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ORIGINAL RESEARCH article

Front. Phys., 25 April 2023
Sec. Mathematical Physics
Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1160767

Numerical approach for the fractional order cable model with theoretical analyses

  • 1Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, Pakistan
  • 2Department of Mathematics, Applied Sciences, Umm-Al-Qura University, Makkah, Saudi Arabia
  • 3Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh, Saudi Arabia
  • 4Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
  • 5School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia

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.

1 Introduction

Real-life phenomena have been modeled in a variety of ways, and partial differential equations (PDEs) and ordinary differential equations (ODEs) can be used to model some of these phenomena. For the phenomena that are not sufficiently modeled by PDEs, fractional PDEs have been developed by replacing the non-integers order derivative [1]. Fractional calculus can be applied to every field of science, such as biology, engineering, image processing, wave propagation, rheology, viscoelasticity, etc.

Fractional diffusion equations are a type of fractional differential equation that has sparked a lot of interest due to their various applications. By adding a variable lower limit of integration Rajkovic [2] generalized the notions of fractional the g-integral and g-derivative, and came up with a q-Taylor definition that contains fractional-order q-derivatives of the function. Yakar [3] considered a fractional boundary value problem with a two-part operator. The main problem’s eigenvalues with Eigen functions are the same as the constructed operator’s eigenvalues and corresponding Eigen functions in Hilbert spaces. The non-integers order Cable model is derived from the circuit model based on intracellular and extracellular space [4]. Vitali [5] introduced a Caputo formula as an extension of the FCE, obtained the analytical solution using the Laplace transform, and obtained results in terms of special functions. Yu [6] used the compact difference method and the Fourier method for stability and convergence in his computational treatment of the two-dimensional FCE. Liu et al. [7] discussed the FCE having two fractional temporal derivatives, and proposed implicit schemes with convergence orders of Oτ+h2 and Oτ2+h2, respectively. The energy approach was used to investigate the stability and convergence analysis. Lin et al. [8] devised a numerical schema for FCE discretization. They analyzed the proposed schema by providing theoretical and error estimates. The schema was unconditionally stable. Liu et al. [9] used a two-grid approach with the finite element scheme to solve the non-linear FCM, and the stability based on the fully discrete two-grid method was derived. A semi-discrete approach was used for time, and the Galerkin finite element approach was used for space Zhuang [10]. To approximate the time of the FCE involving two fractional temporal derivatives, Nikan [11] proposed a computational scheme for the radial basis function-generated finite difference (RBF-FD). The Grünwald–Letnikov expansion was used to discretize the time domain of the TFCM, and the RBF-FD was used to discretize the spatial derivatives. They demonstrated that the method can easily be implemented on such types of fractional PDEs. The orthogonal spline collocation with a complete theoretical analysis with order Oτmin(2γ1,2γ2+hr+1 was used by Zhang [12]. Quintana-Murillo [13] researched two temporal R-LFD for explicit numerical approaches to solve FCE. The numerical solution was obtained by using the forward difference formula, the Grünwald–Letnikov formula for the first-order derivative and Riemann–Liouville derivatives, respectively, and the three-point centered formula for the spatial derivative. The stability was tested by the von Neumann technique. Baleanu [14] proposed computational schemes for the optimal control problems of fractional order in the R-LFD sense. The approximations were replaced into optimal control equations of fractional order, and an algebraic equation was obtained, which can be solved by a numerical technique. To model the electrodiffusion of ions in nerve cells with anomalous sub-diffusion along and through the nerve cells, Henry [15] introduced fractional Nernst–Planck equations and related FCE. They analyzed fundamental solutions after modeling the sub-diffusion in two different ways, leading to two FCE. The solution approaches the normal non-zero steady state with uniform sub-diffusion along and around the nerve cells, but the approach is delayed by the anomalous sub-diffusion. Realistic electrophysiological studies on actual dendrites may be related to these solutions. Langlands [16] introduced fractional Nernst–Planck equations and derived FCE as macroscopic models for the electro diffusion of ions in nerve cells. They calculated the power lessening along dendrites in response to synaptic inputs of the alpha function. Easy integration and fire variants of the models were also used to calculate action potential firing rates. Tomovski [17] discussed Laplace and Fourier transforms to formulate the Green function of the generalized space-time FCE, and then examined the even moments to demonstrate that it can have a negative sign, indicating that the Green function does not always flow in one direction and that the current can switch directions. Bhrawy [18] used the collocation method in combination with the Shifted-Jacobi operational matrix in the sense of the Caputo fractional derivative. The results of their suggested approach are much more efficient for solving variable-order non-linear PDEs with high accuracy. Liu [19] presented a discrete numerical formula obtained by finite difference and finite element approximation in time and space, respectively, for the FCE. Liang-lian [20] considered the finite volume approach to solving the FCE using an implicit difference scheme. The approach was also convergent and unconditionally stable. Zhang [21] suggested an unconditionally numerical approach for the convection-diffusion of the fractional order problem. A novel shifted version of the Grünwald–Letnikov formula for the fractional order derivatives was used to prove the accuracy, and for theoretical analysis. Hu [22] implemented compact schemes for the FCE, and utilized the energy method to prove that the first scheme is stable and convergent in l-norm with the order Oτ+h4, while the second one is an inner product. The computed result indicates that both schemes are accurate and effective. Moshtaghi and Saadatmandi [23] researched the cable model of fractional order and solved using the collocation-type approach. 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 [2535].

The aim of this study is to find out the numerical solution of the fractional-order cable model. The fractional derivative is approximated by the discretized Riemann–Liouville derivative and for the space derivative use the finite difference approximation. For the proposed approach’s complete theoretical analysis as stability and convergence are discussed. The theoretical analysis, confirms the efficiency and effectiveness of the proposed approach.

Suppose the following fractional order cable model [36] as:

wx,tt=Dt1ρ10K2wx,tx2μ2Dt1ρ20wx,t+hx,t,(1)

with initial and boundary conditions

wx,0=βx,0xL,(2)
w0,t=β1t,wL,t=β2t,0<tT,(3)

where 0<ρ1,ρ2<1,K>0andμ are constants and β,β1andβ2 are known functions and the unknown function w is to be determined .

The Dt1ρ10wx,t is the Riemann–Liouville fractional derivative of fraction order 1ρ1 defined by [37]:

0Dt1ρ1wx,t=1Γρ1ddt0twx,ηtη1ρ1dη=tI0ρ1wx,t.(4)

The Riemann–Liouville fractional integral can be discretized [38] as:

I0ρ1wx,t=1Γρ10twx,ηtη1ρ1dη,

discretizing the equation at the grid point xi,tk.

I0ρ1wxi,tk=1Γρ10tktkηρ11wxi,ηdη.

As by Jumarie property [39] as:

=1ρ1Γρ10tkwxi,ηdηρ1,
=1Γ1+ρ1j=0k1tjtj+1wxi,ηdηρ1,
=1Γ1+ρ1j=0k1wxi,tkjtjtj+11dηρ1.

Again, by Jumarie property as:

=1Γ1+ρ1j=0k1wxi,tkjτρ1j+1ρ1τρ1jρ1,
=τρ1Γ1+ρ1j=0k1wxi,tkjj+1ρ1jρ1,
I0ρ1wxi,tk=τρ1Γ1+ρ1j=0k1djρ1wxi,tkj,(5)

where djρ1=j+1ρ1jρ1. The same procedure can be followed for ρ2.

Lemma 1:. The coefficients dkρ1 k=0,1,2, satisfy the following properties [35]:

(i) d0ρ1=1,dkρ1>0,k=1,2,

(ii) dk1ρ1>dkρ1,k=1,2,3,

(iii) There exists a positive constant C>0, such that τCdkρ1τρ1, k=1,2,3,

(iv) j=0kdjρ1τρ1=k+1ρ1Tρ1

2 Methodology

We implement an implicit numerical approximation for the FCE in Eqs 13, utilizing the discretization of the Riemann–Liouville integral with backward difference approximation for the partial derivative using central difference approximation. The steps as xi=ix along x-axis, where i=1,2,,Mx1,x=LMx and the step tk=kτ, k=1,2,3,,N where τ=TN. Letting the obtained numerical solution be wik to wxi,tk, and using Eq. 4 in Eq. 1, we have

wx,tt=KtI0ρ12x2wx,tμ2tI0ρ2wx,t+hx,t.(6)

Further, applying Eq. 5 in Eq. 6, we can write

wxi,tkt=tm1j=0k1djρ1δx2wxi,tkjtm2j=0k1djρ2wxi,tkj+hxi,tkj,(7)

Where

m1=Kτρ1x2Γρ1+1,m2=μ2τρ2Γρ2+1.(8)

By using implicit discretization with respect to time ‘t’, we have

wikwik1=m1j=0k1djρ1δx2wikjwikj1m2j=0k1djρ2wikjwikj1+τhik,(9)

Simplifying Eq. 9, we obtained

wik=wik1+m1d0ρ1wi+1k2wik+wi1km1dk1ρ1wi+102wi0+wi10m1j=1k1dj1ρ1djρ1wi+1kj2wikj+wi1kj+m2dk1ρ2wi0d0ρ2wik+m2j=1k1dj1ρ2djρ2wikj+τhik,(10)

With

wi0=βxi,0xL,(11)
w0k=β1tk,wMxk=β2tk,0tT.(12)

where i=1,2,,Mx1,k=1,2,3,,N.

3 Stability

In this section, we use the Fourier series method to analyze the stability of the implicit numerical scheme. Letting Wik be the approximate solution for Eq. 10, we have

Wik=Wik1+m1d0ρ1Wi+1k2Wik+Wi1km1dk1ρ1Wi+102Wi0+Wi10m1j=1k1dj1ρ1djρ1Wi+1kj2Wikj+Wi1kj+m2dk1ρ2Wi0d0ρ2Wik+m2j=1k1dj1ρ2djρ2Wikj+τhik.(13)

The error is defined as:

Eik=wikWik,(14)

where Eik satisfies 13 and

Eik=Eik1+m1d0ρ1Ei+1k2Eik+Ei1km1dk1ρ1Ei+102Ei0+Ei10m1j=1k1dj1ρ1djρ1Ei+1kj2Eikj+Ei1kj+m2dk1ρ2Ei0d0ρ2Eik+m2j=1k1dj1ρ2djρ2Eikj.(15)

The error and initial conditions are

E0k=EMxk=Ei0=0.(16)

Here, i=1,2,,Mx1.

Here, we need to define grid functions for k=1,2,,N, as the following:

Ekx=Eik,whenxix2<x<xi+x2,0,when0xx2orLx2xL.(17)

Then, Ekx can be written in Fourier series, such as

Ekx=l1=λkl1e21πl1xL,(18)

Here

λkl1=1L0LEkxe21πl1xL.(19)

From the definition of l2 norm and Parseval equality, we have

Ek2=i=1Mx1xEik2=l1=λkl12.(20)

Supposing that

Eik=λke1σix.(21)

where σ=2πl1L and by substituting 1821 in Eq. 15, we have

λk=11+m2+4m1sin2σx2λk1+λ0m2dk1ρ2+m1dk1ρ14sin2σx2+λkjm2j=1k1dj1ρ2djρ2+m14sin2σx2j=1k1dj1ρ1djρ1,(22)

Proposition 1: If λkK=1,2,,N satisfies Eq. 22, then λkλ0.

Proof:. To prove the above equality based on mathematical induction, we take k=1 in Eq. 22.

λ111+m2+4m1sin2σx2λ11+λ0m2d11ρ2+m1d11ρ14sin2σx2+λkjm2j=1k1dj1ρ2djρ2+m14sin2σx2j=1k1dj1ρ1djρ1]],

As d0ρ1=d0ρ2=1 and 0<γ1,γ2<1, we have

λ1λ0.

Now consider,

λmλ0,m=1,2,,k1.

As 0<ρ1,ρ2<1, from 22 and Lemma 1

λk11+m2sin2σΔx2λk1+λ0m2dk1ρ2+m1dk1ρ14sin2σΔx2+λkjm2j=1k1dj1ρ2djρ2+m14sin2σΔx2,11+m2+4m1sin2σΔx21+m2dk1ρ2+m1dk1ρ14sin2σΔx2+m2d0ρ2+m2dk1ρ2+m1d0ρ14sin2σΔx2m1dk1ρ14sin2σΔx2λ0,11+m2+4m1sin2σΔx21+m2d0ρ2+m1d0ρ14sin2σΔx2λ0,

Here, d0ρ1=d0ρ2=1, we have

11+m2+4m1sin2σx21+m2+m14sin2σx2λ0,λkλ0.(23)

By using proposition 1 and Eq. 20

λk2λ02.

The implicit numerical scheme in Eq. 10 is unconditionally stable.

4 Convergence

To investigate the convergence of the proposed implicit scheme. Let wxi,tk be the exact solution represented by Taylor series, then the local truncation error is obtained as

Qik=wxi,tkwxi,tk1m1j=0k1djρ1δx2wxi,tkjwxi,tkj1+m2j=0k1djρ2wxi,tkjwxi,tkj1τhxi,tk,(24)

From Eq. 1

Qik=wxi,tkwxi,tk1τwxi,tkt+Dt1ρ10K2wx,tx2m1j=0k1djρ1δx2wikjwikj1Dt1ρ2wxi,tk+0m2j=0k1djρ2wikjwikj1,=Oτ+x2(25)

Since i and k are finite, then there exist a positive constant C1, then we have

QikC1τ+x2,(26)

The error is defined as

ψik=wxi,tkwik.(27)

From Eq. 24, as

ψik=ψik1+m1d0ρ1ψi+1k2ψik+ψi1km1dk1ρ1ψi+102ψi0+ψi10m1j=1k1dj1ρ1djρ1ψi+1kj2ψikj+ψi1kjm2d0ρ2ψikdk1ρ2ψi0+m2j=1k1dj1ρ2djρ2ψikj+τQik.(28)

With error conditions that are

ψi0=0,ψ0k=ψMk=0,

Next, we define the following grid functions for k=1,2,,N.

ψkx=ψik,whenxix2<x<xi+x2,0,when0xx2orLx2xL.

And

Qkx=Qik,whenxix2<x<xi+x2,0,when0xx2orLx2xL.

Here, ψkx and Qkx can be expanded in Fourier series such as

ψkx=l1=ξkl1e21πl1xL,k=1,2,,N,
Qkx=l1=φkl1e21πl1xL,k=1,2,,N.

where

ξkl1=1L0Lψkxe21πl1xL,(29)
φkl1=1L0LQkxe21πl1xL.(30)

From the definition of l2 norm and the Parseval equality, we have

ψk2l2=i=1Mx1xEik2=l1=ξkl12,(31)
Qk2l2=i=1Mx1xEik2=l1=φkl12.(32)

Based on the above, supposing that

ψik=ξke1σ1ix,(33)
Qik=φke1σ1ix,(34)

where σ1=2πl1L, by using 33 and (34) in Eq. 28, we have

ξk=11+m2+4m1sin2σ1x2ξk1+ξ0m2dk1ρ2+4m1dk1ρ1sin2σ1x2+ξkjm2j=1k1dj1ρ2djρ2+4m1sin2σ1x2j=1k1dj1ρ1djρ1]+τφk],(35)

Proposition 2: If ξk is the solution of 35, then there exists a positive constant C2 such that ξkC2kτφ1.

Proof: From ψ0=0 and Eq. 29 we have

ξ0=ξ0l1=0.(36)

From (29) and (30), there exists positive constant C2, such that

φ1C2φ1l1.(37)

Using mathematical induction, for k=1, then from 35,

ξ111+m2+4m1sin2σ1x2ξ0+ξ0m2d0ρ2+4m1d0ρ1sin2σ1x2+C2τφ1,

From Eq. 36

ξ111+m2+4m1sin2σ1x2C2τφ1,ξ1C2τφ1.

Now suppose

ξmC2mτφ1,m=1,2,,k1.

From Eq. 34 and Lemma 1

ξk11+m2+4m1sin2σ1x2ξk1+ξ0m2dk1ρ2+4m1dk1ρ1sin2σ1x2+m2j=1k1dj1ρ2djρ2ξkj+4m1sin2σ1x2j=1k1dj1ρ1djρ1ξkj+τφk],

from Eq. 36

ξk11+m2+4m1sin2σ1x2k1+4m1k1sin2σ1x2j=1k1dj1ρ1djρ1+m2k1j=1k1dj1ρ2djρ2+1C2τφ1,
ξk11+m2+4m1sin2σ1x2k+4m1sin2σ1x2k11dk1ρ1+m2k11dk1ρ2+]C2τφ1,
ξk11+m2+4m1sin2σ1x2k1+m21dk1ρ2+4m1sin2σ1x21dk1ρ1m21dk1ρ2+4m1sin2σ1x21dk1ρ1]C2τφ1.

Here, 1dk1ρ11, 1dk1ρ21 because dk1ρ10, and dk1ρ10.

ξkk1+m2+4m1sin2σ1x2m2+4m1sin2σ1x2C2τφ11+m2+4m1sin2σ1x2,
ξkkm2+4m1sin2σ1x21+m2+4m1sin2σ1x2C2τφ1,

The value of (m2+4m1sin2σ1x21+m2+4m1sin2σ1x2) is very small, lying between 0 and 1. So, we obtained

ξkkC2τφ1.

5 Numerical tests

In this study, the numerical result of an implicit scheme for one-dimensional FCE are discussed numerically and graphically. The examples are as following.

Example 1. Consider the fractional-order cable model [15] with the closed-form solution is given as:

wx,tt=Dt1ρ10K2wx,tx2μ2Dt1ρ20wx,t+2t+π2t1+ρ1Γ2+ρ1+t1+ρ2Γ2+ρ2sinπx,

with initial and boundary conditions

wx,0=0,0x1,

w0,t=0,w1,t=0,0<t1. Where ρ1,ρ2(0,1 and K=1,μ=1. The closed-form solution is wx,t=t2sinπx.

Example 2. Consider the 1D fractional Stokes’ first problem for the heated generalized second-grade equation [40].

wx,tt=Dt1ρ302wx,tx2+2wx,tx2+2+ρ3t1+ρ3Γ3+ρ3Γ2+2ρ3t1+2ρ3t2+ρ3ex,

with initial and boundary conditions

wx,0=0,0x1,

w0,t=t1+ρ3,w1,t=et1+ρ3,0<t1. Where ρ3(0,1 and K=1,μ=1. The closed-form solution is wx,t=ext1+ρ3.The errors between a numerical and an exact solution are defined as follows:

E=max1<iMx1wxi,tkwik.(38)
E2=k=1Mx1wxi,tkwik2x1/2.(39)

The above problem is solved using the modified implicit scheme. The errors EandE2 at T=1.0 and for different values of x and N. The time step τ is defined by τ=TN.

6 Results and discussion

The modified implicit finite difference approximation is used to solve the numerical example of fractional order, such as fractional cable model and the fractional order Stokes’ first problem for the heated generalized second grade equation. Numerical results are presented in the form of tables and figures for various values of space and time steps in order to demonstrate the efficiency of the suggested numerical scheme.

In Tables 13, the exact and the numerical solution are compared of the given example 1 for fixed values ρ1=0.5 and ρ2=0.75,0.5,0.25, and different values of N and h. The error decreases as the value of N increases. Similarly, as the time and space step size τ and Δx,Δy reduce, the errors decrease for a fixed value of ρ1andρ2. In Table 4, the exact and the numerical solution of example 1 are compared for ρ1,ρ1=0.25, ρ1,ρ1=0.5, and ρ1,ρ1=0.95, and for different values of N and x. The results show that as we increase the value of N, i.e., reduce the time and space step size τ and hx, the errors decrease for different values of ρ1andρ2. In Table 5, the numerical results are explained for example, 2 of the suggested scheme for the fractional order first problem for a heated generalized second-grade fluid for various values of order ρ3, step size τ, and x. Figures 13 show the comparison of the numerical and the exact solution of example 1 in Figure 1 at ρ1, ρ2 = 0.5, T=1.0, x = 1/10, and N=40. For Figure 2, at ρ1, ρ2 = 0.5, T=1.0, x = 1/20, and N=80. For Figure 3, at ρ1, ρ2 = 0.5, T=1.0, x = 1/40, and N=250. Furthermore, added Figure 4 which shows the graphical representation of example 2 for ρ3=.6,Δx=1/8,T=1.0 and N=64, which confirmed our theoretical analysis and demonstrated that the proposed approach is very powerful and efficient.

TABLE 1
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TABLE 1. Numerical results for example, 1 of the modified implicit scheme for various values of , x, and fixed values of ρ1=0.5,ρ2=0.75.

TABLE 2
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TABLE 2. Numerical results example 1 of the modified implicit scheme for various values of , x, and fixed value of ρ1,ρ2=0.5.

TABLE 3
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TABLE 3. Numerical results example 1 of the modified implicit scheme for various values of N,x, and for fixed value of ρ1=0.5,ρ2=0.25.

TABLE 4
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TABLE 4. Numerical results example 1 of the modified implicit scheme for various values of ρ1,ρ2, N, and x.

TABLE 5
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TABLE 5. Numerical results for example 2 of the modified implicit scheme for various values of ρ3, τ, and x.

FIGURE 1
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FIGURE 1. Comparison of the numerical and exact solution of the given example 1 at ρ1, ρ2 = 0.5, T=1.0, x = 1/10, and N=40.

FIGURE 2
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FIGURE 2. Comparison of the numerical and exact solution of the given example 1 at ρ1, ρ2 = 0.5, T=1.0, x = 1/20, and N=80.

FIGURE 3
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FIGURE 3. Comparison of the numerical and exact solution of the given example 1 at ρ1, ρ2 = 0.5, T=1.0, x = 1/40, and N=250.

FIGURE 4
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FIGURE 4. Comparison of the numerical and exact solution of the given example 1 at ρ3 = 0.6, T=1.0, x = 1/8, and N=64.

7 Conclusion

This paper presented the modified implicit numerical approximation for a fractional one-dimensional linear Cable model. The scheme is convergent and unconditionally stable, as seen by the investigation using the Fourier series method. The time-fractional derivative was calculated using the Riemann–Liouville formula. The outcome of an application to specific examples of fractional order one-dimensional linear Cable model and the fractional order Stokes’ first problem for the heated generalized second-grade equation have been explored graphically and numerically. The scheme is verified through the comparison of the numerical solution with the exact solution, which shows an agreement with the theoretical analysis and the numerical experiment, confirming that the approximate solution converges to the exact solution. This modified approach can also extend to other types of two and three dimensional fractional order differential models.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under Grant code: 22UQU4310396DSR58.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: fractional cable equation, implicit approximation, stability, convergence, riemann-liouville fractional derivative

Citation: Ali U, Naeem M, Ganie AH, Fathima D, Salama FM and Abdullah FA (2023) Numerical approach for the fractional order cable model with theoretical analyses. Front. Phys. 11:1160767. doi: 10.3389/fphy.2023.1160767

Received: 07 February 2023; Accepted: 29 March 2023;
Published: 25 April 2023.

Edited by:

Alexander Nepomnyashchy, Technion Israel Institute of Technology, Israel

Reviewed by:

Yuriy Povstenko, Jan Długosz University, Poland
Sunil Kumar, National Institute of Technology, India

Copyright © 2023 Ali, Naeem, Ganie, Fathima, Salama and Abdullah. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Umair Ali, umairkhanmath@gmail.com; Dowlath Fathima, d.fathima@seu.edu.sa

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