Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

1.   Introduction

Fractional calculus is as old as ordinary calculus, i. e. at three centuries; however, it is not widely used in research and engineering. In a letter to Leibniz dated September 30, 1695, L'Hopital introduced the concept of fractional-order derivatives. In Lacroix's writings, P. S. Laplace defined a fractional derivative of arbitrary order in 1812. Several researchers have developed excellent literature on fractional differentiation and integration operators for the purposes of extending scientific and technical areas, including Caputo ^[1], Oldham and Spanier ^[2], Carpinteri and Mainardi ^[3], Samko et al. ^[4], Ahmed et al. ^[5], Podlubny ^[6], Atangana and Alabaraoye ^[7], Kumar et al. ^[8], Yin et al. ^[9] and Arife et al. ^[10]. The beauty of fractional derivatives is that they are not local point properties. The genetic and nonlocal dispersed effects are taken into account in fractional calculus. This property makes it more accurate than the integer-order derivative description.

Due to their proven applications in science and engineering, fractional differential equations have grown in prominence and popularity. These equations, for example, are increasingly being utilized to simulate problems in signal processing, biology, fluid mechanics, acoustics, diffusion, electromagnetism and a wide range of other physical phenomena. The nonlocal quality of fractional differential equations is the most essential advantage of utilizing them in these and other applications. The integer order differential operator is well-known to be a local operator, whereas the fractional order differential operator is not. This indicates that a system's next state is determined not just by its current state, but also by all of its previous states. The theory of fractional differential equations better and more systematically describes natural occurrences ^{[11,12,13,14,15,16,17,18,19]}.

Fractional coupled systems are commonly used to investigate the complex behavior of plasma that contains several components such as atoms, free electrons and ions. Many scholars have tried to assess this behavior. Recently, Paul Kersten and Joseph Krasil'shchik investigated the Korteweg-de Vries (KdV) equation and modified KdV (mKdV) equation, proposing absolute complexity between coupled KdV-mKdV nonlinear systems for the investigation of nonlinear system behavior ^{[20,21,22,23]}. Many scholars have proposed numerous variants of this Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system ^{[24,25,26,27,28]}. Among these variants, the nonlinear fractional Kersten-Krasil'shchik linked KdV-mKdV system provides a mathematical model for understanding the behavior of multi-component plasma for waves travelling along the positive $zeta$ axis:

where $\vartheta$ is a spatial coordinate and $\chi$ is a time coordinate. The fractional operator's order is represented by the factor $\alpha$. The Caputo form is used to study this operator. When $\alpha = 1$, the fractional coupled system becomes a classical system, as follows:

If $\mathbb{G} = 0$, the Kersten-Krasil'shchik linked KdV-mKdV system is converted to the well-known KdV system as follows:

When $\mathbb{F} = 0$, the Kersten-Krasil'shchik coupled KdV-mKdV system becomes the well-known mKdV system as follows:

As a result, the Kersten-Krasil'shchik linked KdV-mKdV system can be considered of as a combination of the KdV system and the mKdV system, which are represented by (1.2) to (1.4). We also investigate the following fractional nonlinear two component homogeneous time fractional coupled third order KdV system in this work as follows:

where $\chi$ is the temporal coordinate, $\vartheta$ is the spatial coordinate and $\alpha$ is the fractional operator's order factor. The Caputo form is used to study this operator. When $\alpha = 1$, the fractional coupled system becomes a classical system, as follows:

To solve the differential equations, we employ the natural decomposition method (NDM), which combines the natural transform (NT) and Adomian decomposition method and offers the approximate solution in series form. The proposed method has been implemented with the aid of two different fractional derivatives to solve two nonlinear systems. Many researchers have employed the NDM to obtain approximate analytical solutions, and it has provided reliable and closely convergent results. The calculations were done in Maple. The convergence of the proposed technique was also achieved by extending the concept discussed in ^[29,30].

The organization of the present paper is as follows: Section 2 gives some basic definitions and the properties of the natural transform that is used in our present work. Section 3 handles the methodology of the proposed technique. In Section 4, we presented the convergence analysis of the suggested technique. Section 5 gives the implementation of the suggested approach to approximate the solution of the above systems. Finally, we discuss the obtained results and conclusion.

2.   Preleminaries

In this section, we recall some basic definitions and results from fractional calculus.

Definition 2.1. The Riemann-Liouville integral of a function $j\in C_{\mu}$, $\mu\geq-1$, having fractional-order is defined as ^[31]

Definition 2.2. The derivative of $j(\rho)$ with fractional-order in the Caputo sense is given as ^[31]

for ${m}-1 < {{\alpha}}\leq {m}, \ \ {m}\in N, \ \ \rho > 0, \; j\in C^{{m}}_{\mu}, \; \mu\geq-1$.

Definition 2.3. The derivative of $j(\rho)$ with fractional-order in the Caputo-Fabrizio (CF) manner is given as ^[31]

where $0 < {{\alpha}} < 1$ and $F({{\alpha}})$ is a normalization function with $F(0) = F(1) = 1$.

Definition 2.4. The derivative of $j(\rho)$ with fractional-order in the Atangana-Baleanu-Caputo operator (ABC) manner is given as ^[31]

where $0 < {{\alpha}} < 1$, $B(\alpha)$ is normalization function and $E_{{{\alpha}}}(z) = \sum_{m = 0}^{\infty}\frac{z^{m}}{\Gamma({\alpha}^{m}+1)}$ is the Mittag-Leffler function.

Definition 2.5. For the function $\mathbb{F}(\chi)$, the natural transformation is given as

and for $\chi\in(0, \infty)$, the natural transformation of $\mathbb{F}(\chi)$ is given as

where $H(\chi)$ is the Heaviside function.

Definition 2.6. For the function $\mathbb{F}({\zeta}, {{\tau}})$, the inverse natural transformation is given as

Lemma 2.1. Let the $\mathbb{F}_{1}(\chi)$ and $\mathbb{F}_{2}(\chi)$ natural transformations be $\mathcal{U}_{1}({\zeta}, {{\tau}})$ and $\mathcal{U}_{2}({\zeta}, {{\tau}})$; so,

where $c_{1}$ and $c_{2}$ constants.

Lemma 2.2. Let the $\mathbb{F}_{1}(\chi)$ and $\mathbb{F}_{2}(\chi)$ inverse natural transformations be $\mathbb{F}_{1}({\zeta}, {{\tau}})$ and $\mathbb{F}_{2}({\zeta}, {{\tau}})$; so,

where $c_{1}$ and $c_{2}$ constants.

Definition 2.7. The NT of $D_{\chi}^{{{\alpha}}}\mathbb{F}(\chi)$ in the Caputo sense is stated as ^[31]

Definition 2.8. The NT of $D_{\chi}^{{{\alpha}}}\mathbb{F}(\chi)$ in the CF sense is defined as ^[31]

Definition 2.9. The NT of $D_{\chi}^{{{\alpha}}}\mathbb{F}(\chi)$ in the ABC manner is given as ^[31]

with $M[{{\alpha}}]$ denoting a normalization function.

3.   Methodology

This section is concerned with the general procedure for numerical treatment of the below equation.

where the initial condition

has $\mathcal{L}$ and ${N}$ linear and nonlinear terms, and where $h(\vartheta, \chi)$ represents the source term.

3.1.   Case I $(NTDM_{CF})$

By means of the NT and CF fractional derivative, Eq (3.1) can be stated as

with

By taking the natural inverse transform, we get

Assume that for $\mathbb{F}(\vartheta, \chi)$, the series form solution is determined as

and the ${N}(\mathbb{F}(\vartheta, \chi))$ decomposition is given as

with $A_{i}$ representing the Adomian polynomials, which is illustrated as

By putting Eqs (3.6) and (3.7) into Eq (3.5), we have

From Eq (3.8), we obtain

In this way, the solution of Eq (3.1) is obtained by putting Eq (3.9) into Eq (3.6) to solve for the $NTDM_{CF}$,

3.2.   Case II $(NTDM_{ABC})$

By means of the NT and ABC fractional derivative, Eq (3.1) can be stated as

with

By taking the natural inverse transform, we get

In terms of the Adomain decomposition, we have

From Eq (3.8), we have the following:

In this way, the solution of Eq (3.1) is obtained to solve for the $NTDM_{ABC}$:

4.   Convergence analysis

This section is concerned with the $NTDM_{ABC}$ and $NTDM_{CF}$ convergence and uniqueness.

Theorem 4.1. Let $|\mathcal{L}(\mathbb{F})-\mathcal{L}(\mathbb{F}^{*})| < \gamma_{1}|\mathbb{F}-\mathbb{F}^{*}|$ and $|{N}(\mathbb{F})-{N}(\mathbb{F}^{*})| < \gamma_{2}|\mathbb{F}-\mathbb{F}^{*}|$, where $\mathbb{F}: = \mathbb{F}(\mu, \chi)$ and $\mathbb{F}^{*}: = \mathbb{F}^{*}(\mu, \chi)$ are two variable functions values, and $\gamma_{1}$, $\gamma_{2}$ are Lipschitz constants.

The operators $\mathcal{L}$ and ${N}$ are given in Eq (3.1). Thus, the solution for the $NTDM_{CF}$ is unique for Eq (3.1) when $0 < (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}} \chi) < 1$ for all $\chi.$

Proof. Let $K = (C[J], ||.||)$, where norm $||\phi(\chi)|| = {max}_{\chi\in J}|\phi(\chi)|$ is the Banach space and $\forall$ continuous on the $J = [0, T]$ interval. Let $I:K\rightarrow K$ be a nonlinear mapping with

So, $I$ is a contraction as $0 < (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}} \chi) < 1$. Thus, by means of the Banach fixed point theorem, the solution of Eq (3.1) is unique.

Theorem 4.2. According to the above theorem, the solution of Eq (3.1) is unique for the $NTDM_{ABC}$ when $0 < (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}}\frac{\chi^{\alpha}}{\Gamma(\alpha+1)}) < 1$ for all $\chi.$

Proof. Now, from the theorem above, let $K = (C[J], ||.||)$ be the Banach space that is $\forall$ continuous on the $J = [0, T]$ interval. Let $I:K\rightarrow K$ be the nonlinear mapping with

So, $I$ is a contraction as $0 < (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}}\frac{\chi^{{{\alpha}}}}{\Gamma{{({\alpha}}+1)}}) < 1$. Thus, by means of the Banach fixed point theorem, the solution of Eq (3.1) is unique.

Theorem 4.3. Let $\mathcal{L}$ and ${N}$ be Lipschitz functions as given in the above theorems; then, the solution for the $NTDM_{CF}$ is convergent for Eq (3.1).

Proof. Let us consider $K$ to be the Banach space as stated above and let $\mathbb{F}_{m} = \sum_{r = 0}^{m}\mathbb{F}_{r}(\mu, \chi)$. To prove that $\mathbb{F}_{m}$ is a Cauchy sequence in H, let

Let $m = n+1$; then,

where $\gamma = (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}} \chi)$. Thus, we have that

As $0 < \gamma < 1$, we have that $1-\gamma^{m-n} < 1$. Thus,

Since $||\mathbb{F}_{1}|| < \infty, \ \ ||\mathbb{F}_{m}-\mathbb{F}_{n}||\rightarrow 0$ when $n \rightarrow \infty$. In this way, $\mathbb{F}_m$ is a Cauchy sequence in K and is convergent.

Theorem 4.4. Let $\mathcal{L}$ and ${N}$ be Lipschitz functions as given in the above theorems; then, the solution for the $NTDM_{ABC}$ is convergent for Eq (3.1).

Proof. Suppose $\mathbb{F}_{m} = \sum_{r = 0}^{m}\mathbb{F}_{r}(\mu, \chi)$. To prove that $\mathbb{F}_{m}$ is a Cauchy sequence in $K$, let

Suppose $m = n+1$; thus,

where $\gamma = (\gamma_{1}+\gamma_{2})(1-{{\alpha}}+{{\alpha}}\frac{\chi^{{{\alpha}}}}{\Gamma({{\alpha}}+1)})$. Thus, we have that

As $0 < \gamma < 1$, we have that $1-\gamma^{m-n} < 1$. Thus,

Since $||\mathbb{F}_{1}|| < \infty, \ \ ||\mathbb{F}_{m}-\mathbb{F}_{n}||\rightarrow 0$ when $n \rightarrow \infty$. In this way, $\mathbb{F}_m$ is a Cauchy sequence in $K$ and convergent.

5.   Applications

Example 5.1. Let us consider the fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system as

where the initial conditions are as follows:

By taking the NT, we get

Thus, we have that

By simplification, we get

By taking the inverse NT, we have that

Solution for the $NDM_{CF}$:

Assume that for the unknown functions $\mathbb{F}(\vartheta, \chi)$ and $\mathbb{G}(\vartheta, \chi)$, the series form solutions are given as

The nonlinear components in terms of the Adomian polynomials are given as $-6{\mathbb{F}} {\mathbb{F}}_{{\vartheta}}+3{\mathbb{G}} {\mathbb{G}}_{3{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{A}_m$, $3{\mathbb{G}}_{{\vartheta}} {\mathbb{G}}_{2{\vartheta}}-3{\mathbb{F}}_{{\vartheta}} {\mathbb{G}}^{2} = \sum_{m = 0}^{\infty}\mathcal{B}_m$, $6{\mathbb{F}} {\mathbb{G}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{C}_m$ and $-3{\mathbb{G}}^{2}{\mathbb{G}}_{{\vartheta}}-3{\mathbb{F}} {\mathbb{G}}_{{\vartheta}}+3{\mathbb{F}}_{{\vartheta}}{\mathbb{G}} = \sum_{m = 0}^{\infty}\mathcal{D}_m$. With the help of these terms, Eq (5.6) can be stated as

By comparing both sides of Eq (5.8), we have that

In this way, given $({l}\geq3)$, the remaining terms for $\mathbb{F}_{{l}}$ and $\mathbb{G}_{{l}}$ are easy to get. Thus, the series form solutions are given as

Solution for the $NDM_{ABC}$:

Assume that for the unknown functions $\mathbb{F}(\vartheta, \chi)$ and $\mathbb{G}(\vartheta, \chi)$, the series form solutions respectively are given as

The nonlinear components in terms of the Adomian polynomials are given as $-6{\mathbb{F}} {\mathbb{F}}_{{\vartheta}}+3{\mathbb{G}} {\mathbb{G}}_{3{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{A}_m$, $3{\mathbb{G}}_{{\vartheta}} {\mathbb{G}}_{2{\vartheta}}-3{\mathbb{F}}_{{\vartheta}} {\mathbb{G}}^{2} = \sum_{m = 0}^{\infty}\mathcal{B}_m$, $6{\mathbb{F}} {\mathbb{G}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{C}_m$ and $-3{\mathbb{G}}^{2}{\mathbb{G}}_{{\vartheta}}-3{\mathbb{F}} {\mathbb{G}}_{{\vartheta}}+3{\mathbb{F}}_{{\vartheta}}{\mathbb{G}} = \sum_{m = 0}^{\infty}\mathcal{D}_m$. With the help of these terms, Eq (5.6) can be stated as

By comparing both sides of Eq (5.13), we have that

In this way, given $({l}\geq3)$, the remaining terms for $\mathbb{F}_{{l}}$ and $\mathbb{G}_{{l}}$ are easy to get. Thus, the series form solutions are given as

When $\alpha = 1$, we get the exact solution as

Example 5.2. Let us consider the time-fractional homogeneous two component coupled third order KdV system as

where the initial conditions are as follows:

By taking the NT, we get

Thus, we have

By simplification, we get

By taking the inverse NT, we have that

Solution for the $NTDM_{CF}$:

Assume that for the unknown functions $\mathbb{F}(\vartheta, \chi)$ and $\mathbb{G}(\vartheta, \chi)$, the series form solution are given as

The nonlinear components in terms of the Adomian polynomials are given as $-{\mathbb{F}} {\mathbb{F}}_{{\vartheta}}-{\mathbb{G}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{A}_m$ and ${\mathbb{F}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{B}_m$. With the help of these terms, Eq (5.23) can be stated as

By comparing both sides of Eq (5.25), we have that

In this way, given $({l}\geq3)$, the remaining terms for $\mathbb{F}_{{l}}$ and $\mathbb{G}_{{l}}$ are easy to get. Thus, the series form solutions are given as

Solution for the $NTDM_{ABC}$:

Assume that for the unknown functions $\mathbb{F}(\vartheta, \chi)$ and $\mathbb{G}(\vartheta, \chi)$, the series form solutions are given as

The nonlinear components in terms of the Adomian polynomials are given as $-{\mathbb{F}} {\mathbb{F}}_{{\vartheta}}-{\mathbb{G}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{A}_m$ and ${\mathbb{F}} {\mathbb{G}}_{{\vartheta}} = \sum_{m = 0}^{\infty}\mathcal{B}_m$. Given these terms, Eq (5.23) can be stated as

By comparing both sides of Eq (5.30), we have that

In this way, given $({l}\geq3)$, the remaining terms for $\mathbb{F}_{{l}}$ and $\mathbb{G}_{{l}}$ are easy to get. Thus, the series form solutions are given as

When $\alpha = 1$, we get the exact solution as

6.   Results and discussion

In Figure 1, the actual and suggested methods solutions of $\mathbb{F}({\vartheta}, {\chi})$ are calculated at $\alpha = 1$. Figure 2 gives the graphical layouts of $\mathbb{F}({\vartheta}, {\chi})$ when $\alpha = 0.8$ and $0.6$. In Figure 3, the 2D and 3D behaviors of $\mathbb{F}({\vartheta}, {\chi})$ for different fractional orders are respectively given. In Figure 4, the actual and suggested methods solutions of $\mathbb{G}({\vartheta}, {\chi})$ are calculated at $\alpha = 1$. Figure 5 gives the graphical layouts of $\mathbb{G}({\vartheta}, {\chi})$ when $\alpha = 0.8$ and $0.6$. In Figure 6, the 2D and 3D behaviors of $\mathbb{G}({\vartheta}, {\chi})$ for different fractional orders are respectively given. The graphical layout of Example 5.1 confirms that our solutions are in a strong agreement with the exact solution. Similarly, in Figures 7, the actual and suggested methods solutions of $\mathbb{F}({\vartheta}, {\chi})$ are respectively presented for $\alpha = 1$. Figures 8 gives the graphical layouts of $\mathbb{F}({\vartheta}, {\chi})$ when $\alpha = 0.8$ and $0.6$, while Figures 9 shows the 2D and 3D behaviors of $\mathbb{F}({\vartheta}, {\chi})$ for different fractional orders, respectively. Also, in Figures 10, the actual and suggested methods solutions of $\mathbb{G}({\vartheta}, {\chi})$ are respectively presented for $\alpha = 1$. Figures 11 gives the graphical layouts of $\mathbb{G}({\vartheta}, {\chi})$ when $\alpha = 0.8$ and $0.6$, while Figures 12 shows the 2D and 3D behaviors of $\mathbb{G}({\vartheta}, {\chi})$ for different fractional orders, respectively. In the same way, the graphical layout of Example 5.2 confirms that our solutions are in a strong agreement with the exact solution. All the figures have been drawn at c = 0.1 and $\chi = 0.01$ within the domain $-5\leq\vartheta\geq5$. Furthermore, Tables 1 and 2 shows the approximate solution of Example 5.1 at different values of $\vartheta$ and $\chi$. Tables 3 and 4 show a numerical comparison of the reduced differential transform method and proposed method in terms of the absolute error for Example 5.1. Finally, Tables 5 and 6 show the approximate solution of Example 5.2 at different values of $\vartheta$ and $\chi$. From the figures and tables, it can be observed that our methods solution and the exact solution are very close to each other and possess a higher degree of accuracy.

7.   Conclusions

The main goal of this work as to develop a fractional-order Kersten-Krasil' shchik coupled KdV-mKdV nonlinear system approximate analytical solution. Using the NDM, we were able to achieve this goal. The numerical solutions can be achieved in two steps. The targeted problems were first simplified using the NT; then, the decomposition approach was employed to get the solutions. The method's fundamental benefit is that it gives the user an analytical approximation, and in many cases an exact solution, in a quickly converging sequence with elegantly computed terms. The suggested method has been applied to obtain the solution of the given two problems. The method's small computational size in comparison to the computational size required by other numerical methods, as well as its rapid convergence, demonstrate that it is reliable and a significant improvement on existing methods in terms of solving the generalized Kersten-Krasil' shchik coupled KdV-mKdV equation. The results we obtained have been illustrated with the help of plots and tables which confirm the validity of the proposed method. Furthermore, the proposed method is simple, straightforward, and requires minimal computational time; it may also be extended to solve other fractional-order partial differential equations.

Acknowledgements

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation.

Conflict of interest

The authors declare that they have no competing interests.