In this thesis, ordinary and partial differential equations with piecewise constant arguments are solved and the qualitative behaviors of solutions are examined. In addition, the solutions of ordinary and partial differential equations with generalized piecewise constant argument and the qualitative behaviors of these solutions are handled. The obtained solutions are graphed using the MATLAB package program and added to the thesis in the form of figures. During these investigations, ordinary differential equations with generalized piecewise constant arguments are solved using the Laplace transform method. In addition, partial differential equations with generalized piecewise constant arguments are solved by Laplace transform method as well. The obtained theoretical results are graphed numerically. These studies are presented in the form of six chapters. In the first chapter, general information about differential equations and differential equations with deviation arguments is given and definitions for the stability of solutions are expressed. In addition, studies related to these subjects that exist in the literature have been addressed. In the sequel, the mass-spring system, a mechanical system in which theoretical results will be applied, is introduced and the parameters of the system are explained. Finally, general information about partial differential equations with piecewise constant arguments is given and the related studies in the literature have been expressed. In the second chapter, the keystone resources used during the study are given briefly. In the next section, the theoretical results given in the previous chapters are applied to the first order ordinary differential equations with piecewise constant argument and with generalized piecewise constant argument and the results are presented in terms of parameters. It is stated in this chapter that in which cases the solutions of the handled differential equations are stable, asymptotically stable, unstable and oscillatory. Validity of these results are supported by numerical simulations through graphing for different values of the parameters. As the fourth chapter, the application of the theoretical results given in the previous chapters to the mass-spring system with a delay effect in the form of a generalized piecewise constant argument and the results depending on the parameters are presented. With these results, it is stated in this section in which cases the solutions of the system are stable, asymptotically stable, unstable and oscillatory. The accuracy of these results has been supported via numerical simulation by graphing for different values of the parameters. In the fifth chapter, the heat equation and a neutral type partial differential equation are discussed together with generalized piecewise constant arguments. The cases in which the solutions of equations are bounded, unbounded, convergent, stable, asymptotically stable, unstable, oscillatory have been examined. In the last chapter, the theoretical and practical results obtained in the third, fourth and fifth chapters are discussed in general. Keywords: Ordinary differential equations, Partial differential equations, Laplace transform, Piecewise constant argument, Mass-spring system, Stability, Boundedness, Oscillation.
Tez (Doktora-PhD) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2022.
Kaynakça var.
In this thesis, ordinary and partial differential equations with piecewise constant arguments are solved and the qualitative behaviors of solutions are examined. In addition, the solutions of ordinary and partial differential equations with generalized piecewise constant argument and the qualitative behaviors of these solutions are handled. The obtained solutions are graphed using the MATLAB package program and added to the thesis in the form of figures. During these investigations, ordinary differential equations with generalized piecewise constant arguments are solved using the Laplace transform method. In addition, partial differential equations with generalized piecewise constant arguments are solved by Laplace transform method as well. The obtained theoretical results are graphed numerically. These studies are presented in the form of six chapters. In the first chapter, general information about differential equations and differential equations with deviation arguments is given and definitions for the stability of solutions are expressed. In addition, studies related to these subjects that exist in the literature have been addressed. In the sequel, the mass-spring system, a mechanical system in which theoretical results will be applied, is introduced and the parameters of the system are explained. Finally, general information about partial differential equations with piecewise constant arguments is given and the related studies in the literature have been expressed. In the second chapter, the keystone resources used during the study are given briefly. In the next section, the theoretical results given in the previous chapters are applied to the first order ordinary differential equations with piecewise constant argument and with generalized piecewise constant argument and the results are presented in terms of parameters. It is stated in this chapter that in which cases the solutions of the handled differential equations are stable, asymptotically stable, unstable and oscillatory. Validity of these results are supported by numerical simulations through graphing for different values of the parameters. As the fourth chapter, the application of the theoretical results given in the previous chapters to the mass-spring system with a delay effect in the form of a generalized piecewise constant argument and the results depending on the parameters are presented. With these results, it is stated in this section in which cases the solutions of the system are stable, asymptotically stable, unstable and oscillatory. The accuracy of these results has been supported via numerical simulation by graphing for different values of the parameters. In the fifth chapter, the heat equation and a neutral type partial differential equation are discussed together with generalized piecewise constant arguments. The cases in which the solutions of equations are bounded, unbounded, convergent, stable, asymptotically stable, unstable, oscillatory have been examined. In the last chapter, the theoretical and practical results obtained in the third, fourth and fifth chapters are discussed in general. Keywords: Ordinary differential equations, Partial differential equations, Laplace transform, Piecewise constant argument, Mass-spring system, Stability, Boundedness, Oscillation.