Search Results (3)

The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples. Full article

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown $u=u(x,t)$, also contain the same functions with dilated or contracted arguments of the form $w=u(px,t)$, $w=u(x,qt)$, and $w=u(px,qt)$, where p and q are the free scaling parameters (for equations with proportional delay we have $0<p<1$, $0<q<1$). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations). Full article

An effective algorithm for solving quadratic Riccati differential equation (QRDE), multipantograph delay differential equations (MPDDEs), and optimal control systems (OCSs) with pantograph delays is presented in this paper. This technique is based on Genocchi polynomials (GPs). The properties of Genocchi polynomials are stated, and operational matrices of derivative are constructed. A collocation method based on this operational matrix is used. The findings show that the technique is accurate and simple to use. Full article