On a Partial Differential Equation with Piecewise Constant Mixed Arguments
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RESEARCH PAPER
On a Partial Differential Equation with Piecewise Constant Mixed Arguments Mehtap Lafci Bu¨yu¨kkahraman1
•
Hu¨seyin Bereketoglu2
Received: 31 May 2020 / Accepted: 31 August 2020 Ó Shiraz University 2020
Abstract So far, although there have been several articles on partial differential equations with piecewise constant arguments, as far as we know, there is no article on neither a heat equation with piecewise constant mixed arguments that includes three extra diffusion terms, delayed arguments ½t 1; ½t and an advanced argument ½t þ 1; or exploring qualitative properties of the equation. With the motivation to investigate elaborate and well-established qualitative properties of such an equation, in this paper, we deal with a problem involving a heat equation with piecewise constant mixed arguments and initial, boundary conditions. By using the separation of variables method, we obtain the formal solution of this problem. Because of the piecewise constant arguments, we get a differential equation and then a difference equation. With the help of qualitative properties of the solutions of the differential equation and with the behavior of the solutions of the difference equation, we investigate the existence of solutions and qualitative properties of the solutions of the problem such as the convergence of the solutions to zero, the unboundedness of the solutions and oscillations of them. In addition, two examples are given to illustrate the application of the results in particular cases. Keywords Partial differential equation Piecewise constant arguments Oscillation Stability Unbounded Mathematics Subject Classification 35B05 35B35 35K05
1 Introduction In recent years, ordinary differential equations with arguments having intervals of constancy have been studied. The studies of these equations were initiated by Cooke and Wiener (1984). Also, the qualitative works such as oscillation, periodicity and convergence of solutions of ordinary differential equations with piecewise constant arguments (EPCAs) have been considered some papers (Aftabizadeh and Wiener 1988; Aftabizadeh et al. 1987; Akhmet 2008; Busenberg and Cooke 1982; Gy}ori 1991; Gy}ori and Ladas 1989; Huang 1990; Liang and Wang 2009; Muroya 2008; Pinto 2009; Wiener 1993; Wiener and Debnath 1991; Yuan 2002). The general theory and basic results for EPCA have & Mehtap Lafci Bu¨yu¨kkahraman [email protected] 1
Faculty of Arts and Sciences, Usak University, 64200 Usak, Turkey
2
Department of Mathematics, Faculty of Science, Ankara University, 06100 Tandog˘an, Ankara, Turkey
by now been thoroughly investigated in the book (Wiener 1993). Such equations represent a hybrid of continuous and discrete dynamical systems and combine properties of both differential and difference equations. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the values of the solution at such points. Therefore, EPCA is intrinsically closer to dif
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