A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation

Abstract

In this paper, a novel numerical scheme is developed using a new construct by non-polynomial spline for solving the time fractional Generalize Fisher equation. The proposed models represent bacteria, epidemics, Brownian motion, kinetics of chemicals and fuzzy systems. The basic concept of the new approach is constructing a non-polynomial spline with different non-polynomial trigonometric and exponential functions to solve fractional differential equations. The investigated method is demonstrated theoretically to be unconditionally stable. Furthermore, the truncation error is analyzed to determine the or-der of convergence of the proposed technique. The presented method was tested in some examples and compared graphically with analytical solutions for showing the applicability and effectiveness of the developed numerical scheme. In addition, the present method is compared by norm error with the cubic B-spline method to validate the efficiency and accuracy of the presented algorithm. The outcome of the study reveals that the developed construct is suitable and reliable for solving nonlinear fractional differential equations.

Keywords

Non-polynomial spline generalize fisher equation truncation error stability analysis

1 Introduction

Nowadays, differential equations, particularly fractional differential equations, have become a popular and appealing topic that is being researched both theoretically and practically. The major advantage of employing fractional order differential equations in mathematical modeling is their non-locality. The fractional order differential equations have received a significant amount of research work in the modern scientific era such as physics, power systems, control theory, nonlinear dynamics, viscoelasticity theory, neurology, electromagnetic acoustics theory, problems with bacterial growth, intelligent robot system and fuzzy soft computing [1 –7]. According to the preceding significance, many researchers have conducted analytical and numerical studies on FDEs, including Analytical solutions for time fractional two-dimensional rection differential equations using Lie symmetry analysis [8]. Time-fractional coupled Klein-Gordon and Korteweg-de Vries equations were investigated numerically the y by the Local meshless approach [9]. nonlinear fractional stochastic differential equations studied using an integro-quadratic scheme [10]. B-spline method for solving Fitzhugh-Nagumo [11].

Fisher 1937 proposed One of the famous and important time-fractional differential equations called the Fisher equation as a model for the temporal and spatial propagation of a virile gene in an infinite medium [12]. Fisher’s equation is extensively employed in various areas, including bacteria and epidemics [13], Brownian motion with branches [14], kinetics of chemicals [15] and fuzzy fractional differential equations [16]. Baranwal et al. studied two-dimensional fractional reaction-diffusion to generalize the Fisher equation [17]. Khalid et al. investigated Product terms in fourth-order fractional boundary value problems [18].

Nonetheless, a considerable number of nonlinear fractional differential equations found in these fields are extremely difficult to solve numerically. Researchers have been particularly interested in the numerical solution of such models because of their wide spectrum of uses in real life. There is a lot of work in the open literature on numerically solving the time-fractional Fisher equation employing various techniques. The scholar in [19] used Atangana-Baleanu fractional derivative. A q-homotopy analysis transform method was used by [20]. Abdul Majeed et al. [21] employed combining cubic B-spline and finite element method. Many authors solved the Fisher equation numerically, see ([19 –34]).

The non-polynomial spline method has recently been investigated for solving any type of differential equations [35 –37]. Consider this: Faraidun et al. solved some fractional differential equations using non-polynomial spline with fractional order [38]. In [39] introduced non-polynomial spline for solving the fractional Bagely-Torvik equation. Qinxu et al. have used quintic non-polynomial spline for solving the Schrödinger equation [40]. Ali et al. have founded an approximation solution for forth order fractional boundary value problem [41]. Recently Hamad et al. solved fuzzy Fredholm integral equation [42]. In this paper, we construct a new numerical scheme using a trigonometric and exponential non-polynomial spline for solving the time-fractional generalized Fisher equation and investigate the order of convergent and stability analysis of the method.

Consider the generalized fractional formula of Fisher’s equation: [17] $\begin{matrix} \frac{\partial^{α} u}{\partial t^{α}} - ν \frac{\partial^{2} u}{\partial x^{2}} - ru (1 - u^{λ}) = f (x, t), \\ a ⩽ x ⩽ b, 0 < α ⩽ 1, t ⩾ 0, \end{matrix}$ (1) with the initial condition: $u (x, 0) = φ (x) a ⩽ x ⩽ b,$ (2) and the boundary conditions: $u (a, t) = ψ_{1} (t), u (b, t) = ψ_{2} (t), t ⩾ 0 .$ (3) where ν = 1, is viscosity parameter, r, λ ⩾ 0, are constants, and Caputo fractional derivative of the function u (x, t) is defined as: $\frac{\partial^{α} u}{\partial t^{α}} = \frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{u^{'} (s)}{{(t - s)}^{α}} ds, 0 < α ⩽ 1 .$ (4)

The remainder of the paper is structured as follows. A description of the numerical scheme using the non-polynomial spline function is given in section 2. In section 3, the Taylor expansion is used to develop the truncation error. The application and stability of the numerical scheme are investigated in sections 4 and 5. To explain the theoretical results in section 6, an illustrated example is carried out. Finally, the last part contains the conclusion.

2 Description of developed method

Let x_j = j (h) , j = 0, 1, ⋯ , M and t_n = n (τ) , n = 0, 1, ⋯ , N, where $\frac{b - a}{M}$ and $τ = \frac{T}{N}$ , are step size of uniform spatial and temporal time respectively. To develop a numerical approach for simulate the solution of Equation (1), consider the quartic non-polynomial spline combine trigonometric and exponential terms: $\begin{matrix} S_{j, n} (x_{j}, t_{n}) = a_{j}^{n} cos k (x - x_{j}) + b_{j}^{n} sin k (x - x_{j}) \\ + c_{j}^{n} e^{k (x - x_{j})} + d_{j}^{n}, \end{matrix}$ (5)

Where S_j,n (x_j, t_n) be the nonpolynomial spline function which are approximate solution to $u_{j}^{n} = u (x_{j}, t_{n})$ and k is frequency for trigonometric functions, and $a_{j}^{n}, b_{j}^{n}, c_{j}^{n}$ and $d_{j}^{n}$ are undetermined coefficient, according to the following conditions: $\begin{matrix} \begin{matrix} S_{j, n} (x_{j}, t_{n}) = u_{j}^{n}, \\ S_{j, n} (x_{j + 1}, t_{n}) = u_{j + 1}^{n}, \end{matrix} \\ \begin{matrix} S_{j, n}^{(2)} (x_{j}, t_{n}) = u_{j}^{n^{(2)}} = M_{j}^{n}, \\ S_{j, n}^{(2)} (x_{j + 1}, t_{n}) = u_{j + 1}^{n^{(2)}} = M_{j + 1}^{n}, \end{matrix} \end{matrix}$ (6)

The coefficient involved Equation (5) using conditions (6), we get: $a_{j}^{n} = \frac{(1 - 2 e^{θ}) M_{j}^{n} + M_{j + 1}^{n} + k^{2} (- u_{j}^{n} + u_{j + 1}^{n})}{2 (- 1 + e^{θ}) k^{2}},$ (7) $\begin{matrix} b_{j}^{n} = - \frac{csc θ}{2 (- 1 + e^{θ}) k^{2}} ((e^{θ} + cos θ - 2 e^{θ} cos θ) M_{j}^{n}, \\ + (- 2 + e^{θ} + cos θ) M_{j + 1}^{n} \\ + k^{2} (e^{θ} - cos θ) (u_{j}^{n} + u_{j + 1}^{n})) \end{matrix}$ (8) $c_{j}^{n} = \frac{- M_{j}^{n} + M_{j + 1}^{n} + k^{2} (- u_{j}^{n} + u_{j + 1}^{n})}{2 (- 1 + e^{θ}) k^{2}},$ (9)

$d_{j}^{n} = - \frac{M_{j + 1}^{n} - e^{θ} (M_{j}^{n} + k^{2} u_{j}^{n}) + k^{2} u_{j + 1}^{n}}{(- 1 + e^{θ}) k^{2}} .$ (10) where, θ = hk,

Using the first derivative continuity equation $S_{j, n}^{'} (x_{i}, t_{n}) = S_{j - 1, n}^{'} (x_{i}, t_{n})$ , The main relationships that may be deduced are as follows: ${kb}_{j}^{n} + {kc}_{j}^{n} = - a_{j - 1}^{n} k sin θ + b_{j - 1}^{n} k cos θ + c_{j - 1}^{n} k e^{θ},$ (11)

After some simplification and collection, we get: $β_{1} M_{j - 1}^{n} + β_{2} M_{j}^{n} + β_{3} M_{j + 1}^{n} = γ_{1} u_{j - 1}^{n} + u_{j}^{n} + γ_{2} u_{j + 1}^{n},$ (12) where, $γ_{1} = \frac{- (e^{θ} (1 + cot θ) - csc θ)}{(1 + e^{θ} + (- 1 + e^{θ}) cot \frac{θ}{2})},$ $γ_{2} = \frac{- (1 - cot θ + e^{θ} csc θ)}{(1 + e^{θ} + (- 1 + e^{θ}) cot \frac{θ}{2})} .$ $β_{1} = \frac{(csc θ + e^{θ} csc θ (- 2 + cos θ + sin θ))}{k^{2} (1 + e^{θ} + (- 1 + e^{θ}) cot \frac{θ}{2})},$ $β_{2} = \frac{(- 1 + 3 (- 1 + e^{θ}) cot θ + csc θ - e^{θ} (1 + csc θ))}{k^{2} (1 + e^{θ} + (- 1 + e^{θ}) cot \frac{θ}{2})},$ $β_{3} = \frac{(1 - cot θ + 2 csc θ - e^{θ} csc θ)}{k^{2} (1 + e^{θ} + (- 1 + e^{θ}) cot \frac{θ}{2})} .$

3 Truncation error

Suppose that T_j is local truncation error for numerical scheme (12) in j^th term as follows: $\begin{matrix} T_{j} = γ_{1} u_{j - 1}^{n} + u_{j}^{n} + γ_{2} u_{j + 1}^{n} \\ - (β_{1} M_{j - 1}^{n} + β_{2} M_{j}^{n} + β_{3} M_{j + 1}^{n}) \\ = γ_{1} u_{j - 1}^{n} + u_{j}^{n} + γ_{2} u_{j + 1}^{n} \\ - β_{1} M_{j - 1}^{n} - β_{2} M_{j}^{n} - β_{3} M_{j + 1}^{n}, \end{matrix}$ (13)

Using Taylor expansion and collecting the derivative coefficients, it holds that: $\begin{matrix} T_{j} = (γ_{1} + 1 + γ_{2}) u_{j}^{n} + (- γ_{1} + γ_{2}) {hu}_{j}^{n^{'}} \\ + (\frac{γ_{1} + γ_{2}}{2} - \frac{β_{1} + β_{2} + β_{3}}{h^{2}}) h^{2} u_{j}^{n} (2) \\ + (\frac{- γ_{1} + γ_{2}}{6} + \frac{β_{1} - β_{3}}{h^{2}}) h^{3} u_{j}^{n} (3) \\ + (\frac{γ_{1} + γ_{2}}{24} - \frac{β_{1} + β_{3}}{2 h^{2}}) h^{4} u_{j}^{n} (4) \\ + (\frac{- γ_{1} + γ_{2}}{120} + \frac{β_{1} - β_{3}}{6 h^{2}}) h^{5} u_{j}^{n} (5) \\ + (\frac{γ_{1} + γ_{2}}{720} - \frac{β_{1} + β_{3}}{24 h^{2}}) h^{6} u_{j}^{n} (6) \\ + O (k^{2 - α} + h^{6}), \end{matrix}$ (14)

According to Equation (14), and equating the coefficient of , we get: $γ_{1} = γ_{2} = - \frac{1}{2}$ $β_{1} = β_{3} = - \frac{h^{2}}{24} and β_{2} = - \frac{10 h^{2}}{24},$

Substituting the above determined coefficients to numerical scheme (12), we have: $\begin{matrix} u_{j - 1}^{n} - 2 u_{j}^{n} + u_{j + 1}^{n} \\ = \frac{h^{2}}{12} M_{j - 1}^{n} + \frac{10 h^{2}}{12} M_{j}^{n} + \frac{h^{2}}{12} M_{j + 1}^{n}, \end{matrix}$ (15) and the local truncation error as follows $t_{i} = \frac{h^{6}}{180} u_{j}^{n} (6) + (2 β_{1} + β_{2}) O (k^{2 - α} + h^{6}) .$ (16)

4 Methodology of numerical scheme to Fisher equation

In this section, we investigate a numerical scheme using quantic non-polynomial spline for solving (1)-(3). The L₁-approximation used by replacing with the time Caputo derivative and the approximation order can be obtained by the following lemma.

Lemma 1. [40] Suppose f (t) ∈ C² [0, t_n] and 0 < α ⩽ 1, it holds that: $| \begin{matrix} \frac{1}{Γ (1 - α)} \int_{0}^{t_{n}} \frac{f^{'} (t)}{{(t_{n} - t)}^{α}} dt \\ - \frac{τ^{- α}}{Γ (2 - α)} [\begin{matrix} a_{0} f (t_{n}) \\ - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) f (t_{n}) - a_{n - 1} f (t_{0}) \end{matrix}] \end{matrix} |$ $\begin{matrix} ⩽ \frac{1}{Γ (2 - α)} [\frac{1 - α}{12} + \frac{2^{2 - α}}{2 - α} - (1 + 2^{- α})] \\ max_{0 ⩽ t ⩽ t_{n}} | f^{″} (t) | τ^{2 - α}, (17) \end{matrix}$

Where a_m = (m+ 1) ^1-α - m^1-α, m = 0, 1, ….

Lemma 2. [40] Let a_m = (m+ 1) ^1-α - m^1-α, m = 0, 1, … and 0 < α ⩽ 1, then $1 = a_{0} > a_{1} > \dots > a_{m} \to 0, asm \to \infty .$

According to Lemma 1, the Caputo fractional derivative can follow as: $\begin{matrix} \frac{\partial^{α} u_{j}^{n}}{\partial t^{α}} = \frac{τ^{- α}}{Γ (2 - α)} [a_{0} u_{j}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) \\ u_{j}^{m} - a_{n - 1} u_{j}^{0}] + O (τ^{2 - α}), \end{matrix}$ (18)

According to non-polynomial spline function, we can replace by second-order space derivative at (x_j, t_n) as follows: $\frac{\partial^{2} u_{j}^{n}}{{\partial x}^{2}} = u_{j}^{n} (2) = M_{j}^{n},$ (19)

At the grid point (x_j, t_n) and using Eqs. (1), (18), and (19), may be written $M_{j}^{n}$ in the following format: $M_{j}^{n} = \frac{\partial^{α} u_{j}^{n}}{\partial t^{α}} - r u_{j}^{n} (1 - u_{j}^{n λ}) - f_{j}^{n} = μ [a_{0} u_{j}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) u_{j}^{m} - a_{n - 1} u_{j}^{0}] - r u_{j}^{n} (1 - u_{j}^{n λ}) - f_{j}^{n},$ (20)

Replacing j in Eq. (20) with j --1 and j + 1 yields: $\begin{matrix} M_{j - 1}^{n} = μ [a_{0} u_{j - 1}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) \\ u_{j - 1}^{m} - a_{n - 1} u_{j - 1}^{0}] - {ru}_{j - 1}^{n} (1 - u_{j - 1}^{n} λ) - f_{j - 1}^{n}, \end{matrix}$ (21) and $\begin{matrix} M_{j + 1}^{n} = μ [a_{0} u_{j + 1}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) \\ u_{j + 1}^{m} - a_{n - 1} u_{j + 1}^{0}] - {ru}_{j + 1}^{n} (1 - u_{j + 1}^{n} λ) - f_{j + 1}^{n}, \end{matrix}$ (22)

Substituting Equations (20)–(22) into Equation (15), we have: $\begin{matrix} u_{j - 1}^{n} - 2 u_{j}^{n} + u_{j + 1}^{n} \\ = \frac{h^{2}}{12} (μ [a_{0} u_{j - 1}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) u_{j - 1}^{m} \\ - a_{n - 1} u_{j - 1}^{0}] - {ru}_{j - 1}^{n} (1 - u_{j - 1}^{n} λ) - f_{j - 1}^{n}) \\ + \frac{10 h^{2}}{12} (μ [a_{0} u_{j}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) \\ u_{j}^{m} - a_{n - 1} u_{j}^{0}] - {ru}_{j}^{n} (1 - u_{j}^{n} λ) - f_{j}^{n}) \\ + \frac{h^{2}}{12} (μ [a_{0} u_{j + 1}^{n} - \sum_{m = 1}^{n - 1} (a_{n - m - 1} - a_{n - m}) u_{j + 1}^{m} \\ - a_{n - 1} u_{j + 1}^{0}] - {ru}_{j + 1}^{n} (1 - u_{j + 1}^{n} λ) - f_{j + 1}^{n}), \end{matrix}$ (23)

Scheme (23) can be rewritten as the following system for ease of implementation: $\begin{matrix} A_{j} u_{j - 1}^{n} + B_{j} u_{j}^{n} + C_{j} u_{j + 1}^{n} \\ = A_{j}^{*} u_{j - 1}^{n - 1} + B_{j}^{*} u_{j}^{n - 1} + C_{j}^{*} u_{j + 1}^{n - 1} - \frac{h^{2}}{12} f_{j - 1}^{n} \\ - \frac{10 h^{2}}{12} f_{j}^{n} - \frac{h^{2}}{12} f_{j + 1}^{n} - W_{j}^{n} \\ j = 2, \dots, N - 1, n ⩾ 2 . \end{matrix}$ (24)

where, $A_{j} = 1 - μ \frac{h^{2}}{12} a_{0} + r \frac{h^{2}}{12} (1 - u_{j - 1}^{n} λ),$ $B_{j} = - 2 - μ \frac{10 h^{2}}{12} a_{0} + r \frac{10 h^{2}}{12} (1 - u_{j}^{n λ}),$ $C_{j} = 1 - μ \frac{h^{2}}{12} a_{0} + r \frac{h^{2}}{12} (1 - u_{j + 1}^{n} λ),$ $A_{j}^{*} = - μ \frac{h^{2}}{12} (a_{0} - a_{1}),$ $B_{j}^{*} = - μ \frac{10 h^{2}}{12} (a_{0} - a_{1}),$ $C_{j}^{*} = - μ \frac{h^{2}}{12} (a_{0} - a_{1}),$ and $\begin{matrix} W_{j}^{n} = μ \frac{h^{2}}{12} (\sum_{m = 1}^{n - 2} (a_{n - m - 1} - a_{n - m}) u_{j - 1}^{m} + a_{n - 1} u_{j - 1}^{0}) \\ + μ \frac{10 h^{2}}{12} (\sum_{m = 1}^{n - 2} (a_{n - m - 1} - a_{n - m}) u_{j}^{m} + a_{n - 1} u_{j}^{0}) \\ + μ \frac{h^{2}}{12} (\sum_{m = 1}^{n - 2} (a_{n - m - 1} - a_{n - m}) u_{j + 1}^{m} + a_{n - 1} u_{j + 1}^{0}) . \end{matrix}$

5 Stability analysis

According the Von Neumann stability, we assume that the solution of Equation (24) has the following form: $u_{j}^{n} = ϒ^{n} e^{i ɛ jh},$ (25)

Where $i = \sqrt{- 1},$ and ɛ is real spatial wave number.

The stability of presented scheme for solving Equation (1), by linearizing the nonlinear term by substituting Equation (25) into Equation (23), we have: $\begin{matrix} ϒ^{n} e^{i ɛ h (j - 1)} - 2 ϒ^{n} e^{i ɛ hj} + ϒ^{n} e^{i ɛ h (j + 1)} - μ \frac{h^{2}}{12} a_{0} \\ (ϒ^{n} e^{i ɛ h (j - 1)} + 10 ϒ^{n} e^{i ɛ hj} + ϒ^{n} e^{i ɛ h (j + 1)}) \\ + \frac{h^{2}}{12} rd (ϒ^{n} e^{i ɛ h (j - 1)} + 10 ϒ^{n} e^{i ɛ hj} + ϒ^{n} e^{i ɛ h (j + 1)}) \\ = - μ \frac{h^{2}}{12} (a_{0} - a_{1}) (ϒ^{n - 1} e^{i ɛ h (j - 1)} + 10 ϒ^{n - 1} \\ e^{i ɛ hj} + ϒ^{n - 1} e^{i ɛ h (j + 1)}) \\ - μ \frac{h^{2}}{12} [(\sum_{m = 1}^{n - 2} (a_{n - m - 1} - a_{n - m}) (ϒ^{m} e^{i ɛ h (j - 1)} \\ + 10 ϒ^{m} e^{i ɛ hj} + ϒ^{m} e^{i ɛ h (j + 1)})) \\ + a_{n - 1} (e^{i ɛ h (j - 1)} + 10 e^{i ɛ hj} + e^{i ɛ h (j + 1)})], \end{matrix}$ (26)

Dividing both sides by term e^iɛhj and putting e^-iɛh + e^iɛh = 2 cos(ɛh) , we have: $\begin{matrix} ϒ^{n} e^{- i ɛ h} - 2 ϒ^{n} + ϒ^{n} e^{i ɛ h} \\ - μ \frac{h^{2}}{12} a_{0} (ϒ^{n} e^{- i ɛ h} + 10 ϒ^{n} + ϒ^{n} e^{i ɛ h}) \\ + \frac{h^{2}}{12} rd (ϒ^{n} e^{- i ɛ h} + 10 ϒ^{n} + ϒ^{n} e^{i ɛ h}) = - μ \frac{h^{2}}{12} \\ (a_{0} - a_{1}) (ϒ^{n - 1} e^{- i ɛ h} + 10 ϒ^{n - 1} + ϒ^{n - 1} e^{i ɛ h}) \\ - μ \frac{h^{2}}{12} [(\sum_{m = 1}^{n - 2} (a_{n - m - 1} - a_{n - m}) \\ (ϒ^{m} e^{- i ɛ h} + 10 ϒ^{m} + ϒ^{m} e^{i ɛ h})) + a_{n - 1} \\ (e^{- i ɛ h} + 10 + e^{i ɛ h})], \end{matrix}$ (27) $\begin{matrix} ϒ^{n} (2 cos (ɛ h) - 2) - μ \frac{h^{2}}{12} ϒ^{n} a_{0} (2 cos (ɛ h) + 10) \\ + \frac{h^{2}}{12} rd ϒ^{n} (2 cos (ɛ h) + 10) \\ = - μ \frac{h^{2}}{12} (a_{0} - a_{1}) ϒ^{n - 1} (2 cos (ɛ h) + 10) - μ \frac{h^{2}}{12} \\ (\sum_{m = 1}^{n - 2} ϒ^{m} (a_{n - m - 1} - a_{n - m}) (2 cos (ɛ h) + 10)) \\ - μ \frac{h^{2}}{12} a_{n - 1} (2 cos (ɛ h) + 10), \end{matrix}$ (28)

After simplification and collection terms, we get: $\begin{matrix} ϒ^{n} = \frac{a_{1} - 1}{σ} ϒ^{n - 1} \\ - \frac{1}{σ} [(\sum_{m = 1}^{n - 2} ϒ^{m} (a_{n - m - 1} - a_{n - m})) + a_{n - 1}], \end{matrix}$ (29) where $σ = \frac{12 (cos (ɛ h) - 1)}{μ h^{2} (cos (ɛ h) + 5)} - a_{0} + \frac{d}{μ} ⩾ 1, a_{1} < 1$ Then $| ϒ^{n} | ⩽ | \frac{a_{1} - 1}{σ} ϒ^{n - 1} | ⩽ | ϒ^{n - 1} |, n = 1, 2, \dots N + 1 .$ (30)

Implies that |ϒⁿ|⩽ |ϒ⁰| . ■

According to above achievements the presented formula (26) is unconditional stable.

6 Numerical illustration and discussion

In this section, the proposed technique is used to solve three fractional Fisher problems, and a comparison of the exact solution and the nonpolynomial spline method is used to illustrate the correctness and effectiveness of the proposed scheme with figures and Tables. The maximum absolute errors and least square errors are calculated and compared to well-known values: $L_{\infty} = max_{1 ⩽ j ⩽ M} | u_{j_{exact}} - u_{j_{numerical}} |,$ (31) $L_{2} = \sqrt{h \sum_{i = 1}^{M} {| u_{j_{exact}} - u_{j_{numerical}} |}^{2}} .$ (32)

Example 1. Consider the Equation (1), when ν = 1, r = 0 and λ = 0, given as: [24] $\frac{\partial^{α} u}{\partial t^{α}} - \frac{\partial^{2} u}{\partial x^{2}} = f (x, t), 0 ⩽ x ⩽ 1, 0 < α ⩽ 1, t ⩾ 0,$ (33)

With the initial condition: $u (x, 0) = 00 ⩽ x ⩽ 1,$ (34) and the boundary conditions: $u (0, t) = u (1, t) = 0, t ⩾ 0 .$ (35) where, $f (x, t) = t^{2} [\begin{matrix} \frac{1}{2} e^{x} x^{2} {(x - 1)}^{2} Γ (3 + α) \\ - t^{α} e^{x} (x^{4} + 6 x^{3} + x^{2} - 8 x + 2) \end{matrix}] .$ (36)

The exact solution analytically for Example 1, is u (x, t) = x² (x - 1) ²e^xt^(2+α), when ν = 1 and r = 0 in Equation (1).

From the Fig. 1, the 2D preview shows the comparison between exact solution and numerical scheme, when x ∈ [0, 1], t = 0.5 and α = 0.5, and it is clear that, both methods are strongly agreed with each other. In Fig. 2, the 3D-mesh plot shows, when x ∈ [0, 1] t ∈ [0, 1] and α = 0.5. Also, from Fig. 3, the contour plot indicates. Figure 4, clearly shows that exact and numerical solutions have comparable behavior for a set temperature t = 0.025, 0.05, 0.075, 0.1, and it concluded that, u (x, t) decreases, when the time increase. The error norms, as shown in Table 1, are used to assess the scheme’s efficiency and it shows clearly that both solutions are relatively close and have insignificant flaws.

Fig. 1

Curve of exact and numerical solution example1, when x ∈ [0, 1], t = 0.5 and α = 0.5.

Fig. 2

Mesh of u (x, t) for example 1, when x ∈ [0, 1] t ∈ [0, 1] and α = 0.5.

Fig. 3

Contour plot of u (x, t) for example 1, when x ∈ [0, 1] , t ∈ [0, 0.1] and α = 0.5.

Fig. 4

Time effect on u (x, t) for example 1, when x ∈ [0, 1] and α = 0.5.

Table 1

Absolute errors comparison of Example 1, when x ∈ [0, 1] and M = 100

t	α = 1		α = 0.25
	L _∞	L ₂	L _∞	L ₂
0.02	4.7087E-07	2.1851E-07	1.0688E-06	5.3056E-07
0.04	9.3476E-07	4.4337E-07	1.9474E-06	9.6757E-07
0.06	1.0583E-06	5.0679E-07	2.1061E-06	1.0469E-06
0.08	6.7440E-07	3.2162E-07	1.3505E-06	6.7132E-07

Example 2. Consider the Equation (1), when ν = 1, r = 1 and λ = 3, given as: [22] $\begin{matrix} \frac{\partial^{α} u}{\partial t^{α}} - \frac{\partial^{2} u}{\partial x^{2}} - u (1 - u^{3}) = f (x, t), \\ 0 ⩽ x ⩽ 1, 0 < α ⩽ 1, t ⩾ 0, \end{matrix}$ (37)

With the initial condition: $u (x, 0) = x^{2} e^{2 x} 0 ⩽ x ⩽ 1,$ (38) and the boundary conditions: $u (0, t) = 0, u (1, t) = e^{2} (1 + t^{2}), t ⩾ 0 .$ (39) where, $\begin{matrix} f (x, t) = \frac{2 x^{2} t^{2 - α} e^{2 x}}{Γ (3 - α)} - 2 (1 + t^{2}) (1 + 4 x + 2 x^{2}) \\ e^{2 x} - ((1 + t^{2}) x^{2} e^{2 x}) (1 - {((1 + t^{2}) x^{2} e^{2 x})}^{3}) . \end{matrix}$ (40)

From the Equation (1), the exact solution analytically is u (x, t) = (1 + t²) x²e^2x.

Figure 5, display the illustrations clearly indicate that numerical and exact solution are indistinguishably comparable to one another in the same time, when x ∈ [0, 1] and α = 1.0. Figure 6, depicts the 3D-mesh plot for when x ∈ [0, 1] t ∈ [0, 0.1] and α = 1.0. In addition, the contour plot in Fig. 7, indicates. In Fig. 8, illustrates the study of the effect of time concentrations and it is established that, the time has no influence on u (x, t) especially when x ⩽ 0.9. From Table 2, shows error norms computed to estimate the validity of the present scheme and it is clear that, the numerical scheme and analytical solutions are completely in agreement with each other. In addition, Table 3, compared by using L₂ and L_∞ norm errors, the present scheme with Cubic B-spline method [22]. This comparison shows exactly meticulous the proposed method is. Additionally, a remarkable convention between the new approach and the previous published work is apparent.

Fig. 5

Curve of exact and numerical solution example 2, when x ∈ [0, 1], t = 0.05 and α = 1.0.

Fig. 6

Mesh of u (x, t) for example 2, when x ∈ [0, 1] t = ∈ [0, 0.1] and α = 1.0.

Fig. 7

Contour plot of u (x, t) for example 2, when x ∈ [0, 1], t = ∈ [0, 0.1] and α = 1.0.

Fig. 8

Time effect on u (x, t) for example 2, when x ∈ [0, 1] and α = 1.0.

Table 2

Absolute errors comparison of Example 2, when x ∈ [0, 1] and M = 100

t	α = 1		α = 0.8
	L _∞	L ₂	L _∞	L _∞
0.02	5.3870E-09	2.4281E-09	0.02	5.3870E-09
0.04	3.3119E-08	1.4959E-08	0.04	3.3119E-08
0.06	9.6173E-08	4.2047E-08	0.06	9.6173E-08
0.08	5.1927E-07	3.2842E-07	0.08	5.1927E-07

Table 3

Absolute errors comparison between present method and cubic B - spline method [22] for Example 2, when x ∈ [0, 1], M = 100 and α = 0.95.

t	Present method		Cubic B - spline method [22]
	L _∞	L ₂	L _∞	L ₂
0.5	6.4391E-06	2.0123E-06	2.12E-05	2.49E-06
0.75	2.2862E-06	1.0450E-06	2.13E-05	3.05E-06
1.0	9.6983E-07	2.9659E-07	3.3E-06	5.105E-07

Example 3. Consider the Equation (1), when ν = 1, r = 1 and λ = 2, given as: [22] $\begin{matrix} \frac{\partial^{α} u}{\partial t^{α}} - \frac{\partial^{2} u}{\partial x^{2}} - u (1 - u^{2}) = f (x, t), \\ 0 ⩽ x ⩽ 1, 0 < α ⩽ 1, t ⩾ 0, \end{matrix}$ (41)

With the initial condition: $u (x, 0) = 0, 0 ⩽ x ⩽ 1,$ (42) and the boundary conditions: $u (0, t) = u (1, t) = 0, t ⩾ 0 .$ (43) where, $\begin{matrix} f (x, t) = \frac{24 t^{4 - α}}{Γ (5 - α)} sin (2 π x) + 4 π^{2} t^{4} sin (2 π x) \\ - t^{4} sin (2 π x) (1 - t^{4} sin (2 π x)) . \end{matrix}$ (44) and exact solution analytically is u (x, t) = t⁴ sin(2πx).

In Fig. 9, illustrates the comparability of exact solution obtained analytically and numerical solution obtained by developed numerical scheme for example 3, when x ∈ [0, 1] and t = 0.5, and it clear the, the methods behave similarly for fixed values α = 0.5. The three-dimensional preview for mesh and contour plots is shown in Figs. 10 11. Also, the action of time on u (x, t) in various time t = 0.25 : 0.25 : 1, illustrates in Fig. 12, and the graph clearly shows that, the solutions of u (x, t) having the opposite direction in x ⩽ 0.5 and x ⩾ 0.5.

Fig. 9

Curve of exact and numerical solution example 3, when x ∈ [0, 1], t = 0.5 and α = 0.5.

Fig. 10

Mesh of u (x, t) for example 3, when x ∈ [0, 1] t ∈ [0, 1] and α = 0.5.

Fig. 11

Contour plot of u (x, t) for example 3, when x ∈ [0, 1] , t ∈ [0, 1] and α = 0.5.

Fig. 12

Time effect on u (x, t) for example 3, when x ∈ [0, 1] and α = 0.5.

Table 4, examined the norm error comparison of analytical and approximate solutions obtained by the proposed approach and the tabular data clearly show that both results strongly agree with each other. In addition, Table 5, compared the present scheme with Cubic B-spline method [22] by using L₂and L_∞norm errors,

Table 4

Absolute errors comparison of Example 3, when x ∈ [0, 1] and M = 100

t	α = 1		α = 0.5
	L _∞	L ₂	L _∞	L ₂
0.02	8.8416E-07	3.6729E-07	7.7373E-07	2.8300E-07
0.04	6.0616E-07	2.5180E-07	5.3030E-07	1.9398E-07
0.06	5.0945E-07	2.1165E-07	4.4636E-07	1.6320E-07
0.08	9.2092E-07	3.8259E-07	8.0656E-07	2.9494E-07

Table 5

Error norm comparison for Example 3 between present method and Cubic B - spline method [22], when x ∈ [0, 1] and α = 0.96

t	Present method		Cubic B - spline method [22]
	L _∞	L ₂	L _∞	L ₂
0.6	8262E-05	3.397E-06	1.97E-04	2.541E-05
0.8	6.732E-04	2.789E-05	6.366E-03	6.371E-04
1.0	3.520E-05	1.466E-06	3.9E-04	5.383E-05

This comparison appears the meticulousness for the proposed method. Moreover, an excellent convention is clearly noted between developed scheme and the previous published work.

7 Conclusion

In this paper, according to the trigonometric and exponential nonpolynomial spline has been employed and effective procedure for solving time fractional generalized Fisher equation. The truncation error is developed and the proposed numerical scheme is proved to be unconditional stable. The effect of time on the problem is studied. The numerical results are appeared to be more accurate as compared to the exact solution and pervious published work using cubic B-spline method. The figures and tables of comparison is also shown applicability and feasibility of our method.

Author declarations

Conflict of Interest: The authors have no conflicts to disclose.

Author Contributions: The manuscript was written through contributions from all authors. All authors have approved the final version of the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

Guo

, Pu

and Huang

, Fractional Calculus and Fractional Differential Equations, 2015. https://doi.org/10.1142/9789814667050_0002

Oliveira EdC de , Solved Exercises in Fractional Calculus: Studies in Systems, Decision and Control 240 (2019).

Agarwal

, Baleanu

, Chen

, Momani

and Machado

, 2019 Book FractionalCalculus.pdf. 2018.

Machado

C.M.

, GDJT, Introduction to Fractional Differential Equations, 25 (2019).

Glass

and Murray

J.D.

, Interdisciplinary Applied Mathematics, Mathematical Biology I 17 (2003).

Allahviranloo

, Studies in Fuzziness and Soft Computing Fuzzy Fractional Differential Operators and Equations Fuzzy Fractional Differential Equations, n.d.

Zhou

and Gu

, Numerical Study of Some Intelligent Robot Systems Governed by the Fractional Differential Equations, IEEE Access 7 (2019), 138548–138555. https://doi.org/10.1109/ACCESS.2019.2943089

Jannelli

and Speciale

M.P.

, Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries, Nonlinear Dyn 105 (2021), 2375–2385. https://doi.org/10.1007/s11071-021-06697-5

Khan

M.N.

, Ahmad

, Akgül

, Ahmad

and Thounthong

, Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method, Journal of Physics 95 (2021). https://doi.org/10.1007/s12043-020-02025-5

10.

Moghaddam

B.P.

, Mostaghim

Z.S.

, Pantelous

A.A.

and MachadoAn

J.A.T.

, integro quadratic spline-based scheme for solving nonlinearfractional stochastic differential equations with constant timedelay, Commun Nonlinear Sci Numer Simul 92 (2021). https://doi.org/10.1016/j.cnsns.2020.105475

11.

Khater

M.M.A.

, Attia

R.A.M.

, Qin

, Kadry

, Kharabsheh

and Lu

, On the stable computational, semi-analytical, and numericalsolutions of the Langmuir waves in an ionized plasma, Journalof Intelligent and Fuzzy Systems 38 (2020), 2847–2858. https://doi.org/10.3233/JIFS-179569

12.

Fisher

R.A.

, The wave of advance of advantageous genes , 7 (1937), 353–369.

13.

Kenkre

V.M.

, Results from variants of the Fisher equation in the study of epidemics and bacteria, Physica A: Statistical Mechanics and Its Applications 342 (2004), 242–248. https://doi.org/10.1016/j.physa.2004.04.084

14.

Mehmet Merdan , Solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative, International Journal of Physical Sciences 7 (2012), 2317–2326. https://doi.org/10.5897/ijps12.027

15.

Ross

, Villaverde

A.F.

, Banga

J.R.

, Vázquez

and Morán

, A generalized Fisher equation and its utility in chemical kinetics, Proc Natl Acad Sci U S A 107 (2010), 12777–12781. https://doi.org/10.1073/pnas.1008257107

16.

Verma

and Meher

, Solution for generalized fuzzy time-fractional Fisher’s equation using a robust fuzzy analytical approach, Journal of Ocean Engineering and Science 2022. https://doi.org/10.1016/j.joes.2022.03.019

17.

Baranwal

V.K.

, Pandey

R.K.

, Tripathi

M.P.

and Singh

O.P.

, An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on a new iterative method, Proc Natl Acad Sci U S A 107 (2012), 3906–3921. https://doi.org/10.1073/pnas.1008257107

18.

Khalid

, Abbas

and Iqbal

M.K.

, Non-polynomial quintic splinefor solving fourth-order fractional boundary value problemsinvolving product terms, Appl Math Comput 349 (2019), 393–407. https://doi.org/10.1016/j.amc.2018.12.066

19.

Saad

K.M.

, Khader

M.M.

, Gómez-Aguilar

J.F.

and Baleanu

, Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos 29 (2019). https://doi.org/10.1063/1.5086771

20.

Veeresha

, Prakasha

D.G.

and Baskonus

H.M.

, Novel simulations to the time-fractional Fisher’s equation, Mathematical Sciences 13 (2019), 33–42. https://doi.org/10.1007/s40096-019-0276-6

21.

Majeed

, Kamran

, Iqbal

M.K.

and Baleanu

, Solving time fractional Burgers’ and Fisher’s equations using cubic B-spline approximation method, Adv Differ Equ 2020 (2020). https://doi.org/10.1186/s13662-020-02619-8

22.

Majeed

, Kamran

, Abbas

and Singh

, An Efficient Numerical Technique for Solving Time-Fractional Generalized Fisher’s Equation, Front Phys 8 (2020). https://doi.org/10.3389/fphy.2020.00293

23.

Angstmann

C.N.

and Henry

B.I.

, Time fractional Fisher-KPP and Fitzhugh-Nagumo equations, Entropy 22 (2020), 1–13. https://doi.org/10.3390/e22091035

24.

Tamsir

, Dhiman

, Nigam

and Chauhan

, Approximation of caputo time-fractional diffusion equation using redefined cubic exponential b-spline collocation technique, AIMS Mathematics 6 (2021), 3805–3820. https://doi.org/10.3934/math.2021226

25.

Akram

, Abbas

and Ali

, A numerical study on time fractional fisher equation using an extended cubic b-spline approximation, Journal of Mathematics and Computer Science 22 (2020), 85–96. https://doi.org/10.22436/jmcs.022.01.08

26.

Khalid

, Abbas

, Iqbal

M.K.

, Singh

and Ahmad

A.I.

, A computational approach for solving time fractional differential equation via spline functions, Alexandria Engineering Journal 59 (2020), 3061–3078. https://doi.org/10.1016/j.aej.2020.06.007

27.

Mohyud-Din

S.T.

, Akram

, Abbas

, Ismail

A.I.

and Ali

N.H.M.

, A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation, Adv Differ Equ 2018 (2018). https://doi.org/10.1186/s13662-018-1537-7

28.

Demir

, Bayrak

M.A.

and Ozbilge

, An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method, Math Probl Eng 2018 (2018). https://doi.org/10.1155/2018/9471910

29.

Khader

M.M.

and Saad

K.M.

, A numerical approach for solving thefractional Fisher equation using Chebyshev spectral collocationmethod, Chaos Solitons Fractals 110 (2018), 169–177. https://doi.org/10.1016/j.chaos.2018.03.018

30.

Verma

and Meher

31.

Zhang

, Xia

and N’gbo

P.R.

, Global existence and uniqueness of a periodic wave solution of the generalized Burgers–Fisher equation, Appl Math Lett 121 (2021), 107353. https://doi.org/10.1016/j.aml.2021.107353

32.

Gupta

A.K.

and Ray

S.S.

, On the solutions of fractional burgers-fisher and generalized fisher’s equations using two reliable methods, Int J Math Math Sci 2014 (2014). https://doi.org/10.1155/2014/682910

33.

Rohila

and Mittal

R.C.

, Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method, Mathematical Sciences 12 (2018), 79–89. https://doi.org/10.1007/s40096-018-0247-3

34.

Rosa

, Bruzón

M.S.

and Gandarias

M.L.

, Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates, Commun Nonlinear Sci Numer Simul 25 (2015), 74–83. https://doi.org/10.1016/j.cnsns.2015.01.010

35.

El-Danaf

T.S.

, Ramadan

M.A.

and Abd Alaal

F.E.I.

, Numerical studies of the cubic non-linear Schrodinger equation, Nonlinear Dyn 67 (2012), 619–627. https://doi.org/10.1007/s11071-011-0014-6

36.

Isvand

, Numerical solution of the regularized long wave equation using nonpolynomial splines(2012), 459–471. https://doi.org/10.1007/s11071-011-0277-y

37.

Yousif

M.A.

and Hamasalh

F.K.

, Novel simulation of the time fractional Burgers–Fisher equations using a non-polynomial spline fractional continuity method, AIP Adv 12 (2022), 115018. https://doi.org/10.1063/5.0128819

38.

Hamasalh

F.K.

and Headayat

M.A.

, The applications of non-polynomial spline to the numerical solution for fractional differential equations, AIP Conf Proc 2334 (2021), https://doi.org/10.1063/5.0042319

39.

Hamasalh

F.K.

and Muhammed

P.O.

, Computational Non-Polynomial Spline Function for Solving Fractional Bagely-Torvik Equation, Mathematical Sciences Letters 6 (2017), 83–87. https://doi.org/10.18576/msl/060113

40.

Ding

and Wong

P.J.Y.

, Quintic non-polynomial spline for time-fractional nonlinear Schrödinger equation, Adv Differ Equ 2020 (2020). https://doi.org/10.1186/s13662-020-03021-0

41.

Akgül

, A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems (2019), 1–13. https://doi.org/10.3390/fractalfract3020033

42.

Hammad

D.A.

, Semary

M.S.

and Khattab

A.G.

, Ten non-polynomial cubic splines for some classes of Fredholm integral equations, Ain Shams Engineering Journal (2022), 13. https://doi.org/10.1016/j.asej.2021.101666