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Yuldashev, Tursun K.; Kadirkulov, Bakhtiyor J.

Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator. (English) Zbl 1448.35341

Ural Math. J. 6, No. 1, 153-167 (2020).
Summary: In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Cited in 12 Documents

MSC:

35M12 Boundary value problems for PDEs of mixed type
35R11 Fractional partial differential equations
35R09 Integro-partial differential equations

Keywords:

non-self-adjoint boundary value problem; Hilfer operator; Mittag-Leffler function; spectral parameter; solvability
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\textit{T. K. Yuldashev} and \textit{B. J. Kadirkulov}, Ural Math. J. 6, No. 1, 153--167 (2020; Zbl 1448.35341)
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References:

[1] Abdullaev O. Kh., Sadarangani K., “Non-local problems with integral gluing condition for loaded mixed type equations involving the Caputo fractional derivative”, Electron. J. Differential Equations, 2016:164 (2016), 1-10 · Zbl 1383.35239
[2] Hilfer R. (ed.), Application of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000, 472 pp. · Zbl 0998.26002
[3] Agarwal P., Berdyshev A., Karimov E., “Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative”, Results Math., 71 (2017), 1235-1257 · Zbl 1375.35282
[4] Aziz S., Malik S. A., “Identification of an unknown source term for a time fractional fourth-order parabolic equation”, Electron. J. Differential Equations, 2016:293 (2016), 1-20 · Zbl 1358.80004
[5] Berdyshev A. S., Cabada A., Kadirkulov B. J., “The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator”, Comput. Math. Appl., 62 (2011), 3884-3893 · Zbl 1236.35197
[6] Berdyshev A. S., Kadirkulov J. B., “On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan-Nersesyan operator”, Differ. Equ., 52:1 (2016), 122-127 · Zbl 1381.35209
[7] Beshtokov M. Kh., “Boundary value problems for the generalized modified moisture transfer equation and difference methods for their numerical implementation”, Appl. Math. Phys., 52:2 (2020), 128-138 (in Russian)
[8] Gel’fand I. M., “Some questions of analysis and differential equations”, Uspekhi Matem. Nauk, 14:3 (1959), 3-19 (in Russian) · Zbl 0091.08805
[9] Tenreiro Machado J. A. (ed.), Handbook of Fractional Calculus with Applications, v. 1-8, Walter de Gruyter GmbH, Berlin, Boston, 2019
[10] Hilfer R., “Experimental evidence for fractional time evolution in glass forming materials”, Chem. Phys., 284:1-2 (2002), 399-408
[11] Hilfer R., “On fractional relaxation”, Fractals. Part III: Scaling, 11, Supp. 01. (2003), 251-257 · Zbl 1140.82315
[12] Hilfer R., Luchko Y., Tomovski Ž., “Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives”, Fract. Calc. Appl. Anal., 12:3 (2009), 299-318 · Zbl 1182.26011
[13] Kal’menov T. S., Sadybekov M. A., “On a Frankl-type problem for a mixed parabolic-hyperbolic equation”, Sib. Math. J., 58:2 (2017), 227-231 · Zbl 1367.35094
[14] Kerbal S., Kadirkulov B. J., Kirane M., “Direct and inverse problems for a Samarskii-Ionkin type problem for a two dimensional fractional parabolic equation”, Progr. Fract. Differ. Appl., 4:3 (2018), 147-160
[15] Kim M.-Ha, Ri G.-Chol, O. H.-Chol, “Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives”, Fract. Calc. Appl. Anal., 17:1 (2014), 79-95 · Zbl 1312.34018
[16] Kumar D., Baleanu D., “Editorial: fractional calculus and its applications in physics”, Front. Phys., 7:6 (2019), 1-2
[17] Malik S. A., Aziz S., “An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions”, Comput. Math. Appl., 73:12 (2017), 2548-2560 · Zbl 1386.35479
[18] Nakhushev A. M., Zadachi so smeshcheniem dlya uravnenij v chastnyh proizvodnyh [Problems with Displacement for Partial Differential Equations], Nauka, Moscow, 2006, 287 pp. (in Russian)
[19] Patnaik S., Hollkamp J. P., Semperlotti F., “Applications of variable-order fractional operators: a review”, Proc. A, 476:2234 (2020), 1-32 · Zbl 1439.26028
[20] Sabitov K. B., “Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain”, Math. Notes, 89:4 (2011), 562-567 · Zbl 1234.35152
[21] Sabitov K. B., K teorii uravnenij smeshannogo tipa [On the Theory of Mixed Type Equations], Fizmatlit, Moscow, 2014, 304 pp. (in Russian)
[22] Sabitov K. B., Sidorov S. N., “On a nonlocal problem for a degenerating parabolic-hyperbolic equation”, Differ. Equ., 50:3 (2014), 352-361 · Zbl 1295.35343
[23] Sandev T., Metzler R., Tomovski Ž., “Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative”, J. Phys. A. Math. Theor., 44:25 (2011), 255203 · Zbl 1301.82042
[24] Sandev T., Tomovski Ž., Fractional Equations and Models: Theory and Applications, Dev. Math., 61, Springer Nature Switzerland AG, Cham, Switzerland, 2019, 345 pp. · Zbl 1473.35002
[25] Saxena R. K., Garra R., Orsingher E., “Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives”, Integral Transforms Spec. Funct., 27:1 (2015), 30-42 · Zbl 1337.26015
[26] Sun H., Chang A., Zhang Y., Chen W., “A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”, Fract. Calc. Appl. Anal., 22:1 (2019), 27-59 · Zbl 1428.34001
[27] Terlyga O., Bellout H., Bloom F., “A hyperbolic-parabolic system arising in pulse combustion: existence of solutions for the linearized problem”, Electron. J. Differential Equations, 2013:46 (2013), 1-42 · Zbl 1288.35202
[28] Uflyand Ya. S., “K voprosu o rasprostranenii kolebanij v sostavnyh elektricheskih liniyah [On oscillation propagation in compound electric lines]”, Inzhener.-Phis. Zhurn. [J. of Engineering Physics and Thermophysics], 7:1 (1964), 89-92 (in Russian)
[29] Yuldashev T. K., “Nonlocal problem for a mixed type differential equation in rectangular domain”, Proc. Yerevan State Univ. Phys. Math. Sci., 2016, no. 3, 70-78 · Zbl 1364.35205
[30] Yuldashev T. K., “Solvability of a boundary value problem for a differential equation of the Boussinesq type”, Differ. equ., 54:10 (2018), 1384-1393 · Zbl 1411.35211
[31] Yuldashev T. K., “A coefficient determination in nonlocal problem for Boussinesq type integro-differential equation with degenerate kernel”, Vladikavkaz. Mat. Zh., 21:2 (2019), 67-84 (in Russian)
[32] Yuldashev T. K., “On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument”, Lobachevskii J. Math., 41:1 (2020), 111-123
[33] Yuldashev T. K., “Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations”, Axioms, 9:2 (2020), 45, 1-21
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