Non-Local Partial Differential Equations for Engineering and Biology

2018 ◽  
Author(s):  
Nikos I. Kavallaris ◽  
Takashi Suzuki
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Non Local

Some non-local logistic population model with non-zero boundary condition

2018 ◽  
Vol 20 (08) ◽  
pp. 1750075 ◽  
Author(s):  
Patricio Cerda ◽  
Marco Souto ◽  
Pedro Ubilla
Keyword(s):  
Boundary Condition ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Population Model ◽  
A Priori ◽  
Partial Differential ◽  
Non Local

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions.

scholarly journals Stochastic partial differential equations driven by space-time fractional noises

2019 ◽  
Vol 19 (02) ◽  
pp. 1950012
Author(s):  
Ying Hu ◽  
Yiming Jiang ◽  
Zhongmin Qian
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Space Time ◽  
Partial Differential ◽  
The Family ◽  
Non Local ◽  
Fractional Noises

In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying at first the non-local SPDEs and thereafter showing the convergence of the family of these equations. The limit gives the solution of the SPDE.

scholarly journals Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation

2020 ◽  
Vol 82 (10) ◽  
Author(s):  
P. Aceves-Sanchez ◽  
P. Degond ◽  
E. E. Keaveny ◽  
A. Manhart ◽  
S. Merino-Aceituno ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Scale ◽  
Coupled System ◽  
Model Parameters ◽  
Partial Differential ◽  
Pattern Size ◽  
Non Local ◽  
Model Derivation ◽  
Macroscopic Equations

Abstract We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of nonlinear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.

scholarly journals Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

2020 ◽  
Vol 21 (01) ◽  
pp. 2150004
Author(s):  
Hanxiao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Parabolic Equations ◽  
Natural Extension ◽  
Volterra Integral Equations ◽  
Degenerate Parabolic Equations ◽  
Partial Differential ◽  
Non Local ◽  
Quasilinear Parabolic

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

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