and $$a_2(x)$$ by $$x$$. The functional shifts $$a_1(x)$$ and $$a_2(x)$$ are the orientation preserving mutually inverse diffeomorphisms. The integration domain is $$D=D^+\cup D^-\cup I$$. The “parabolicity” domain $$D^+$$ is the set of $$(x,y)$$ such that $$x_0<x<x_3$$, $$y>0$$. The ellipticity domain is $$D^-=D_0^-\cup D_1^-\cup D_2^-$$, where $$D_k^-$$ is the set of $$(x,y)$$ such that $$x_k<x<x_{k+1}$$, $$-\rho_k(x)<y<0$$, and $$\rho_k=\sqrt{a_1^k(x)(x_1-a_1^k(x))}$$, $$\rho_k(x)=\rho_0(a_1^k(x))$$, $$k=0, 1, 2$$. A general solution to this Tricomi problem is found. The uniqueness and existence theorems are proved.

### MSC:

 35R10 Partial functional-differential equations 35M13 Initial-boundary value problems for PDEs of mixed type 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Full Text:

### References:

 [1] Muravnik A. B., “On the Cauchy problem for differential-difference equations of the parabolic type”, Dokl. Math., 66:1 (2002), 107-110 · Zbl 1148.39300 [2] Zarubin A. N., “Tricomi problem for a functional-differential advanced-retarded Lavrent’ev-Bitsadze equation”, Differ. Equ., 53:10 (2017), 1329-1339 · Zbl 1387.35425 [3] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh [Some classes of partial differential equations], Nauka, Moscow, 1981, 448 pp. (In Russian) · Zbl 0511.35001 [4] Zarubin A. N., Uravneniia smeshannogo tipa s zapazdyvaiushchim argumentom [Equations of mixed type with retarded argument], Orel State Univ., Orel, 1999, 255 pp. (In Russian) [5] Korpusov M. O., Pletner Yu. D. Sveshnikov A. G., “On the existence of a steady-state oscillation mode in the Cauchy problem for a composite-type equation”, Comput. Math. Math. Phys., 41:4 (2001) · Zbl 1023.35083 [6] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poriadka i nekotorye ikh prilozheniia [Integrals and derivatives of fractional order and some of their applications], Nauka i tekhnika, Minsk, 1987, 688 pp. (In Russian) · Zbl 0617.26004 [7] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, riadov i proizvedenii [Tables of integrals, sums, series and products], Nauka, M., 1971, 1108 pp. (In Russian) [8] Ter-Krikorov A. M., Shabunin M. I., Kurs matematicheskogo analiza [A course of mathematical analysis], Nauka, Moscow, 1988, 816 pp. (In Russian) · Zbl 0702.26002 [9] Nakhushev A. M., Elementy drobnogo ischisleniia i ikh primeneniia [Elements of Fractional Calculus and their Application], Kabardino-Balkarsk. Nauchn. Tsentr Ross. Akad. Nauk, Nalchik, 2000, 253 pp. (In Russian) [10] Zarubin A. N., “Boundary value problem for a mixed type equation with an advanced-retarded argument”, Differ. Equ., 48:10 (2012), 1384-1391 · Zbl 1259.35146 [11] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006, xvi+523 pp. · Zbl 1092.45003 [12] Ditkin V. A., Prudnikov A. P., Integral’nye preobrazovaniia i operatsionnoe ischislenie [Integral transforms and operational calculus], Nauka, Moscow, 1974, 521 pp. (In Russian) · Zbl 0298.44007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.