Constructing Solutions to a Fourth-Order Partial Integro-Differential Equation
DOI:
https://doi.org/10.58649/am1w0n06Keywords:
differential, integral, reduction, additional argument, operator, notation, partial derivativesAbstract
There is considered solution of an integro-differential equation in which the coefficients of the derivatives of the unknown function up to the fourth order inclusive are known functions. This equation is studied in an operator form, and the solution to the original problem is obtained using the method of additional argument. The prominent feature of this methodology is its ability to determine analytical solutions in some cases. Overall, the method of additional argument represents a robust contemporary framework capable of producing mathematically rigorous results.
References
Z.Z. Abzalimov, E.V. Salyakhov, Difference-analytical method for calculating eigenvalues for fourth-order equations with separated boundary conditions, Izvestiya VUZov, Mathematics, 11 (2008), 3–11.
B.S. Ablabekov, A.R. Asanov, A.K. Kurmanbaeva, Inverse problems for differential equations of the third order, Bishkek: Ilim, (2011).
A.Zh. Ashirbaeva, Construction of solutions of nonlinear integro-differential equations in partial derivatives of higher order with a degenerate kernel, Research on Integro-Differential Equations, 27 (1998), 45–50.
A.Zh. Ashirbaeva, M. Bilal kyzy, G.A. Karimova, G.K. Sadykova, Construction of the Solution to a Second-Order Partial Differential Equation of Parabolic Type with Multiple Variables by the Method of Additional Argument, J. Mod. Math. Stud., 1(1) (2025), 75–81.
A.Zh. Ashirbaeva, M.R. Bekieva, G.K. Sadykova, Solution of a system of integro-differential equations in partial derivatives of the second order by the method of additional argument, AIP Conference Proceedings, 3244 (2024), 02002.
A.Zh. Ashirbaeva, G.K. Sadykova, Solution of a system of first-order partial operator equations, Eurasian Scientific Association, 11-1(81) (2021), 1–5.
A.Zh. Ashirbaeva, G.K. Sadykova, Solution of a system of nonlinear partial differential equations with n + 1 independent variables, Eurasian Scientific Association, 8-1(78) (2021), 6–8.
A.Zh. Ashirbaeva, G.K. Sadykova, Development of the method of additional argument for a system of nonlinear differential equations, International Research Journal, 4-1(82) (2019), 6–10.
A.Zh. Ashirbaeva, G.K. Sadykova, Development of the method of additional argument for a system of nonlinear partial differential equations, Science, New Technologies and Innovations of Kyrgyzstan, 12 (2019), 35–39.
A.Zh. Ashirbaeva, G.K. Sadykova, E. Mamaziaeva, Z. Mambetov, Construction of Solutions of Non-Linear Integro-Differential Equations of the Second Order with Derivative Under the Integral Sign, AIP Conference Proceedings, 3085 (2024), 020022.
A.Zh. Ashirbaeva, G.K. Sadykova, E. Mamaziaeva, Z. Mambetov, Solving of System of Partial Operator Equations of the First Order by the Method of Additional Argument, AIP Conference Proceedings, 3085 (2024), 020019.
S.N. Glazatov, Some problems for nonlinear pseudoparabolic equations in non-cylindrical regions, Siberian Math. Journal, 48(2) (2007), 272–289.
M.I. Imanaliev, S.N. Alekseenko, On the theory of nonlinear integro-differential partial differential equations of Whitham type, Doklady Rossiiskoi Akademii Nauk, 323(7) (1992), 410–414.
M.I. Imanaliev, S.N. Alekseenko, On the theory of systems of nonlinear integro-differential equations of Whitham type, Reports AN, 325(6) (1992), 1111–1115.
M.I. Imanaliev, P.S. Pankov, T.M. Imanaliev, The method of additional argument in the theory of nonlinear wave partial differential equations, Doklady Rossiiskoi Akademii Nauk, 343(5) (1995), 596–598.
E.A. Mamaziaeva, Construction of solutions of nonlinear integro-differential equations of the second order with a partial derivative under the integral sign, Research on Integro-Differential Equations, 47 (2014), 44–48.
A.D. Polyanin, V.F. Zeitsev, A.I. Zhurov, Methods of solving nonlinear equations in mathematical physics and mechanics, Moscow: Fizmatlit, (2005).
K.B. Sabitov, On the theory of equations of mixed type, Moscow: Fizmatlit, (2014).
G.K. Sadykova, Constructing a solution to a system of nonlinear integro-differential equations in partial derivatives, Science, New Technologies and Innovations of Kyrgyzstan, 7 (2021), 10–13.
G.K. Sadykova, Study of the solution of one system of nonlinear partial differential equations of the first order, News of Universities of Kyrgyzstan, 11 (2019), 15–19.
M.M. Smirnov, Model equations of the fourth order of the mixed type, Leningrad: LenGU, (1972).
A. Sopuev, N.K. Arkabaev, Conjugation problems for linear pseudoparabolic equations of the third order, Vestnik TomGU. Mathematics. Mechanics, 21 (2013), 16–23.
A.N. Tychonov, A.A. Samarskii, Equations of mathematical physics, Moscow: Nauka, (1977).
F. Tricomi, Lectures on equations in partial derivatives, Moscow: Foreign English Publishing House, (1957).
J.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons (Wiley-Interscience), New York, (1974).
Published
Data Availability Statement
No datasets were generated or analysed during the current study.
Issue
Section
License
Copyright (c) 2026 A.Zh. Ashirbaeva, C.B. Zholdoshova, G.K. Sadykova (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Articles published in Journal of Modern Mathematical Studies are distributed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Authors retain copyright and grant the journal the right of first publication.
