Fractional cable problem in the frame of meshless singular boundary method

Aslefallah, Mohammad; Abbasbandy, Saeid; Shivanian, Elyas

doi:10.1016/j.enganabound.2019.08.003

Cited by 17 publications

(9 citation statements)

References 50 publications

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“…We choose γ = 0.0001, ε = 2.4, τ = 0.01, and n s = 31 for the present method. From the table we can see that the L ∞ errors of the present method are smaller than the ones of [28], while RMS errors of [28] are smaller than the results of the present method. We plotted the numerical solution and absolute error in Figure 6 for γ = 0.000155, ε = 2.1, τ = 0.001, α = 0.4, β = 0.7, and N = 1517 at final time T = 1.…”

Section: Numerical Resultsmentioning

confidence: 59%

“…In first problem we consider 2D fractional cable equation as follows

\begin{array}{l} \frac{\partial u ((), x, y, t)}{\partial t} =_{0} D_{t}^{1 - α} Δ u ((), x, y, t) -_{0} D_{t}^{1 - β} u ((), x, y, t) \\ + ([], 2 t + \frac{2 t^{1 + β}}{Γ ((), 2 + β)} + \frac{4 π^{2} t^{1 + α}}{Γ ((), 2 + α)}) normalsin ((), π x) normalsin ((), π y), x, y \in Ω, 0 < t \leq T, \end{array}

with initial condition

u (x, y, 0) = 0, (x, y) \in Ω

and boundary condition

u (x, y, t) = 0, (x, y) \in ∂Ω

in which Ω = [0, 1] × [0, 1]. This problem is also studied in [22, 23, 27, 28] and its exact solution is given as u ( x , y , t ) = t 2 sin( πx )sin( πy ). In Table 2, we give error norms, CPU times and condition numbers κ for decreasing values of time step width τ with N = 1225, γ = 0.0001, ε = 1.0, and n s = 51 at final time T = 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…We select γ = 0.000155 and ε = 2.1 and calculate the results for different stencil sizes and report them in Table 8. We compare the current results with the results of meshless singular boundary method [28] for α = 0.15 and β = 0.55 in Table 9. We choose γ = 0.0001, ε = 2.4, τ = 0.01, and n s = 31 for the present method.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The fractional cable equation is studied by many researchers with various numerical techniques. Some of these numerical techniques can be sorted as follows: an element free Galerkin method is proposed by Dehghan and Abbaszadeh [22], an orthogonal spline collocation method is used in [23], a discontinuous Galerkin finite element method is considered in [24], some second‐order θ schemes combined with finite element method are proposed in [25], Galerkin spectral method is used by Liu and Lü [26], an improved meshless algorithm is developed by Shivanian and Jafarabadi [27] and recently meshless singular boundary method is used in [28]. As far as we know the current work is the first attempt to use local hybrid kernels for fractional problems.…”

Section: Introductionmentioning

confidence: 99%

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A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

Oruç¹

2020

Numerical Methods Partial

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This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L ∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians. KEYWORDS 2D fractional cable equation, irregular domain, local hybrid kernel meshless method 1 INTRODUCTION Fractional calculus and fractional differential equations have attracted more attention recently due to their ability of handling many problems from various branches of physics, engineering, and chemistry in a more realistic approach [1, 2]. It has been recognized that fractional derivative has more superiorities than integer order derivative. For instance fractional derivatives can be used to describe memory and hereditary properties of physical materials [3-5]. This feature was not identified in integer order derivatives. There are a lot of applications of fractional calculus and fractional partial differential equations (PEDs) in different branches of science such as viscoelasticity, physics, chemistry, fluid mechanics, control, continuous-time random walks, and finance [6]. Some classic books about fractional calculus and its applications are [7-10].

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Section: Numerical Resultsmentioning

confidence: 59%

“…In first problem we consider 2D fractional cable equation as follows

\begin{array}{l} \frac{\partial u ((), x, y, t)}{\partial t} =_{0} D_{t}^{1 - α} Δ u ((), x, y, t) -_{0} D_{t}^{1 - β} u ((), x, y, t) \\ + ([], 2 t + \frac{2 t^{1 + β}}{Γ ((), 2 + β)} + \frac{4 π^{2} t^{1 + α}}{Γ ((), 2 + α)}) normalsin ((), π x) normalsin ((), π y), x, y \in Ω, 0 < t \leq T, \end{array}

with initial condition

u (x, y, 0) = 0, (x, y) \in Ω

and boundary condition

u (x, y, t) = 0, (x, y) \in ∂Ω

Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

Oruç¹

2020

Numerical Methods Partial

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“…They converted the fractional order model into a set of algebraic equations and presented two numerical examples to confirm the accuracy and efficiency. Aslefallah et al [24] studied the 2D time-fractional order cable model with Dirichlet boundary conditions and implemented the singular boundary method to split the solution of the inhomogeneous governing equation. More studies related to the fractional order differential equation can be seen in [25][26][27][28][29][30][31][32][33][34][35].…”

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confidence: 99%

Numerical approach for the fractional order cable model with theoretical analyses

et al. 2023

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This study, considers the fractional order cable model (FCM) in the sense of Riemann–Liouville fractional derivatives (R-LFD). We use a modified implicit finite difference approximation to solve the FCM numerically. The Fourier series approach is used to examine the proposed scheme’s theoretical analysis, including stability and convergence. The scheme is shown to be unconditionally stable, and the approximate solution converges to the exact solution. To demonstrate the application and feasibility of the proposed approach, a numerical example is provided.

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On new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients

Hashemi

Akgül

2020

Numerical Methods Partial

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Fractional cable problem in the frame of meshless singular boundary method

Cited by 17 publications

References 50 publications

A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

Numerical approach for the fractional order cable model with theoretical analyses

On new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients

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