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 *n*th 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 *n*th 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 *n*th 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 *n*th 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.