On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations

The initial value problem for the third order delay differential equation in a Hilbert space with an unbounded operator is investigated. The absolute stable three-step difference scheme of a first order of accuracy is constructed and analyzed. This difference scheme is built on the Taylor’s decomposition method on three and two points. The theorem on the stability of the presented difference scheme is proven. In practice, stability estimates for the solutions of three-step difference schemes for different types of delay partial differential equations are obtained. Finally, in order to ensure the coincidence between experimental and theoretical results and to clarify how efficient the proposed scheme is, some numerical experiments are tested.

[1]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[2]  Eugenio Sinestrari,et al.  On a class of retarded partial differential equations , 1984 .

[3]  S. Grace.  Oscillation criteria for third order nonlinear delay differential equations with damping , 2015 .

[4]  Ping Li,et al.  Numerical Algorithm for the Third-Order Partial Differential Equation with Three-Point Boundary Value Problem , 2014 .

[5]  Yu. P. Apakov,et al.  Boundary-Value Problem for a Degenerate High-Odd-Order Equation , 2015 .

[6]  P. Ricciardi,et al.  Existence and regularity for linear delay partial differential equations , 1980 .

[7]  Blanka Baculíková,et al.  Oscillation of third order trinomial delay differential equations , 2012, Appl. Math. Comput..

[8]  Yusufjon P. Apakov,et al.  On a boundary value problem to third order PDE with multiple characteristics , 2011 .

[9]  Baruch Cahlon,et al.  Stability criteria for certain third-order delay differential equations , 2006 .

[10]  On the solution of a boundary-value problem for a third-order equation with multiple characteristics , 2012 .

[11]  Deniz Agirseven,et al.  Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition , 2012 .

[12]  Allaberen Ashyralyev,et al.  On convergence of difference schemes for delay parabolic equations , 2013, Comput. Math. Appl..

[13]  Muhammet Köksal,et al.  Taylor's decomposition on four points for solving third-order linear time-varying systems , 2009, J. Frankl. Inst..

[14]  G A Pikina.  Predictive time optimal algorithm for a third-order dynamical system with delay , 2017 .

[15]  Stabilization of third-order differential equation by delay distributed feedback control , 2018, Journal of inequalities and applications.

[16]  A. Ashyralyev,et al.  Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations , 2019, Mathematics.

[17]  Allaberen Ashyralyev,et al.  Well-posedness of delay parabolic difference equations , 2014 .

[18]  Gabriella Di Blasio,et al.  Delay differential equations with unbounded operators acting on delay terms , 2003 .

[19]  C. Latrous,et al.  A three-point boundary value problem with an integral condition for a third-order partial differential equation , 2005 .

[20]  A. Ashyralyev,et al.  A numerical algorithm for the third-order partial differential equation with time delay , 2019, THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019).

[21]  S. M. Shah,et al.  On the exponential growth of solutions to non-linear hyperbolic equations , 1989 .

[22]  Doghonay Arjmand,et al.  A note on the Taylor's decomposition on four points for a third-order differential equation , 2007, Appl. Math. Comput..

[23]  Method of Lines for Third Order Partial Differential Equations , 2014 .