A Linearized Compact ADI Scheme for Semilinear Parabolic Problems with Distributed Delay

Linearized compact difference schemes applied to nonlinear variable coefficient parabolic equations with distributed delay

“…which is an important model in ecology and population dynamics [7,13,25]. In the following, we set the parameters 𝜀 = r = 1 and apply Methods I-II with Δt = 4h 2 to solve problem (63).…”

“…These numerical results further verify the computational effectiveness of Method I and Method II and the theoretical results stated in Theorem 3.5 and 3.6.Example This example considers the single‐species population model with $$$ \varepsilon $$$ and $$$ r $$$ being two positive constants $$$$ \left\{\begin{array}{c}{u}_t-\varepsilon {u}_{xx}= ru\left(1-{\int}_{t-1}^tu\left(x,s\right) ds\right),\kern0.75em \left(x,t\right)\in \left(0,1\right)\times \left(0,1\right],\\ {}u\left(x,t\right)=\sin \left(\pi x\right),\kern0.75em \left(x,t\right)\in \left[0,1\right]\times \left[-1,0\right],\\ {}u\left(0,t\right)=0,\kern0.5em u\left(1,t\right)=0,\kern0.75em t\in \left(0,1\right],\end{array}\right. $$$$ which is an important model in ecology and population dynamics [7, 13, 25]. In the following, we set the parameters $$$ \vareps...$…”

“…For solving numerically partial differential equations with distributed delay, He et al [8] studied one‐dimensional semi‐linear parabolic problems by using the linearized compact scheme. Subsequently, Qin et al [13] further applied the linearized compact scheme to solve two and three‐dimensional semi‐linear parabolic problems with distributed delay.…”

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

“…Soon afterwards, Zhang et al [27][28][29][30] extended this type of investigation to nonlinear delay convection-reaction-diffusion equations, delay parabolic equations with variable coefficient, 2D semilinear multidelay parabolic equations and non-Fickian delay reaction-diffusion equations. Moreover, Deng [31] derived a compact multistep method combined with the Richardson extrapolation for 2D nonlinear delay reaction-diffusion equations, Xie and Zhang [32] considered the error estimate of a high-order compact multistep ADI method for 2D nonlinear delay reaction-diffusion equations with variable coefficients, and Qin et al [33] studied linearized compact ADI methods for semilinear parabolic problems with distributed delay.…”

Solvability for a Class of Integro-Differential Inclusions Subject to Impulses on the Half-Line

“…Currently, many researchers around the world are investigating the semilinear integrodifferential equations or inclusions, as witnessed, for example, by the recent articles [1][2][3][4][5][6][7][8]. One of the main reasons for this research is that these equations are well suited to serve as a model for real phenomena such as heat transfer or the spread of epidemics or population dynamics, in which it is significant to take into account the spatial diffusion of the phenomenon or the past of the phenomenon itself (e.g., [9,10]).…”