Abstract
Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Nowacki, Theory of Asymmetric Elasticity (Pergamon, Oxford, 1986; Mir, Moscow, 1975).
V. S. Vladimirov, Equations of Mathematical Physics (Fizmatlit, Moscow, 1971; Imported Publ., 1985).
V. I. Gorbachev, “Green’s tensor method for solving boundary value problems of the theory of elasticity of inhomogeneous bodies,” Vychisl. Mekh. Deform. Tverd. Tela, No. 2, 61–76 (1991).
E. I. Grigolyuk and A. A. Fil’shtinskii, Perforated Plates and Shells (Nauka, Moscow, 1970) [in Russian].
E. Sanchez-Palensija, Non-Homogeneous Media and the Theory of Oscillations (Springer, New York, 1980).
V. I. Gorbachev, “On the effective elasticity coefficients of a heterogeneous body,” Izv. Akad. Nauk, Mekh. Deform. Tverd. Tela, No. 4, 114–125 (2018).
L. V. Olekhova, “Torsion of heterogeneous anisotropic core,” Cand. Sci. (Phys.-Math.) Dissertation (Mosc. State Univ., Moscow, 2009).
V. I. Gorbachev and O. B. Moskalenko, “Stability of a straight bar of variable rigidity,” Mech. Solids 46, 645–655 2011.
V. I. Gorbachev and O. B. Moskalenko, “Stability of bars with variable rigidity,” Mosc. Univ. Mech. Bull. 65, 147–150 2010.
V. I. Gorbachev and O. B. Moskalenko, “Stability of bars with variable rigidity compressed by a distributed force,” Mosc. Univ. Mech. Bull. 67, 5–10 2012.
V. I. Gorbachev, “Oscillations in a non-uniform elastic body,” in Elasticity and Inelasticity, Proceedings of the International Symposium on Problems of Mechanics of Deformable Bodies, Dedicated to the 100th Anniversary of A. A. Ilyushin, Moscow, Jan. 20–21, 2011 (Mosk. Gos. Univ., Moscow, 2011), pp. 319–326.
V. I. Gorbachev, “Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod,” Mosc. Univ. Mech. Bull. 71, 7–15 2016.
V. I. Gorbachev, “Heat propagation in a nonuniform rod of variable cross section,” Mosc. Univ. Mech. Bull. 72 (2), 48–53 (2017).
B. E. Pobedrja and V. I. Gorbachev, “Stress and strain concentrations in composites,” Mekh. Kompoz. Mater. 20, 207–214 1984.
V. I. Gorbachev, “Operators of stress and strain in composites,” in Strength Calculations, Collection of Articles (Mashinostroenie, Moscow, 1989), Vol.30, pp. 124–130 [in Russian].
V. I. Gorbachev and A. L. Mikhaylov, “Stress concentration tensor for the case of n-dimensional elastic space with spherical inclusion,” Vestn. Mosk. Univ., Mekh., No. 2, 78–83 (1993).
V. I. Gorbachev and B. E. Pobedrja, “On some criteria for the destruction of composites,” Izv. Akad. Nauk Arm. SSR, No. 4, 30–37 (1985).
B. E. Pobedrja and V. I. Gorbachev, “Strength criteria for laminated and fiber composites,” Probl. Mashinostr. Avtomatiz., No. 21, 65–68 (1988).
V. I. Gorbachev, “Engineering resistance theory of heterogeneous rods made of composite materials,” Vestn. MGTU im. N. E. Baumana, No. 6, 56–72 (2016).
V. I. Gorbachev and T. M. Melnik, “Formulation of problems in the Bernoulli-Euler theory of anisotropic inhomogeneous beams,” Mosc. Univ. Mech. Bull. 73, 18–26 2018.
V. I. Gorbachev, “Engineering theory of inhomogeneous plates made of composite materials,” Mekh. Kompoz. Mater. Konstrukts. 22, 585–601 2016.
V. I. Gorbachev and L. A. Kabanova, “Formulation of problems in the general Kirchhoff-Love theory of inhomogeneous anisotropic plates,” Mosc. Univ. Mech. Bull. 73 (3), 60–66 (2018).
V. I. Gorbachev, “Integral formulas of the main linear differential equations of mathematical physics with variable coefficients,” Chebyshev. Sb. 18, 138–160 (2017).
L. I. Sedov, Mechanics of Continuous Media (Nauka, Moscow, 1970), Vol. 2 [in Russian].
N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media (Nauka, Moscow, 1984) [in Russian].
B. E. Pobedrya, Mechanics of Composite Materials (Mosk. Gos. Univ., Moscow, 1984) [in Russian].
Funding
The work was performed by the Federal State Budgetary Higher Professional Educational Institution L.N. Tolstoy Tula State Pedagogical University with financial support from the Ministry of Education and Science of the Russian Federation (projects 14.577.21.0271, ID RFMEFI57717X0271). The L.N. Tolstoy Tula State Pedagogical University is the recipient of the subsidy from the Ministry of Education and Science of the Russian Federation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gorbachev, V.I. About one Approach to a Solution of Linear Differential Equations with Variable Coefficients. Lobachevskii J Math 40, 969–980 (2019). https://doi.org/10.1134/S1995080219070126
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080219070126