In this section, we present a new variation of the Discrete Temimi–Ansari method DTAM for handling stochastic linear electrical circuits with fractional order, which includes resistances, inductances, capacitances, and voltage sources for a new fractional operator with a singular kernel of Caputo type and nonsingular kernels of Caputo–Fabrizio and Atangana–Baleanu ABC types.
Consider the following differential equation in the form
$$\begin{aligned} &L \bigl[ v ( t ) \bigr] +N \bigl[ v ( t ) \bigr] +g ( t ) =0, \end{aligned}$$
(3a)
$$\begin{aligned} &\text{with initial conditions}\quad I \bigl( v, \bigl(d^{j} v\bigr)/ \bigl(d t^{j}\bigr) \bigr) =0 , \end{aligned}$$
(3b)
where L and N exemplify the linear and the nonlinear operators, respectively, and \(g(t)\) exemplifies the nonhomogeneous term. The Temimi–Ansari method was used to solve the differential Eq. (1) as follows:
To obtain the initial approximate function \(v_{0}(t)\), this is the solution to the following initial-value problem
$$\begin{aligned} L \bigl[ v_{0} ( t ) \bigr] +g ( t ) =0, \qquad I \bigl( v_{0}, \bigl(d^{j} v_{0}\bigr)/\bigl(d t^{j}\bigr) \bigr) =0. \end{aligned}$$
(4)
To acquire the next sacrificial function \(v_{1}(t)\), the following problem must be solved
$$\begin{aligned} L \bigl[ v_{1} ( t ) \bigr] +N \bigl[ v_{0} ( t ) \bigr] +g ( t ) =0, \qquad I \bigl( v_{1}, \bigl(d^{j} v_{1}\bigr)/ \bigl(d t^{j}\bigr) \bigr) =0. \end{aligned}$$
(5)
Also, the nth approximate functions \(v_{n} ( t ) \) can be evaluated in the same way. Then,
$$\begin{aligned} L \bigl[ v_{n} ( t ) \bigr] +N \bigl[ v_{n-1} ( t ) \bigr] +g ( t ) =0,\quad n=2,3,\ldots,\qquad I \bigl( v_{n}, \bigl(d^{j} v_{n}\bigr)/\bigl(d t^{j}\bigr) \bigr) =0. \end{aligned}$$
(6)
The produced iterative solution becomes close to the exact solution as the number of iterations increases
$$\begin{aligned} v ( t ) = \lim_{n\rightarrow \infty} v_{n} ( t ). \end{aligned}$$
(7)
The authors of [20, 22, 23] present an expanded investigation of error analysis and convergence standards for the TAM approach applied to an ordinary differential equation and its extension to systems of differential equations.
To present a study of convergence, we will start by:
$$\begin{aligned} \textstyle\begin{cases} \xi _{0} = v_{0} (t),\\ \xi _{1} =\Psi [ \xi _{0} ],\\ \xi _{2} =\Psi [ \xi _{0} + \xi _{1} ],\\ \vdots \\ \xi _{n} =\Psi [ \xi _{0} + \xi _{1} + \cdots + \xi _{n-1} ]. \end{cases}\displaystyle \end{aligned}$$
(8)
By defining the factor \(\Psi [v(t)]\) as
$$\begin{aligned} \Psi \bigl[ \xi _{n} ( t ) \bigr] =v {}_{n} (t) - \sum _{i=0}^{n-1} v_{i} ( t ),\quad i=1,2,3, \ldots, \end{aligned}$$
(9)
given that \(v_{n} ( t )\) is the solution for TAM.
Using these standards, convenient provisions for the convergence of TAM are discussed by the following theorems.
Theorem 1
The chain solution \(v ( t ) = \lim_{n\rightarrow \infty} v {}_{n} ( t )\) will appear as the exact solution to the given problem if this chain solution is convergent.
Proof
See [18, 22]. □
Theorem 2
Assume that Ψ stated in Eq. (9), is a factor from H to H, where H is a Hilbert space. The chain solution \(v(t)= \lim_{n\rightarrow \infty} v {}_{n} (t)\) converges if \(\exists 0 < \eta < 1\) such that
$$\begin{aligned} H \bigl\Vert \Psi [ \xi _{0} + \xi _{1} + \cdots + \xi _{n} ] \bigr\Vert \leq \eta \bigl\Vert \Psi [ \xi _{0} + \xi _{1} + \cdots + \xi _{n-1} ] \bigr\Vert \quad \forall \eta \mathbb{\in N\cup} \{ 0 \}. \end{aligned}$$
This notion is a specific state of the fixed-point notion and it is enough to prove the convergence of TAM.
Proof
See [18, 22]. □
Theorem 3
Whether the chain solution \(\sum_{i=0}^{\infty} v_{i} ( t ) \) is convergent to \(v(t)\), then the maximum error \(E_{n} (t)\) will be
$$\begin{aligned} E_{n} (t)\leq \frac{1}{1-\rho} \rho ^{n} \Vert v _{0} \Vert , \end{aligned}$$
(10)
where the chain \(\sum_{i=0}^{n-1} v_{i} (t)\) is employed to solve a wide class of nonlinear problems.
Proof
See [18, 22]. □
The acquired solution by the TAM converges to the exact solution as: \(\exists\ 0 < \eta < 1\) such that
$$\begin{aligned} D_{n} = \textstyle\begin{cases} \frac{ \Vert \xi _{n} \Vert }{ \Vert \xi _{n-1} \Vert }& \Vert \xi _{n} \Vert \neq 0,\\ 0,& \Vert \xi _{n} \Vert =0. \end{cases}\displaystyle \end{aligned}$$
(11)
The power-chain solution \(\sum_{n=0}^{\infty} v_{n} (t)\) converges to the exact solution \(v(t)\) when \(0\leq D_{n} < 1,\forall n=0,1,2, \ldots \)
Only for \(\alpha =1\), can the solution to the stochastic Eq. (1) be achieved using the TAM technique, however, due to the complexity of integrating random functions, only a few iterations are possible. As the fractional time derivative \({}_{0} D_{t}^{\alpha} \) t can be singular or nonsingular of type \({}_{0}^{C} D_{t}^{\alpha} \) or \({}_{0}^{CF} D_{t}^{\alpha} \) or \({}_{0}^{ABC} D_{t}^{\alpha} \), we propose the fractional discrete Temimi–Ansari (FDTAM) method to solve stochastic nonlinear differential Eq. (1) for a fractional operator with a local singular kernel type Caputo and a nonsingular kernel of Caputo–Fabrizio and Atangana–Baleanu ABC types as follows:
Liouville–Caputo sense
The FDTAM scheme for a Caputo fractional operator to approximate the solution for problem (1) will be as follows:
Assume: an n-point uniform mesh on \([0, T]\) as \(\{ i: i =1,\ldots, n \}\), \(0< t_{1} < t_{2}< \cdots < t_{n} = T \) with \(t_{i} - t_{i -1} = q\). Let \(h \in ( 0, q ]\) be a fixed constant for a fixed h.
The finite-difference form to approximate \(\frac{d \omega ( t_{j+1} )}{ dt}\) is given by
$$\begin{aligned} \frac{d \omega ( t_{i+1} )}{dt} = \frac{\omega _{i+1} - \omega _{i}}{ h}. \end{aligned}$$
(12)
The generalized Euler’s scheme to approximate \({}_{0}^{C} D_{t}^{\alpha} v_{0} ( t_{j +1} )\) is given by
$$\begin{aligned} v_{0} ( t_{i+1} ) = v_{0} ( t_{i} ) + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[G ( t_{i} ) +f ( t_{i} ) \frac{\omega ( t_{i+1} ) - \omega ( t_{i} )}{h} \biggr], \end{aligned}$$
(13)
where \(v_{0} ( t_{i+1} ) = v_{0}^{i+1}, v_{0} ( t_{i} ) =v_{0}^{i}, \omega ( t_{i+1} ) = \omega _{i+1}\) and \(\omega ( t_{i} ) = \omega _{i}\). Therefore, the first refined equation to approximate the initial approximate function \(v_{0}\) (\(t_{i+1}\)) is
$$\begin{aligned} v_{0}^{i+1} = v_{0}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[ G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr]. \end{aligned}$$
(14a)
The next discrete approximate function \(v_{1}\) (\(t_{i+1}\)) and the nth discrete approximate functions \(v_{n}\) (\(t_{i+1}\)) can be computed as follows
$$\begin{aligned} &v_{1}^{i+1} = v_{1}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[F \bigl( t_{i}, v_{0}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr], \end{aligned}$$
(14b)
$$\begin{aligned} &v_{n}^{i+1} = v_{n}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[ F \bigl( t_{i}, v_{n-1}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr]. \end{aligned}$$
(14c)
The solution will be calculated through k runs of various patterns of the Wiener process \(\omega (\mathrm{t})\), and then the refined scheme (14a)–(14c) can be written in the following form:
$$\begin{aligned} &v_{0,k}^{i+1} = v_{0,k}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[ G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr], \end{aligned}$$
(15a)
$$\begin{aligned} &v_{1,k}^{i+1} = v_{1,k}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[F \bigl( t_{i}, v_{0,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr], \end{aligned}$$
(15b)
$$\begin{aligned} &v_{n,k}^{i+1} = v_{n,k}^{i} + \frac{h^{\alpha}}{\Gamma ( \alpha +1 )} \biggl[ F \bigl( t_{i}, v_{n-1,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr]. \end{aligned}$$
(15c)
We must select a time step h to guarantee system convergence of (15a)–(15c). Setting the convergence gauge of the fixed-point iteration by dividing the right-hand side of Eq. (15b) we obtain:
$$\begin{aligned} &\biggl( \frac{h^{\alpha}}{\Gamma (\alpha +1)} \biggr) \frac{\partial F( t_{i}, v_{0,k}^{i} )}{\partial v_{0,k}^{i}} < 1, \end{aligned}$$
(16)
$$\begin{aligned} &h < \biggl( \frac{\Gamma (\alpha +1)}{\frac{\partial F( t_{i}, v_{0,k}^{i} )}{\partial v_{0,k}^{i}}} \biggr)^{\frac{1}{\alpha}}. \end{aligned}$$
(17)
Let \(g_{1} = \frac{\partial F( t_{i}, v_{0,k}^{i} )}{\partial v_{0,k}^{i}} \), then we have:
$$\begin{aligned} h < \biggl( \frac{\Gamma (\alpha +1)}{g_{1}} \biggr)^{\frac{1}{\alpha}}. \end{aligned}$$
(18)
The condition \(h < ( \frac{\Gamma (\alpha +1)}{g_{1}} )^{\frac{1}{\alpha}} \) is an appropriate condition for the time step used in the Liouville–Caputo sense for convergence. The other Eqs. (15a) and (15c) can use the same condition because of symmetry.
Caputo–Fabrizio sense
The FDTAM scheme for a Caputo–Fabrizio fractional operator to approximate the solution for the problem (1) will be as follows:
The Caputo–Fabrizio scheme to approximate \({}_{0}^{CF} D_{t}^{\alpha} v_{0} ( t_{j +1} )\) is given by
$$\begin{aligned} v_{0} ( t_{i+1} ) ={}& v_{0} ( t_{i} ) + \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{3\alpha h}{2\beta ( \alpha )} \biggr) \biggl[ G ( t_{i} ) +f ( t_{i} ) \frac{\omega ( t_{i+1} ) - \omega ( t_{i} )}{h} \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega ( t_{i} ) - \omega ( t_{i-1} )}{h} \biggr], \end{aligned}$$
(19)
where \(v_{0} ( t_{i+1} ) = v_{0}^{i+1}, v_{0} ( t_{i} ) =v_{0}^{i}, \omega ( t_{i+1} ) = \omega _{i+1}\), and \(\omega ( t_{i} ) = \omega _{i}\). Therefore, the first iterative equation to approximate the initial approximate function \(v_{0}\) (\(t_{i+1}\)) is
$$\begin{aligned} v_{0}^{i+1} ={}& v_{0}^{i} + \biggl( \frac{1-\alpha}{\beta (\alpha )} + \frac{3\alpha h}{2\beta (\alpha )} \biggr) \biggl[G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1} }{h} \biggr]. \end{aligned}$$
(20a)
The next Caputo–Fabrizio approximate function \(v_{1}\) (\(t_{i+1}\)) and the nth Caputo–Fabrizio approximate functions \(v_{n}\) (\(t_{i+1}\)) can be computed as follows
$$\begin{aligned} v_{1}^{i+1} ={}& v_{1}^{i} + \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{3\alpha h}{2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i}, v_{0}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{0}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{h} \biggr) \biggr], \end{aligned}$$
(20b)
$$\begin{aligned} v_{n}^{i+1} ={}& v_{n}^{i} + \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{3\alpha h}{2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i}, v_{n-1}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{n-1}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{h} \biggr) \biggr]. \end{aligned}$$
(20c)
The solution will be calculated using k runs of various Wiener process \(\omega (\mathrm{t})\) patterns, and the refined scheme (20a)–(20c) can be written as follows:
$$\begin{aligned} v_{0,k}^{i+1} ={}& v_{0,k}^{i} + \biggl( \frac{1-\alpha}{\beta (\alpha )} + \frac{3\alpha h}{2\beta (\alpha )} \biggr) \biggl[G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega ( t_{i} ) - \omega ( t_{i-1} )}{h} \biggr], \end{aligned}$$
(21a)
$$\begin{aligned} v_{1,k}^{i+1} ={}& v_{1,k}^{i} + \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{3\alpha h}{2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i}, v_{0,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{0,k}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{h} \biggr) \biggr], \end{aligned}$$
(21b)
$$\begin{aligned} v_{n,k}^{i+1} ={}& v_{n,k}^{i} + \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{3\alpha h}{2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i}, v_{n-1,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{1-\alpha}{\beta ( \alpha )} + \frac{\alpha h}{ 2\beta ( \alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{n-1,k}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{h} \biggr) \biggr]. \end{aligned}$$
(21c)
For facilitation we take \(\beta(\alpha)=1\).
Atangana–Baleanu sense
The FDTAM scheme for the Atangana–Baleanu fractional operator will be as follows to approximate the solution to problem (1):
The Atangana–Baleanu scheme to approximate \({}_{0}^{ABC} D_{t}^{\alpha} v_{0} ( t_{j +1} )\) is given by
$$\begin{aligned} v_{0} ( t_{i+1} ) ={}& v_{0} ( t_{i} ) + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[ G ( t_{i} ) +f ( t_{i} ) \frac{\omega ( t_{i+1} ) - \omega ( t_{i} )}{h} \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega ( t_{i} ) - \omega ( t_{i-1} )}{h} \biggr], \end{aligned}$$
(22)
where \(v_{0} ( t_{i+1} ) = v_{0}^{i+1}, v_{0} ( t_{i} ) =v_{0}^{i}, \omega ( t_{i+1} ) = \omega _{i+1}\), and \(\omega ( t_{i} ) = \omega _{i}\). Therefore, the first iterative equation to approximate the initial approximate function \(v_{0}\) (\(t_{i+1}\)) is
$$\begin{aligned} v_{0}^{i+1} ={}& v_{0}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega ( t_{i} ) - \omega ( t_{i-1} )}{h} \biggr]. \end{aligned}$$
(23a)
The next Atangana–Baleanu approximate function \(v_{1}\) (\(t_{i+1}\)) and the nth Atangana–Baleanu approximate functions \(v_{n}\) (\(t_{i+1}\)) can be computed as follows:
$$\begin{aligned} v_{1}^{i+1} ={}& v_{1}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[ F \bigl( t_{i}, v_{0}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{0}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{ h} \biggr) \biggr], \end{aligned}$$
(23b)
$$\begin{aligned} v_{n}^{i+1} ={}& v_{n}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[ F \bigl( t_{i}, v_{n-1}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{n-1}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{ h} \biggr) \biggr]. \end{aligned}$$
(23c)
The solution will be calculated using k runs of various Wiener process \(\omega (\mathrm{t})\) patterns, and the refined scheme (23a)–(23c) can be written as follows:
$$\begin{aligned} v_{0,k}^{i+1} ={}& v_{0,k}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega ( t_{i} ) - \omega ( t_{i-1} )}{h} \biggr], \end{aligned}$$
(24a)
$$\begin{aligned} v_{1,k}^{i+1} ={}& v_{1,k}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[ F \bigl( t_{i}, v_{0,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{0,k}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{ h} \biggr) \biggr], \end{aligned}$$
(24b)
$$\begin{aligned} v_{n,k}^{i+1} ={}& v_{n,k}^{i} + \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl( 1+ \frac{(1-\alpha )\Gamma (\alpha )}{h^{\alpha}} \biggr) \biggl[ F \bigl( t_{i}, v_{n-1,k}^{i} \bigr) + \biggl( G ( t_{i} ) +f ( t_{i} ) \frac{\omega _{i+1} - \omega _{i}}{h} \biggr) \biggr] \\ &{}+ \biggl( \frac{h^{\alpha}}{\beta ( \alpha ) \Gamma (\alpha )} \biggr) \biggl[ F \bigl( t_{i-1}, v_{n-1,k}^{i-1} \bigr) + \biggl( G ( t_{i-1} ) +f ( t_{i-1} ) \frac{\omega _{i} - \omega _{i-1}}{ h} \biggr) \biggr]. \end{aligned}$$
(24c)
Computing the mean and variance of the resulting sequences \(\{v_{n,1}, v_{n,2}, v_{n,3},\ldots, v_{n,k}\}\) will yield the solution’s mean and variance.
Finally, we obtain the mean and variance of the solution by taking the mean and variance of the solution sequences \(\{v_{n,1}, v_{n,2}, v_{n,3},\ldots, v_{n,k}\}\). These smart proposed numerical schemes are more effective than conventional TAM in solving fractional stochastic nonlinear differential equations with different fractional operators.