arXiv subjects

Yoshitaro Tanaka

Publications and source records attributed to Yoshitaro Tanaka.

7 recordsLinked to original sources

arXiv2025

Reaction, diffusion and nonlocal interactions in high-dimensional space

In this paper we consider the mathematical relationship between nonlocal interactions of convolution type and multiple diffusive substances in high dimensions. Motivated by that the nonlocal evolution equations reproduce similar patterns to those in reaction-diffusion systems, we approximate nonlocal interactions in evolution equations by the solution to a reaction-diffusion system in any dimensional Euclidean space. The key aspect of this approach is that any absolutely integrable radial kernels can be approximated by a linear combination of specific Green functions. This enables us to demonstrate that any nonlocal interactions of convolution type can be approximated by a linear sum of auxiliary diffusive substances. Moreover, we show that the parameters in the reaction-diffusion system can be specified depending on the kernel shape up to three dimensions. Our results establish a connection between a broad class of nonlocal interactions and diffusive chemical reactions in dynamical systems.

math.AP↗

arXiv2025

On the approximation of spatial convolutions by PDE systems

This paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can eliminate the analytical difficulties arising from integral formulations in one-dimensional space. In this paper, we establish a PDE system approximation for spatial convolutions in higher spatial dimensions. We derive an appropriate approximation function for given arbitrary radial integral kernels as a linear sum of Green functions. In establishing the validity of this methodology, we introduce an appropriate integral transformation to show the completeness of the basis constructed by the Green functions. This framework enables the approximation of nonlocal convolution-type operators with arbitrary radial integral kernels using linear sums of PDE solutions. Finally, we present numerical examples that illustrate the effectiveness of our proposed method.

math.AP↗

arXiv2025

Relationship between haptotaxis and chemotaxis in cell dynamics

Cell sorting mediated by direct cell--cell contact or cellular protrusions can be described by nonlocal cell--cell adhesion models of haptotaxis type, whereas communication through diffusible chemical signals is described by Keller--Segel type chemotaxis systems. We investigate the mathematical relationship between these two descriptions. We first prove that, in the fast signal diffusion limit, subsequences of weak solutions of a Keller--Segel type parabolic--parabolic chemotaxis system with possibly degenerate nonlinear diffusion and density saturation converge to weak solutions of the corresponding parabolic--elliptic system. Eliminating the elliptic chemical fields rewrites the population equation as a nonlocal adhesion model whose interaction kernel is a finite linear combination of Green functions. We then prove, in arbitrary spatial dimensions, that the periodization of the gradient of a prescribed radially symmetric interaction potential can be approximated by finite linear combinations of gradients of Green functions. Combining these results, we prove that suitable weak solutions of the nonlocal adhesion model and a parabolic--parabolic chemotaxis system can be chosen with an arbitrarily small difference over any fixed finite time interval. Numerical simulations compare the nonlocal, parabolic--elliptic, and parabolic--parabolic models and examine how the number of kernel terms and the relaxation time affect their differences.

math.AP↗

arXiv2024

Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analyzing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker-Planck equation can be approximated using the Keller-Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker-Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller-Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker-Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

math.AP↗

arXiv2023

Existence of spiky stationary solutions to a mass-conserved reaction-diffusion model

We deal with a mass-conserved three-component reaction-diffusion system which is proposed by a model describing the dynamics of wavelike actin polymerization in the macropinocytosis and numerically exhibits dynamical patterns such as annihilation, crossover, and nucleation of pulses (Yochelis-Beta-Giv 2020). In this article we first establish the condition for the diffusion driven instability in the system. Then we rigorously prove the existence of spiky stationary solutions to the system in a bounded interval with the Neumann condition. By numerics these solutions play a crucial role in the nucleation of pulses. Reducing the stationary problem to a scalar second order nonlinear equation with a nonlocal term, we construct the desired solution by converting the equation to an integral equation.

math.AP↗

arXiv2020

Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains

The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying Green's theorem to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, $N^2a^N$ order, where $a$ is a positive constant less than one and $N$ is the number of collocation points. Furthermore, it is demonstrated that the error tends to $0$ in exponential order in the numerical simulations with increasing number of collocation points $N$.

math.NA↗

arXiv2020

Effective nonlocal kernels on Reaction-diffusion networks

A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called "effective equation" including the reduced integral kernel (called "effective kernel" ) in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations.

math.AP↗