THE NUMERICAL SOLUTION OF SOME CLASSES OF THE SEMILINEAR SOBOLEV-TYPE EQUATIONS

Evgeniy Viktorovich Bychkov

Abstract


A unique solvability of the Cauchy problem for a class of semilinear Sobolev type equations of the second order is proved. The ideas and techniques, developed by G.A. Sviridyuk for the investigation of the Cauchy problem for a class of semilinear Sobolev type equations of the first order and by A.A. Zamyshlyaeva for the investigation of the high-order linear Sobolev type equations are used. We also used theory of the differential manifolds which was finally formed in S. Leng's works. In article we considered two cases. The first one is when an operator at the highest time derivative is continuously invertible. In this case for any point from a tangent bundle of an original Banach space there exists a unique solution lying in this space as trajectory. The second case when the operator isn't continuously invertible is of great interest for us. Hence we used the phase space method. Besides the Cauchy problem we considered the Showalter - Sidorov problem. The last generalizes the Cauchy problem and is more natural for Sobolev-type equation. In the last section described an algorithm of the numerical solution of Showalter - Sidorov problem for Sobolev-type equation of the second order.


Keywords


Sobolev-type equation, phase space, Showalter - Sidorov problem, algorithm of the numerical solution

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References


Sviriduyk G.A., Zagrebina S.A. Nonclassical Mathematical Physics Models. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming Computer Software", 2012, no.40(299), pp. 7-18. (in Russian)

SviridyukG.А., Sukacheva T.G. The Phase Space of a Class of Operator Equations of Sobolev Type]. Differential Equations, 1990, vol.26, no.2, pp.250-258. (in Russian)

Sviriduyk G.A., Zamyshlyaeva A.A. The Phase Space of a Class of Linear Higher-order Sobolev Type Equations. Differential Equations, 2006,vol.42, no.2, pp.269-278. mboxDOI: 10.1134/S0012266106020145

Zamyshlyaeva A.A., Bychkov E.V. The Phase Space of Modified Boussinesq Equation. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming Computer Software", 2012, no.18(277), issue12, pp.13-19. (in Russian)

Arkhipov D.G., Khabakhpashev G.A. The New Equation to Describe the Inelastic Interaction of Nonlinear Localized Waves in Dispersive Media. JTEP Letters, 2011, vol.93, no.8, pp.423-426. mboxDOI: 10.1134/S0021364011080042

Wang S., Chen G. Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Applications, 2002, vol.274, issue2, pp.846-866.

ManakovaN.А., Bogatyreva E.A. On a Solution of the Dirichlet - Cauchy Problem for the Barenblatt - Gilman Equation The Bulletin of Irkutsk State University. Series "Mathematics", 2014, vol.7, pp.52-60. (in Russian)

Nirenberg L. Topics in Nonlinear Functional Analysis. New York, New ed. (AMS), 2001.

Leng S. Introduction to Differentiable Manifolds. New York, Springer-Verlag, 2002.

Hassard B.D. Тheory and Applications of Hopf Bifurcation. Cambridge, Cambridge University Press, 1981.

Keller A.V. Algorithm of Numerical Solution of Shoulter - Sidorov Problem for Leontiev-type Systems. Bulliten of the South Ural State University. Series "Computer Technologies, automatic control radioelectronics", 2009, no.26(159), pp.82-86. (in Russian)


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