First order partial differential equations with time delay and retardation of a state variable

Solodushkin, S. I.; Iumanova, Irina; Staelen, Rob De

doi:10.1016/j.cam.2014.12.032

Cited by 20 publications

(11 citation statements)

References 14 publications

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“…In this section, we present some numerical examples to illustrate the efficiency and high accuracy of the proposed multistep LGR collocation method. All computations were performed using the package of Mathematica 12.Example Consider the following first order partial differential equation with discrete time delay

\tau =\frac{\pi }{2}

and retardation of a state variable with biasing constant

\alpha =\frac{1}{2}

[1, 23]:

\frac{\partial u}{\partial t}\left(x,t\right)+2\frac{\partial u}{\partial x}\left(x,t\right)=u\left(x,t\right)-{e}^{\frac{x}{2}}u\left(\frac{x}{2},t-\frac{\pi }{2}\right)+{e}^x\sin t,\kern1.6em 0<x<2,\kern1.6em 0<t\le 2\pi,

with the initial function

u\left(x,t\right)={e}^x\sin t,\kern1.6em 0\le x\le 2,\kern1.6em -\frac{\pi }{2}\le t\le 0,

and the boundary condition

u\left(0,t\right)=\sin t,\kern1.6em 0\le t\le 2\pi .

…”

Section: Numerical Examplesmentioning

confidence: 99%

A multistep collocation method for solving advection equations with delay

Sadeghi

Maleki

Kamali

et al. 2022

Numerical Methods Partial

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Advection equations with delay are appeared in the modeling of the dynamics of structured cell populations. In this article, we construct an efficient two‐dimensional multistep collocation method for the numerical solution of a class of advection equations with delay. Equations with aftereffect and equations with both aftereffect and retardation of a state variable are considered. Computability of the algorithm and convergence properties of the proposed numerical method are analyzed for solutions in appropriate Sobolev spaces, and it is shown that the proposed scheme enjoys the spectral accuracy. Numerical examples are given and comparison with other existing methods in the literature is made to demonstrate the efficiency, superiority and high accuracy of the presented method.

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\tau =\frac{\pi }{2}

and retardation of a state variable with biasing constant

\alpha =\frac{1}{2}

[1, 23]:

\frac{\partial u}{\partial t}\left(x,t\right)+2\frac{\partial u}{\partial x}\left(x,t\right)=u\left(x,t\right)-{e}^{\frac{x}{2}}u\left(\frac{x}{2},t-\frac{\pi }{2}\right)+{e}^x\sin t,\kern1.6em 0<x<2,\kern1.6em 0<t\le 2\pi,

with the initial function

u\left(x,t\right)={e}^x\sin t,\kern1.6em 0\le x\le 2,\kern1.6em -\frac{\pi }{2}\le t\le 0,

and the boundary condition

u\left(0,t\right)=\sin t,\kern1.6em 0\le t\le 2\pi .

…”

Section: Numerical Examplesmentioning

confidence: 99%

A multistep collocation method for solving advection equations with delay

Sadeghi

Maleki

Kamali

et al. 2022

Numerical Methods Partial

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“…Analytical methods for solving some linear and nonlinear PDEs with proportional delays are discussed in [86][87][88]. In [89], a finite-difference scheme for the numerical integration of firstorder PDEs with constant delay in t and proportional delay in x is constructed. Papers [90][91][92] are devoted to numerical methods for solving pantograph-type PDEs with proportional delay [90,91] and more complex varying delay [92].…”

Section: Pantograph-type Odes and Pdes And Their Applicationsmentioning

confidence: 99%

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Polyanin

Сорокин

2021

Mathematics

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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).

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“…The literature on numerically solving PDDE with non-local boundary conditions is rather limited [25]. However, the problem structure permits to use the cohort strategy presented in [26] to discretize the solution space (Fig.…”

Section: Mathematical Frameworkmentioning

confidence: 99%

Stochastic Process for the Availability Assessment of Single-Feeder Industrial Energy System Sections

Acker

Hertem

2018

IEEE Trans. Rel.

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Given the high interruption costs of industrial processes, dependent on the cause, duration and time, accurate availability assessment of the industrial energy systems supplying those processes is crucial in order to avoid unnecessary investments or indicate necessary improvements. A method is presented to sectionalize an industrial energy system based on its radial topology and recoverability options. Additionally, this paper introduces a novel non-renewal stochastic process able to compute the (un)availability of single-feeder sections with respect to cause, duration and time, taking into account competing risks, general transition rates and both minimal and perfect corrective maintenance. An accompanying solution strategy is developed which allows decoupling of the very diverse time constants. Finally, a numerical illustration is provided validating the proposed approach against the Markov and semi-Markov processes under their respective assumptions and illustrating the possible errors when modeling the problem using the aforementioned stochastic processes.

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First order partial differential equations with time delay and retardation of a state variable

Cited by 20 publications

References 14 publications

A multistep collocation method for solving advection equations with delay

A multistep collocation method for solving advection equations with delay

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Stochastic Process for the Availability Assessment of Single-Feeder Industrial Energy System Sections

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