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Fig. 5 Spiral waves in the FitzHugh-Nagumo model for varying a where we consider the following choice of model parameters, a = 0.1, € = 0.01, B = 0.5, y = 1, 5 = 0, which is known to generate stable patterns in the system in the form of re-entrant spiral waves. In our simulations, the trivial state (vu, v) = (0, 0) was perturbed by setting the lower-left quarter of the domain to u = | and the upper half part to v = 0.1, which allows the initial condition to curve and rotate clockwise generating the spiral pattern. The domain is taken to be [0, 2.5]”, discretised using N = 256 points in each spatial coordinate, with a diffusion coefficient K, = 107+. Stable rotating solutions at t = 2,000 are presented in Fig. 5 to illustrate the effect of fractional diffusion in the FitzHugh-Nagumo model. The width of the excitation wavefront (red areas) is markedly reduced for decreasing a, so is the wavelength of the system, with the domain being able to accommodate a larger number of wavefronts for smaller a.

Figure 5 Spiral waves in the FitzHugh-Nagumo model for varying a where we consider the following choice of model parameters, a = 0.1, € = 0.01, B = 0.5, y = 1, 5 = 0, which is known to generate stable patterns in the system in the form of re-entrant spiral waves. In our simulations, the trivial state (vu, v) = (0, 0) was perturbed by setting the lower-left quarter of the domain to u = | and the upper half part to v = 0.1, which allows the initial condition to curve and rotate clockwise generating the spiral pattern. The domain is taken to be [0, 2.5]”, discretised using N = 256 points in each spatial coordinate, with a diffusion coefficient K, = 107+. Stable rotating solutions at t = 2,000 are presented in Fig. 5 to illustrate the effect of fractional diffusion in the FitzHugh-Nagumo model. The width of the excitation wavefront (red areas) is markedly reduced for decreasing a, so is the wavelength of the system, with the domain being able to accommodate a larger number of wavefronts for smaller a.

Related Figures (11)

Fig. 1 Selection of appropriate internal nodes for mesh discretisation for homogeneous Dirichlet (/eff) and homogeneous Neumann (right) boundary conditions, in order to ensure continuity of the periodic extension of the solutions at the discrete level (dashed lines)
Fig. 1 Selection of appropriate internal nodes for mesh discretisation for homogeneous Dirichlet (/eff) and homogeneous Neumann (right) boundary conditions, in order to ensure continuity of the periodic extension of the solutions at the discrete level (dashed lines)
Fig. 2 Convergence results for the fractional heat equation in one space dimension for Fourier (solid lines) and finite differences (dashed lines) methods. A value of k = 25 was used in both cases. Right panel illustrates evolution of initial data (dashed black lines) at t = 0.1, for K = | and varying a OL (2.1) LOT Valyins Values OL UIC IWaCuONdl DOWCT &. Convergence results in the €°°-norm are presented in Fig. 2 for the Fourier anc finite differences approximations of (2.1) in x € [—1, 1] att = 0.1, with K = 1 Two different initial conditions were considered: a smooth Gaussian profile, subjec to homogeneous Dirichlet conditions (Fig. 2a); and a sigmoid exhibiting sharper gradi. ents, with homogeneous Neumann data (Fig. 2b). Reference solutions were calculatec by evaluating (2.4) with 2!* Fourier modes, with coefficients ui ; computed by adaptive Gauss-Kronrod quadrature. In both situations, the Fourier approach is able to achieve
Fig. 2 Convergence results for the fractional heat equation in one space dimension for Fourier (solid lines) and finite differences (dashed lines) methods. A value of k = 25 was used in both cases. Right panel illustrates evolution of initial data (dashed black lines) at t = 0.1, for K = | and varying a OL (2.1) LOT Valyins Values OL UIC IWaCuONdl DOWCT &. Convergence results in the €°°-norm are presented in Fig. 2 for the Fourier anc finite differences approximations of (2.1) in x € [—1, 1] att = 0.1, with K = 1 Two different initial conditions were considered: a smooth Gaussian profile, subjec to homogeneous Dirichlet conditions (Fig. 2a); and a sigmoid exhibiting sharper gradi. ents, with homogeneous Neumann data (Fig. 2b). Reference solutions were calculatec by evaluating (2.4) with 2!* Fourier modes, with coefficients ui ; computed by adaptive Gauss-Kronrod quadrature. In both situations, the Fourier approach is able to achieve
Results are independent of a Table 1 Timing results (seconds) for solving Eq. (2.1) in one space dimension for different discretisations
Results are independent of a Table 1 Timing results (seconds) for solving Eq. (2.1) in one space dimension for different discretisations
N indicates points in each domain direction. Results are independent on a Table 2 Timing results (seconds) for solving Eq. (2.1) in two space dimensions for different discretisations
N indicates points in each domain direction. Results are independent on a Table 2 Timing results (seconds) for solving Eq. (2.1) in two space dimensions for different discretisations
Fig. 3_ Numerical solutions of the fractional heat equation in two space dimensions at t = 0.1, for K = 1 and varying a. The right panel illustrates convergence results for the Fourier method (solid lines) compared to finite differences (dashed lines). N indicates the number of discretisation points in each domain direction
Fig. 3_ Numerical solutions of the fractional heat equation in two space dimensions at t = 0.1, for K = 1 and varying a. The right panel illustrates convergence results for the Fourier method (solid lines) compared to finite differences (dashed lines). N indicates the number of discretisation points in each domain direction
conditions). Two-dimensional Sine Transforms (dst2/idst2) as indicated in Code 3. Code 4 Fractional heat equation in 2D with source term (homogeneous Dirichlet boundary — << ee ee ee ee ee Oe ee ee re An additional advantage of the proposed time-space stencil is the fully impli reatment of the fractional Laplacian, resulting in an unconditionally stable scheme erms of the non-local operator. This can be verified in (2.10) in the absence of reacti ources, since for f;(u, tf) = 0 all Fourier coefficients monotonically decay to zero — oo (with the exception of the zero-frequency mode associated to A; = Oint ‘ase of homogeneous Neumann boundary cond yossible time step restrictions will therefore be r itions, which remains constant). Ai elated only to the explicit treatment ource terms. Hence, the convergence of the proposed fixed point iteration is depende Ipon the time step, although in practice, even w hen using reasonably large time ster onvergence was always experienced. Alternative linearisations of these terms a idapted time stepping may be constructed to improve both the time step restricti ind the convergence rates of the ones presented ; this constitutes ongoing research.
conditions). Two-dimensional Sine Transforms (dst2/idst2) as indicated in Code 3. Code 4 Fractional heat equation in 2D with source term (homogeneous Dirichlet boundary — << ee ee ee ee ee Oe ee ee re An additional advantage of the proposed time-space stencil is the fully impli reatment of the fractional Laplacian, resulting in an unconditionally stable scheme erms of the non-local operator. This can be verified in (2.10) in the absence of reacti ources, since for f;(u, tf) = 0 all Fourier coefficients monotonically decay to zero — oo (with the exception of the zero-frequency mode associated to A; = Oint ‘ase of homogeneous Neumann boundary cond yossible time step restrictions will therefore be r itions, which remains constant). Ai elated only to the explicit treatment ource terms. Hence, the convergence of the proposed fixed point iteration is depende Ipon the time step, although in practice, even w hen using reasonably large time ster onvergence was always experienced. Alternative linearisations of these terms a idapted time stepping may be constructed to improve both the time step restricti ind the convergence rates of the ones presented ; this constitutes ongoing research.
Table 3. Time convergence in the solution of the fractional heat Eq. (2.7) in two space dimensions with source term given by (2.11) att = 1 (a = 1.5, N = 51, K = 10), subject to u(x, y,0) = O and homogeneous Dirichlet boundary conditions
Table 3. Time convergence in the solution of the fractional heat Eq. (2.7) in two space dimensions with source term given by (2.11) att = 1 (a = 1.5, N = 51, K = 10), subject to u(x, y,0) = O and homogeneous Dirichlet boundary conditions
Fig. 4 Metastability of solutions of the Allen—Cahn equation for varying a 3.2 FitzHugh—Nagumo model: excitable media
Fig. 4 Metastability of solutions of the Allen—Cahn equation for varying a 3.2 FitzHugh—Nagumo model: excitable media
Fig. 6 Solutions of the FitzHugh-Nagumo model for varying diffusion coefficient and a = 2 However, it is important to emphasise here that the role of reducing the fractional power a is not equivalent to the influence of a decreased diffusion coefficient in the pure diffusion case (Fig. 6). This can be clearly observed by comparison of Figs. 5b, c and 6b, c: for approximately the same width of the excitation wavefront, the wavelength of the system is larger in the fractional diffusion case, due to the long-tailed mechanisms of the fractional Laplacian operator. These results are therefore consistent with those of Engler [12] for the fractional reaction-diffusion Fisher equation, showing distinct effects of fractional diffusion to those of reduced conductivity for a family of travelling wave solutions. They also illustrate the use of fractional diffusion as a modelling tool to characterise intermediate dynamic states not solely described by pure diffusion mechanisms.
Fig. 6 Solutions of the FitzHugh-Nagumo model for varying diffusion coefficient and a = 2 However, it is important to emphasise here that the role of reducing the fractional power a is not equivalent to the influence of a decreased diffusion coefficient in the pure diffusion case (Fig. 6). This can be clearly observed by comparison of Figs. 5b, c and 6b, c: for approximately the same width of the excitation wavefront, the wavelength of the system is larger in the fractional diffusion case, due to the long-tailed mechanisms of the fractional Laplacian operator. These results are therefore consistent with those of Engler [12] for the fractional reaction-diffusion Fisher equation, showing distinct effects of fractional diffusion to those of reduced conductivity for a family of travelling wave solutions. They also illustrate the use of fractional diffusion as a modelling tool to characterise intermediate dynamic states not solely described by pure diffusion mechanisms.
Fig. 8 Isosurfaces u = 0.65 for the Gray—Scott model for k = 0.061
Fig. 8 Isosurfaces u = 0.65 for the Gray—Scott model for k = 0.061
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