The New Runge-Kutta Fehlberg Method for the Numerical Solution of Second-Order Fuzzy Initial Value Problems

We will discuss the fundamental terms and definitions related to fuzzy sets theory in this section, which will make it easier for us to understand the work in the following sections, such as the fuzzy fractional numbers, which are a generalization of the traditional crisp number [20-23]. It is important to note that the FDEs also employ the idea of fuzzy leve sets. The α-cut or fuzzy level sets are fuzzy numbers that convert the whole fuzzy system to crisp system with more genralzation transforms an imprecise data into precise data this called defuzzification [21]. On the other hand, it's important to remember the following basic concept of fuzzy sets:

Definition 2.2 [20]: $\tilde{\mathrm{f}}(\mathrm{x})$ is a fuzzy function if $\mathrm{f}: \mathbb{R} \rightarrow \widetilde{\mathrm{E}}$, where, if $\widetilde{E}$ be the set of all fuzzy numbers.

Definition 2.3 [20]: $\mathrm{f}: \mathrm{K} \rightarrow \widetilde{\mathrm{E}}$ called a fuzzy function process on interval $K \subseteq \widetilde{\mathrm{E}}$, then the $\alpha$-level set is:

$[\tilde{\mathrm{f}}(\mathrm{x})]_\alpha=[\underline{\mathrm{f}}(\mathrm{x} ; \alpha), \overline{\mathrm{f}}(\mathrm{x} ; \alpha)], \mathrm{x} \in \mathrm{K}, \alpha \in[0,1]$.

Using the α-level sets, we can characterize and describe fuzzy sets more effectively. They provide a comprehensive view of the fuzzy set's membership distribution, allowing us to observe the gradual change in membership grades. Moreover, α-level sets enable comparisons and analysis between different fuzzy sets based on their overlapping or containment relationships at different levels of resolution.

Definition 2.5 [24]: Each function $f: X \rightarrow Y$ induces another function $\tilde{f}: F(X) \rightarrow F(Y)$ defined for each fuzzy interval $\mathrm{U}$ in $\mathrm{X}$ by:

$\tilde{f}(\mathrm{U})(v)= \begin{cases}\operatorname{Sup}_{\mathrm{x} \in \mathrm{f}^{-1}(\mathrm{y})} \mathrm{U}(\mathrm{x}) & , \text { if } v \in \operatorname{range}(\mathrm{f}) \\ 0 & , \text { if } v \notin \operatorname{range}(\mathrm{f})\end{cases}$

This is called the Zadeh extension principle.

Definition 2.6 [25]: Consider $\widetilde{x}, \widetilde{v} \in \widetilde{E}$. If there exists $\widetilde{z} \in \widetilde{E}$ such that $\widetilde{x}=\widetilde{v}+\widetilde{z}$, then $\widetilde{z}$ is called the H-difference (Hukuhara difference) of $\mathrm{x}$ and $\mathrm{y}$ and is denoted by $\widetilde{z}=\widetilde{x} \ominus \widetilde{v}$.

Definition 2.7 [20]: If $\tilde{\mathrm{f}}: \mathrm{I} \rightarrow \widetilde{\mathrm{E}}$ and $\mathrm{y}_0 \in \mathrm{I}$, where $\mathrm{I} \in\left[\mathrm{x}_0, \mathrm{~K}\right]$. We say that $\tilde{f}$ Hukuhara differentiable at $y_0$, if there exists an element $\left[\tilde{\mathrm{f}}^{\prime}\right]_\alpha \in \widetilde{\mathrm{E}}$ such that for all $\mathrm{h}>0$ sufficiently small (near to 0$)$, exists $\tilde{\mathrm{f}}\left(v_0+\mathrm{h} ; \alpha\right) \ominus \tilde{\mathrm{f}}\left(v_0 ; \alpha\right), \tilde{\mathrm{f}}\left(v_0 ; \alpha\right) \ominus \tilde{\mathrm{f}}\left(\mathrm{y}_0-\mathrm{h} ; \alpha\right)$ and the limits are taken in the metric $(\widetilde{\mathrm{E}}, \mathrm{D})$:

$\lim _{\mathrm{h} \rightarrow 0+} \frac{\tilde{\mathrm{f}}\left(v_0+\mathrm{h} ; \alpha\right) \ominus \tilde{\mathrm{f}}\left(\mathrm{y}_0 ; \alpha\right)}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0+} \frac{\tilde{\mathrm{f}}\left(v_0 ; \alpha\right) \ominus \tilde{\mathrm{f}}\left(v_0-\mathrm{h} ; \alpha\right)}{\mathrm{h}}$

The fuzzy set $\left[\widetilde{\mathrm{f}}^{\prime}\left(v_0\right)\right]_\alpha$ is called the Hukuhara derivative of $\left[\tilde{\mathrm{f}}^{\prime}\right]_\alpha$ at $v_0$.

These limits are taken in the space $(\widetilde{\mathrm{E}}, \mathrm{D})$ if $\mathrm{x}_0$ or $\mathrm{K}$, then we consider the corresponding one-side derivation. Recall that $\tilde{\mathrm{x}} \Theta \tilde{v}=\tilde{\mathrm{z}} \in \widetilde{\mathrm{E}}$ are defined on $\alpha$-level set, where $[\tilde{\mathrm{x}}]_\alpha \ominus[\tilde{v}]_\alpha=[\tilde{\mathrm{z}}]_\alpha, \forall \alpha \in[0,1]$. By consideration of definition of the metric D all the r-level set $[\tilde{\mathrm{f}}(0)]_\alpha$ are Hukuhara differentiable at $y_0$, with Hukuhara derivatives $\left[\widetilde{\mathrm{f}^{\prime}}\left(v_0\right)\right]_\alpha$, when $\tilde{\mathrm{f}}: \mathrm{I} \rightarrow \widetilde{\mathrm{E}}$ is Hukuhara differentiable at $y_0$ with Hukuhara derivative $\left[\widetilde{\mathrm{f}^{\prime}}\left(v_0\right)\right]_\alpha$ it' lead to that $\tilde{\mathrm{f}}$ is Hukuhara differentiable for all $\alpha \in[0,1]$ which satisfies the above limits i.e. if $\mathrm{f}$ is differentiable at $\mathrm{x}_0 \in\left[\mathrm{x}_0+\alpha, \mathrm{K}\right]$ then all its $\alpha$-levels $\left[\tilde{\mathrm{f}}^{\prime}(\mathrm{x})\right]_\alpha$ are Hukuhara differentiable at $\mathrm{x}_0$.

Definition 2.8 [20]: Define the mapping $\widetilde{\mathrm{f}^{\prime}}: \mathrm{I} \rightarrow \widetilde{\mathrm{E}}$ and $\mathrm{y}_0 \in \mathrm{I}$, where $\mathrm{I} \in\left[\mathrm{x}_0, \mathrm{~K}\right]$. We say that $\widetilde{\mathrm{f}^{\prime}}$ Hukuhara differentiable $\mathrm{x} \in \widetilde{\mathrm{E}}$, if there exists an element $\left[\tilde{\mathrm{f}}^{(\mathrm{n})}\right]_\alpha \in \widetilde{\mathrm{E}}$ such that for all $\mathrm{h}>0$ sufficiently small (near to 0$)$, exists $\tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0+\mathrm{h} ; \alpha\right) \ominus \tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0 ; \alpha\right), \tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0 ; \alpha\right) \ominus \tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0-\mathrm{h} ; \alpha\right)$ and the limits are taken in the metric $(\widetilde{\mathrm{E}}, \mathrm{D})$.

$\lim _{\mathrm{h} \rightarrow 0+} \frac{\tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0+\mathrm{h} ; \alpha\right) \Theta \tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0 ; \alpha\right)}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0+} \frac{\tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0 ; \alpha\right) \Theta \tilde{\mathrm{f}}^{(\mathrm{n}-1)}\left(v_0-\mathrm{h} ; \alpha\right)}{\mathrm{h}}$

exists and equal to $\tilde{\mathrm{f}}^{(\mathrm{n})}$ and for $\mathrm{n}=2$, we have second order Hukuhara derivative.

Theorem 2.2 [20]: Let $\tilde{\mathrm{f}}:\left[\mathrm{x}_0+\alpha, \mathrm{K}\right] \rightarrow \widetilde{\mathrm{E}}$ be Hukuhara differentiable and denote.

$\left[\widetilde{\mathrm{f}}^{\prime}(\mathrm{x})\right]_\alpha=\left[\underline{\mathrm{f}}^{\prime}(\mathrm{x}), \overline{\mathrm{f}}^{\prime}(\mathrm{x})\right]_\alpha=\left[\underline{\mathrm{f}}^{\prime}(\mathrm{x} ; \alpha), \overline{\mathrm{f}}^{\prime}(\mathrm{x} ; \alpha)\right]$.

Then the boundary functions $\underline{f}^{\prime}(x ; \alpha), \bar{f}^{\prime(x ; \alpha)}$ are differentiable we can write for second order fuzzy derivative:

$\left[\tilde{\mathrm{f}}^{\prime \prime}(\mathrm{x})\right]_\alpha=\left[\left(\underline{\mathrm{f}}^{\prime \prime}(\mathrm{x} ; \alpha)\right)^{\prime},\left(\overline{\mathrm{f}}^{\prime \prime}(\mathrm{x} ; \alpha)\right)^{\prime}\right], \forall \alpha \in[0,1]$.

Consider the exact solution of Eq. (2) that can define by:

$\left[\widetilde{\mathrm{V}}\left(\mathrm{x}_{\mathrm{n}}\right)\right]_\alpha=\left[\underline{\mathrm{V}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \overline{\mathrm{V}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right]$

while the numerical solution defines as:

$\left[\tilde{v}\left(x_n\right)\right]_\alpha=\left[\underline{v}\left(x_n ; \alpha\right), \bar{v}\left(x_n ; \alpha\right)\right]$

By iteratively applying the RKF5 formulas, RKF5 is an explicit method which calculates the approximate solution to the ODE at the desired future time. It adjusts the step size dynamically based on error estimates to maintain accuracy while minimizing computational effort. The method provides an efficient and reliable approach to numerically solve ODEs, particularly those with smooth solutions [26-28]. The basis of all Runge-Kutta methods in is to express the difference between the value of $\left[\tilde{y}\left(x_n\right)\right]_\alpha$ at $x_{n+1}$ and $x_n$ as in such that for each $\alpha \in[0,1]$:

$\tilde{y}_{n+1}\left(x_{n+1} ; \alpha\right)-\tilde{y}_n\left(x_n ; \alpha\right)=\sum_{i=1}^N c_n k_n$     (3)

where, for n=0,1, 2..., N the c_n are the constants and:

$\mathrm{k}_{\mathrm{i}}=\mathrm{f}\left(\mathrm{x}_{\mathrm{n}}+\mathrm{h} \mathrm{r}_{\mathrm{n}}, \tilde{\mathrm{y}}\left(\mathrm{x}_{\mathrm{n}} ; \tilde{\mathrm{y}}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\mathrm{h} \sum_{\mathrm{j}=1}^{\mathrm{s}-1} \gamma_{\mathrm{js}} \mathrm{k}_{\mathrm{i}}\right)$     (4)

To specify a particular method, one needs to provide the integer N (the number of stages), and the coefficients γ_jn, c_n and r_n (for n, j = 1, 2..., N). The matrix [γ_jn] is called the Runge–Kutta matrix, while the c_n and r_n are known as the weights and the nodes [14]. These data are usually arranged in a mnemonic device, known as a Butcher tableau [29]:

tu_.png

Eq. (2) is to be exact for powers of h through h^p, where p is the order of the Runge-Kutta methods, because it is to be coincident with Taylor series of order p [29]. Therefore from [14], the truncation error T_N in of order p+1 can be written:

$\mathrm{T}_{\mathrm{N}}=\delta_{\mathrm{N}} \mathrm{h}^{\mathrm{p}+1}+\mathrm{O}\left(\mathrm{h}^{\mathrm{p}+2}\right)$     (5)

where, the value of $\delta_{\mathrm{N}}$ will generally be much less than the bound. The nonzero constants $r_n, \gamma_{j n}$ in RKF5 given in the study [30] as $r_1=0, r_2=\frac{1}{4}, r_3=\frac{3}{8}, r_4=\frac{12}{13}, r_5=1, r_6=\frac{1}{2}$ and $\gamma_{21}=\frac{1}{4}, \gamma_{31}=\frac{3}{32}, \gamma_{41}=\frac{3}{16}, \gamma_{51}=\frac{439}{216}, \gamma_{61}=-\frac{8}{27}, \gamma_{32}=\frac{9}{32}, \gamma_{42}=$ $-\frac{7200}{2197}, \gamma_{43}=\frac{7296}{2197}, \gamma_{52}=-8, \gamma_{53}=\frac{3680}{513}, \gamma_{54}=-\frac{845}{4104}, \gamma_{62}=2, \gamma_{63}=-\frac{3544}{2565}, \gamma_{64}=\frac{1859}{4104}, \gamma_{64}=\frac{-11}{40}, c_1=\frac{16}{135}, c_2=0, c_3=$ $\frac{6656}{12825}, c_4=\frac{28561}{56430}, c_5=\frac{-9}{50}, c_6=\frac{2}{55}$, hence:

$\tilde{v}_{\mathrm{n}+1}\left(\mathrm{x}_{\mathrm{n}+1} ; \alpha\right)=\tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{16}{135} k_1+\frac{6656}{12825} k_3+\frac{28561}{56430} k_4-\frac{9}{50} k_5+\frac{2}{55} k_6$     (6)

where,

$\begin{gathered}\mathrm{k}_1=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right), \\ \mathrm{k}_2=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}+\frac{\mathrm{h}}{4}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{\mathrm{k}_1}{4}\right), \\ \mathrm{k}_3=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}+\frac{3 \mathrm{~h}}{8}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{3 \mathrm{k}_1}{32}+\frac{9 \mathrm{k}_2}{32}\right), \\ \mathrm{k}_4=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}+\frac{\mathrm{h}}{2}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{1932 \mathrm{k}_1}{2197}-\frac{7002 \mathrm{k}_2}{2197}+\frac{7296 \mathrm{k}_3}{2197}\right), \\ \mathrm{k}_5=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}+\frac{3 \mathrm{~h}}{4}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{439 \mathrm{k}_1}{216}-8 \mathrm{k}_2+\frac{3680 \mathrm{k}_3}{513}+\frac{845 \mathrm{k}_4}{4104}\right), \\ \mathrm{k}_6=\mathrm{hf}\left(\mathrm{x}_{\mathrm{n}}+\mathrm{h}, \tilde{v}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)+\frac{8 \mathrm{k}_1}{27}+2 \mathrm{k}_2-\frac{3544 \mathrm{k}_3}{2565}+\frac{1859 \mathrm{k}_4}{4104}-\frac{11 \mathrm{k}_4}{40}\right) .\end{gathered}$

To solve Eq. (2) in linear and nonlinear forms numerically the new from RKF5 is obtained from the following fuzzy analysis as below:

$\begin{aligned} & v\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\underline{v}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\mathrm{h} \sum_{\mathrm{j}=1}^6 \mathrm{w}_{\mathrm{j}} \underline{\mathrm{k}}_{\mathrm{j}, 1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{y}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right), \\ & \overline{\mathrm{y}}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\bar{v}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\mathrm{h} \sum_{\mathrm{j}=1}^6 \mathrm{w}_{\mathrm{j}} \overline{\mathrm{k}}_{\mathrm{j}, 2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right), \\ & \underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\mathrm{h} \sum_{\mathrm{j}=1}^6 \mathrm{w}_{\mathrm{j}} \underline{\mathrm{L}}_{\mathrm{j}, 1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right), \\ & \overline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\overline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\mathrm{h} \sum_{\mathrm{j}=1}^6 \mathrm{w}_{\mathrm{j}} \overline{\mathrm{L}}_{\mathrm{j}, 2}\left(\mathrm{x}_{\mathrm{n}}, \bar{v}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right), \mathrm{z}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)\right),\end{aligned}$

where, the $\mathrm{w}_{\mathrm{j}}$ are constants, $\underline{\mathrm{k}}_{\mathrm{j}, 1}, \overline{\mathrm{k}}_{\mathrm{j}, 2}, \underline{\mathrm{L}}_{\mathrm{j}, 1}$, and $\overline{\mathrm{L}}_{\mathrm{j}, 2}$ for $\mathrm{j}=1,2,3,4,5,6$ are defined as follows for $\tilde{v} \in\left[\underline{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \bar{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right], \tilde{\mathrm{z}} \in$ $\left[\underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \overline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right]:$

$\begin{aligned} & \underline{\mathrm{k}}_{1,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)=\min \left\{\underline{\mathrm{f}}\left(\mathrm{x}_{\mathrm{n}}, \underline{v}, \underline{\mathrm{z}}\right)\right\} \\ & \overline{\mathrm{k}}_{1,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{y}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)=\max \left\{\overline{\mathrm{f}}\left(\mathrm{x}_{\mathrm{n}}, \bar{v}, \overline{\mathrm{z}}\right)\right\} \\ & \underline{\mathrm{L}}_{1,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{y}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)=\min \left\{\underline{\mathrm{g}}\left(\mathrm{x}_{\mathrm{n}}, \underline{v}, \underline{\mathrm{z}}\right)\right\} \\ & \overline{\mathrm{L}}_{1,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)=\max \left\{\overline{\mathrm{g}}\left(\mathrm{x}_{\mathrm{n}}, \bar{v}, \overline{\mathrm{z}}\right)\right\}\end{aligned}$

${{\underline{\text{k}}}_{2,1}}\left( {{\text{x}}_{\text{n}}},\widetilde{\text{v}}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)=\min \left\{ \underline{\text{f}}\left( \begin{matrix}   {{\text{x}}_{\text{n}}}+\frac{\text{h}}{4},\underset{\scriptscriptstyle-}{v}+\frac{\text{h}}{4}{{\underline{\text{k}}}_{1,1}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\   \underline{\text{z}}+\frac{\text{h}}{4}{{\underline{\text{L}}}_{1,1}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\\end{matrix} \right) \right\}$

${{\overline{\text{k}}}_{2,2}}\left( {{\text{x}}_{\text{n}}},\widetilde{\text{v}}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)=\max \left\{ \overline{\text{f}}\left( \begin{matrix}   {{\text{x}}_{\text{n}}}+\frac{\text{h}}{4},\bar{v}+\frac{\text{h}}{4}{{\underline{\text{k}}}_{1,2}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right),  \\   \overline{\text{z}}+\frac{\text{h}}{4}{{\underline{\text{L}}}_{1,2}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\\end{matrix} \right) \right\}$

${{\underline{\text{L}}}_{2,1}}\left( {{\text{x}}_{\text{n}}},\widetilde{\text{v}}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)=\min \left\{ \underline{\text{g}}\left( \begin{matrix}   {{\text{x}}_{\text{n}}}+\frac{\text{h}}{4},\underset{\scriptscriptstyle-}{v}+\frac{\text{h}}{4}{{\underline{\text{k}}}_{1,1}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\   \underline{\text{z}}+\frac{\text{h}}{4}{{\underline{\text{L}}}_{1,1}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\\end{matrix} \right) \right\}$

${{\overline{\text{K}}}_{2,2}}\left( {{\text{x}}_{\text{n}}},\widetilde{\text{v}}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)=\max \left\{ \overline{\text{g}}\left( \begin{matrix}   {{\text{x}}_{\text{n}}}+\frac{\text{h}}{4},\bar{v}+\frac{\text{h}}{4}{{\overline{\text{k}}}_{1,2}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right),  \\   \overline{\text{z}}+\frac{\text{h}}{4}{{\overline{\text{L}}}_{1,2}}\left( {{\text{x}}_{\text{n}}},\tilde{v}\left( {{\text{x}}_{\text{n}}};\alpha  \right),\widetilde{\text{z}}\left( {{\text{x}}_{\text{n}}};\alpha  \right) \right)  \\\end{matrix} \right) \right\}$

k31.png

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Now, the RKF5 fuzzy formula is define as follows:

$\begin{gathered}\underline{v}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\underline{v}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\frac{16}{135} \underline{\mathrm{k}}_{1,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{6656}{12825} \underline{\mathrm{k}}_{3,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right) \\ +\frac{28561}{56430} \underline{\mathrm{k}}_{4,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{9}{50} \underline{\mathrm{k}}_{5,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{2}{55} \underline{\mathrm{k}}_{6,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)\end{gathered}$

$\begin{gathered}\quad \bar{v}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\bar{v}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\frac{16}{135} \overline{\mathrm{k}}_{1,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{6656}{12825} \overline{\mathrm{k}}_{3,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right) \\ +\frac{28561}{56430} \overline{\mathrm{k}}_{4,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{9}{50} \overline{\mathrm{k}}_{5,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{2}{55} \overline{\mathrm{k}}_{6,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)\end{gathered}$

$\begin{gathered}\underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\frac{16}{135} \underline{\mathrm{L}}_{1,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{6656}{12825} \underline{\mathrm{L}}_{3,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right) \\ +\frac{28561}{56430} \underline{\mathrm{L}}_{4,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{9}{50} \underline{\mathrm{L}}_{5,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{2}{55} \underline{\mathrm{L}}_{6,1}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)\end{gathered}$

$\begin{gathered}\overline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}+1}, \alpha\right)=\underline{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}}, \alpha\right)+\frac{16}{135} \overline{\mathrm{L}}_{1,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{\mathrm{v}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{6656}{12825} \overline{\mathrm{L}}_{3,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right) \\ +\frac{28561}{56430} \overline{\mathrm{L}}_{4,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{9}{50} \overline{\mathrm{L}}_{5,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)+\frac{2}{55} \overline{\mathrm{L}}_{6,2}\left(\mathrm{x}_{\mathrm{n}}, \tilde{v}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right), \tilde{\mathrm{z}}\left(\mathrm{x}_{\mathrm{n}} ; \alpha\right)\right)\end{gathered}$

where, the step size $\mathrm{h}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{N}}$ and $\mathrm{N}$ is number of numerical iterations. The congruence and error analysis for fuzzy fifth order RangeKutta methods is illustrated [18] such that numerical solution $\left[\tilde{v}\left(\mathrm{x}_n\right)\right]$ convergent to the exact solution $\left[\widetilde{\mathrm{V}}\left(\mathrm{x}_n\right)\right]$ as $\mathrm{h} \rightarrow 0$ each $\alpha \in$ [0,1].

This research proposes a computational approach for solving fuzzy differential equations (FDEs). The approach is based on using the RKF5 method to solve linear and nonlinear second-order FIVPs with fuzzy initial conditions. To make the RKF5 method suitable for solving these problems, a complete fuzzy analysis is presented, which evaluates the method's performance under fuzzy properties using principles of fuzzy set theory. The proposed method is tested on several examples with different numbers of iterations and input domains, using various levels of fuzzy sets. The results show that the RKF5 method performs better than other existing methods in terms of accuracy. The findings are presented in tables that follow the characteristics of fuzzy numbers, which validate the RKF5 fuzzy numerical solution. Overall, this research presents a promising approach for solving fuzzy FDEs using the RKF5 method and provides insights into the behaviour of fuzzy systems under different input conditions.