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.


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
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[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
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