zbMATH — the first resource for mathematics

Generalizations to some integro-differential equations embodying powers of a differential operator. (Russian. English summary) Zbl 07293586
Summary: The abstract equations containing the operators of the second, third and fourth degree are investigated in this work.
The necessary conditions for the solvability of the abstract equations, containing the operators of the second and fourth degree, are proved without using linear independence of the vectors included in these equations. Previous authors have essentially used the linear independence of the vectors to prove the necessary solvability condition.
The present paper also gives the correctness criterion for the abstract equation, containing the operators of the third degree with arbitrary vectors, and its exact solution in terms of these vectors in a Banach space.
The theory presented here, can be useful for investigation of Fredholm integro-differential equations embodying powers of an ordinary differential operator or a partial differential operator.
47Axx General theory of linear operators
92Dxx Genetics and population dynamics
35Rxx Miscellaneous topics in partial differential equations
45Kxx Integro-partial differential equations
35Kxx Parabolic equations and parabolic systems
Full Text: DOI MNR
[1] Apreutesei N., Ducrot A., Volpert V., “Travelling waves for integro-differential equations in population dynamics”, Discrete and Continuous Dynamical Systems, Ser. B, 11:3 (2009), 541-561 (in English) · Zbl 1173.35541
[2] Baiburin M. M., Providas E., “Exact Solution to Systems of Linear First-Order Integro-Differential Equations with Multipoint and Integral Conditions”, Mathematical Analysis and Applications, Springer Optimization and Its Applications book series, 154, eds. Rassias T., Pardalos P., 2019, 1-16 (in English) · Zbl 1462.45010
[3] Bloom F., Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory, SIAM Studies in Applied Mathematics, Philadelphia, 1981, 231 pp. · Zbl 0465.45010
[4] Cushing J. M., Integrodifferential equations and delay models in population dynamics, Springer-Verlag, Berlin-Heidelberg, 1977 (in English) · Zbl 0363.92014
[5] Medlock J., Kot M., “Spreading disease: integro-differential equations old and new”, Mathematical Biosciences, 184, August (2003), 201-222 (in English) · Zbl 1036.92030
[6] Oinarov R. O., Parasidi I. N., “Correct extensions of operators with finite defect in Banach spases”, Izvestiya Akademii Nauk Kazakhskoi SSR, 5 (1988), 42-46 (in Russian) · Zbl 0661.47014
[7] Parasidis I. N., Providas E., “Integro-differential equations embodying powers of a differential operator”, Vestnik of Samara University. Natural Science Series, 25:3 (2019), 13-21 (in English)
[8] Parasidis I. N., Providas E., “On the Exact Solution of Nonlinear Integro-Differential Equations”, Applications of Nonlinear Analysis, 2018, 591-609 (in English) · Zbl 07008034
[9] Parasidis I. N., Tsekrekos P. C., Lokkas Th. G., “Correct and self-adjoint problems for biquadratic operators”, Journal of Mathematical Sciences, 166:2 (2010), 420-427 (in English) · Zbl 1402.47002
[10] Parasidis I. N., Providas E., “Extension Operator Method for the Exact Solution of Integro-Differential Equations”, Contributions in Mathematics and Engineering, eds. Pardalos P., Rassias T., Springer, Cham, 2016, 473-496 (in English) · Zbl 1405.45002
[11] Polyanin A. D., Zhurov A. I., “Exact solutions to some classes of nonlinear integral, integro-functional, and integro-differential equations”, Doklady Mathematics, 77 (2008), 315-319 (in English) · Zbl 1153.45007
[12] Sachs E. W., Strauss A. K., “Efficient solution of a partial integro-differential equation in finance”, Applied Numerical Mathematics, 58 (2008), 1687-1703 (in English) · Zbl 1155.65109
[13] Shishkin G. A., Linear Fredholm integro-differential equations, Buryat State University, Ulan-Ude, 2007 (in Russian)
[14] Shivanian E., “Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics”, Engineering Analysis with Boundary Elements, 37 (2003), 1693-1702 (in English) · Zbl 1287.65091
[15] Vassiliev N. N., Parasidis I. N., Providas E., “Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method”, Information and Control Systems, 2018, no. 6, 14-23 (in English)
[16] Vassiliev N. N., Parasidis I. N., Providas E., “Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators”, Information and Control Systems, 2019, no. 2, 2-9 (in English)
[17] Wazwaz A. M., Linear and nonlinear integral equations, methods and applications, Springer, Berlin-Heidelberg, 2011 (in English) · Zbl 1227.45002
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.