We study the existence and uniqueness of solutions for an analog of the Frankl-type boundary-value problem for a parabolic-hyperbolic-type equation. The uniqueness of the solution is proved by using the principle of extremum, whereas the existence is proved by the method of integral equations.
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O. Kh. Abdullayev and F. Begimqulov, “On the research tasks, such as the Frankl problem with discontinuous matching condition for a degenerate equation of mixed type,” Rep. Adyghe (Circassian) Int. Acad. Sci., 14, No. 1, 9–21 (2012).
S. H. Akbarova, “Analogs of the Tricomi problem for equations of parabolic-hyperbolic-type with two lines and various orders of degeneracy,” Rep. ASGL, 3–5 (1991).
G. Beytman and A. Erdeyi, Higher Transcendental Functions [Russian translation], Nauka, Moscow (1965), Vol. 1; (1966), Vol. 2.
A. V. Bitsadze, Differential Equations of Mixed Type, Pergamon Press, New York (1964).
A. V. Bitsadze, Some Classes of Partial Differential Equations, Gordon & Beach, New York (1988).
I. M. Gel’fand, “Some problems of analysis and differential equations,” Uspekhi Mat. Nauk, 14, 3–19 (1959).
G.-C. Wen and H. Begehr, “Existence of solutions of Frankl problem for general Lavrent’ev–Bitsadze equations,” Rev. Roumaine Math. Pures Appl., 45, 141–160 (2000).
A. M. Gordeev, “Some boundary-value problems for the generalized Euler–Poisson–Darboux,” Volzhsky Math. Comp., 6, 56–61 (1968).
A. M. Il’in, A. S. Kalashnikov, and O. A. Olejnik, “Linear parabolic-type equations of the second order,” Uspekhi Mat. Nauk, 17, No. 3, 3–141 (1962).
M. S. Salakhitdinov and B. I. Islomov, “Tricomi problem for a general linear mixed-type equation with nonsmooth degeneration line,” Rep. AS USSR, 289, No. 3, 549–563 (1986).
B. I. Islomov and N. K. Ochilova, “About a boundary value problem for the parabolic-hyperbolic-type equation with two lines and various order of degeneration,” http://uzmj.mathinst.uz/files/uzmj, 3, 42–53 (2005).
A. A. Kilbas and O. A. Repin, “An analogy of the Bitsadze–Samarskii problem for a mixed-type equation with a fractional derivative,” Different. Equat., 39, No. 5, 674–680 (2003).
A. A. Kilbas and O. A. Repin, “An analog of the Tricomi problem for a mixed-type equation with a partial fractional derivative,” Fract. Calc. Appl. Anal., 13, No. 1, 69–84 (2010).
A. A. Kilbas and O. A. Repin, “Solvability of a boundary-value problem for a mixed-type equation with a partial Riemann–Liouville fractional derivative,” Different. Equat., 46, No. 10, 1457–1464 (2010).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Math. Stud., 204 (2006).
M. Mirsaburov, “Problem with lacking Tricomi condition on a characteristic and an analog of the Frankl condition on a segment of the degeneration line for a class of mixed-type equations,” Different. Equat., 50, Issue 1, 79–87 (2014).
M. Mirsaburov and U. E. Bobomurodov, “Problem with Frankl and Bitsadze–Samarskii condition on the degeneration line and on parallel characteristics for the Gellerstedt equation with a singular coefficient,” Different. Equat., 48, Issue 5, 737–744 (2012).
S. G. Mikhlin, Integral Equations, Gostekhteorizdat, Moscow; Leningrad (1947).
N. K. Ochilova, “Problem Trikomi–Frankl for the equation of parabolic-hyperbolic type with two lines of degeneration,” Theses Rep. Rep. Sci. Conf. Particip. Sci. CIS Countries Modern Problems Different. Equat. and Their Appendix, Tashkent, 21–23 (2013).
N. K. Ochilova, “Research of unique solvability of the Frankl-type boundary-value problem for the mixed-type equation degenerating on a border and in the domain,” J. Collect. Sci. Works Kras., 1, No. 8, 20–32 (2014).
L. V. Pestun, “Solution of the Cauchy problem for the equation yβuxx− xαuyy = 0,” Volzhsky Math. Comp., 3, 289–295 (1965).
J. M. Rassias, “Mixed-type partial differential equations with initial and boundary values in fluid mechanics,” Int. J. Appl. Math. Statist., 77–107 (2008).
M. Kh. Ruziev, “A problem with Frankl condition on the line of degeneration for a mixed-type equation with singular coefficient,” Algebraic Structures in Partial Differential Equations Related to Complex and Clifford Analysis, Ho Chi Minh City University. Education Press, 247–259 (2011).
M. Kh. Ruziev, “A problem with conditions given on the inner characteristics and on the line of degeneracy for a mixed-type equation with singular coefficients,” Bound. Value Probl., 2013 (2013).
M. S. Smirnov, Mixed-Type Equations, Nauka, Moscow (2000).
S. A. Tersenov, Introduction to the Theory of Equations Degenerating on the Boundary, Novosibirsk State University, Novosibirsk (1973).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 10, pp. 1347–1359, October, 2019.
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Islomov, B.I., Ochilova, N.K. & Sadarangani, K.S. On a Frankl-Type Boundary-Value Problem for a Mixed-Type Degenerating Equation. Ukr Math J 71, 1541–1554 (2020). https://doi.org/10.1007/s11253-020-01730-z
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DOI: https://doi.org/10.1007/s11253-020-01730-z