On the improved thinning risk model under a periodic dividend barrier strategy

Fuyun Sun; Yuelei Li; Fuyun Sun; Yuelei Li

doi:10.3934/math.2021779

AIMS Mathematics

2021, Volume 6, Issue 12: 13448-13463. doi: 10.3934/math.2021779

Previous Article Next Article

Research article

On the improved thinning risk model under a periodic dividend barrier strategy

Fuyun Sun ¹,
Yuelei Li ^{2
,
,}

1.
School of Mathematics, Tianjin University, Tianjin 300350, China
2.
College of Management and Economics, Tianjin University, Tianjin 300072, China

Received: 14 April 2021 Accepted: 14 September 2021 Published: 18 September 2021
MSC : 91B30, 97M30

In this study, we consider a periodic dividend barrier strategy in an improved thinning risk model, which indicates that insurance companies randomly receive premiums and pay dividends. In the improved model, the premium is stochastic, and the claim counting process is a p-thinning process of the premium counting process. The integral equations satisfied by the Gerber-Shiu function and the expected discounted cumulative dividend function are derived. Explicit expressions of those actuarial functions are obtained when the claim and premium sizes are exponentially distributed. We analyze and illustrate the impact of various parameters on them and obtain the optimal barrier. Finally, a conclusion is drawn.

Keywords:

thinning model,
periodic dividend strategy,
Gerber-Shiu function,
stochastic premium,
expected discounted cumulative dividend function

Citation: Fuyun Sun, Yuelei Li. On the improved thinning risk model under a periodic dividend barrier strategy[J]. AIMS Mathematics, 2021, 6(12): 13448-13463. doi: 10.3934/math.2021779

Related Papers:

[1]	Jehad Shaikhali, Gunnar Wingsle . Redox-regulated transcription in plants: Emerging concepts. AIMS Molecular Science, 2017, 4(3): 301-338. doi: 10.3934/molsci.2017.3.301
[2]	Amedea B. Seabra, Halley C. Oliveira . How nitric oxide donors can protect plants in a changing environment: what we know so far and perspectives. AIMS Molecular Science, 2016, 3(4): 692-718. doi: 10.3934/molsci.2016.4.692
[3]	Vittorio Emanuele Bianchi, Giancarlo Falcioni . Reactive oxygen species, health and longevity. AIMS Molecular Science, 2016, 3(4): 479-504. doi: 10.3934/molsci.2016.4.479
[4]	Luís J. del Valle, Lourdes Franco, Ramaz Katsarava, Jordi Puiggalí . Electrospun biodegradable polymers loaded with bactericide agents. AIMS Molecular Science, 2016, 3(1): 52-87. doi: 10.3934/molsci.2016.1.52
[5]	Isabella Martins Lourenço, Amedea Barozzi Seabra, Marcelo Lizama Vera, Nicolás Hoffmann, Olga Rubilar Araneda, Leonardo Bardehle Parra . Synthesis and application of zinc oxide nanoparticles in Pieris brassicae larvae as a possible pesticide effect. AIMS Molecular Science, 2024, 11(4): 351-366. doi: 10.3934/molsci.2024021
[6]	Vahid Pouresmaeil, Marwa Mawlood Salman Al-zand, Aida Pouresmaeil, Seyedeh Samira Saghravanian, Masoud Homayouni Tabrizi . Loading diltiazem onto surface-modified nanostructured lipid carriers to evaluate its apoptotic, cytotoxic, and inflammatory effects on human breast cancer cells. AIMS Molecular Science, 2024, 11(3): 231-250. doi: 10.3934/molsci.2024014
[7]	Giulia Ambrosi, Pamela Milani . Endoplasmic reticulum, oxidative stress and their complex crosstalk in neurodegeneration: proteostasis, signaling pathways and molecular chaperones. AIMS Molecular Science, 2017, 4(4): 424-444. doi: 10.3934/molsci.2017.4.424
[8]	Davide Lovisolo, Marianna Dionisi, Federico A. Ruffinatti, Carla Distasi . Nanoparticles and potential neurotoxicity: focus on molecular mechanisms. AIMS Molecular Science, 2018, 5(1): 1-13. doi: 10.3934/molsci.2018.1.1
[9]	Zhaoping Qin, Patrick Robichaud, Taihao Quan . Oxidative stress and CCN1 protein in human skin connective tissue aging. AIMS Molecular Science, 2016, 3(2): 269-279. doi: 10.3934/molsci.2016.2.269
[10]	Morgan Robinson, Brenda Yasie Lee, Zoya Leonenko . Drugs and drug delivery systems targeting amyloid-β in Alzheimer's disease. AIMS Molecular Science, 2015, 2(3): 332-358. doi: 10.3934/molsci.2015.3.332

Abstract

1. Introduction

Nonlinear partial differential equation is a very important branch of the nonlinear science, which has been called the foreword and hot topic of current scientific development. In theoretical science and practical application, nonlinear partial differential is used to describe the problems in the fields of optics, mechanics, communication, control science and biology ^{[1,2,3,4,5,6,7,8,9]}. At present, the main problems in the study of nonlinear partial differential equations are the existence of solutions, the stability of solutions, numerical solutions and exact solutions. With the development of research, especially the study of exact solutions of nonlinear partial differential equations has important theoretical value and application value. In the last half century, many important methods for constructing exact solutions of nonlinear partial differential equations have been proposed, such as the planar dynamic system method ^[10], the Jacobi elliptic function method ^[11], the bilinear transformation method ^[12], the complete discriminant system method for polynomials ^[13], the unified Riccati equation method ^[14], the generalized Kudryashov method ^[15], and so on ^{[16,17,18,19,20,21,22,23,24]}.

There is no unified method to obtain the exact solution of nonlinear partial differential equations. Although predecessors have obtained some analytical solutions with different methods, no scholar has studied the system with complete discrimination system for polynomial method.

The Fokas system is a very important class of nonlinear partial differential equations. In this article, we focus on the Fokas system, which is given as follows ^{[25,26,27,28,29,30,31,32,33,34,35,36,37]}

$\left\{ \begin{array}{l} i{p_t} + {r_1}{p_{xx}} + {r_2}pq = 0, \hfill \\ {r_3}{q_y} - {r_4}{(|p{|^2})_x} = 0, \hfill \end{array} \right.$

(1.1)

where $p = p(x, y, t)$ and $q = q(x, y, t)$ are the complex functions which stand for the nonlinear pulse propagation in monomode optical fibers. The parameters ${r_1}, {r_2}, {r_3}$ and ${r_4}$ are arbitrary non-zero constants, which are coefficients of nonlinear terms in Eq (1.1) and reflect different states of optical solitons.

This paper is arranged as follows. In Section 2, we describe the method of the complete discrimination system for polynomial method. In Section 3, we substitute traveling wave transformation into nonlinear ordinary differential equations and obtain the different new single traveling wave solutions for the Fokas system by complete discrimination system for polynomial method. At the same time, we draw some images of solutions. In Section 4, the main results are summarized.

2. Description of the method

First, we consider the following partial differential equations:

$\left\{ \begin{array}{l}F(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\\ G(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\end{array} \right.$

(2.1)

where $F$ and $G$ is polynomial function which is about the partial derivatives of each order of $u(x, t)$ and $v(x, t)$ with respect to $x$ and $t$ .

Step 1: Taking the traveling wave transformation $u(x, t) = u(\xi), v(x, t) = v(\xi), \xi = kx + ct$ into Eq (2.1), then the partial differential equation is converted to an ordinary differential equation

$\left\{ \begin{array}{l} F(u, v, u', v', u'', v'', \cdots ) = 0, \hfill \\ G(u, v, u', v', u'', v'', \cdots ) = 0. \hfill \end{array} \right.$

(2.2)

Step 2: The above nonlinear ordinary differential equations (2.2) are reduced to the following ordinary differential form after a series of transformations:

${(u')^2} = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}.$

(2.3)

The Eq (2.3) can also be written in integral form as:

$\pm (\xi - {\xi _0}) = \int {\frac{{du}}{{\sqrt {{u^3} + {d_2}{u^2} + {d_1}u + {d_0}} }}} .$

(2.4)

Step 3: Let $\phi (u) = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}$ . According to the complete discriminant system method of third-order polynomial

$\left\{\begin{array}{l} \Delta=-27(\frac{2 d_2^3}{27}+d_0-\frac{d_1 d_2}{3})^2-4(d_1-\frac{d_2^2}{3})^3, \\ D_1=d_1-\frac{d_2^2}{3}, \end{array}\right.$

(2.5)

the classification of the solution of the equation can be obtained, and the classification of traveling wave solution of the Fokas system will be given in the following section.

3. Traveling wave solution of Eq (1.1)

In the current part, we obtain all exact solutions to Eq (1.1) by complete discrimination system for polynomial method. According to the wave transformation

$p(x, y, t) = \varphi (\eta ){e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, q(x, y, t) = \phi (\eta ), \eta = x + y - vt,$

(3.1)

where ${\lambda _1}, {\lambda _2}, {\lambda _3}, {\lambda _4}$ and $v$ are real parameters, and $v$ represents the wave frame speed.

Substituting the above transformation Eq (3.1) into Eq (1.1), we get

$\left\{ \begin{array}{l} ( - v + 2{r_1}{\lambda _1})i\varphi ' - {\lambda _3}\varphi + {r_1}\varphi '' - {r_1}\lambda _1^2\varphi + {r_2}\varphi \phi = 0, \hfill \\ {r_3}\phi ' - 2{r_4}\varphi \varphi ' = 0. \hfill \end{array} \right.$

(3.2)

Integrating the second equation in (3.2) and ignoring the integral constant, we get

$\phi (\eta ) = \frac{{{r_4}{\varphi ^2}(\eta )}}{{{r_3}}}.$

(3.3)

Substituting Eq (3.3) into the first equation in (3.2) and setting $v = 2{r_1}{\lambda _1},$ we get the following:

${r_1}\varphi '' - ({\lambda _3} + {r_1}\lambda _1^2)\varphi + \frac{{{r_2}{r_4}{\varphi ^3}}}{{{r_3}}} = 0.$

(3.4)

Multiplying $\varphi '$ both sides of the Eq (3.4), then integrating once to get

${(\varphi ')^2} = {a_4}{\varphi ^4} + {a_2}{\varphi ^2} + {a_0},$

(3.5)

where ${a_4} = - \frac{{{r_2}{r_4}}}{{2{r_1}{r_3}}}, {a_2} = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_1}}},$ ${a_0}$ is the arbitrary constant.

${\rm{Let}} \ \ \varphi = \pm \sqrt {{{(4{a_4})}^{ - \frac{1}{3}}}\omega } , \ {b_1} = 4{a_2}{(4{a_4})^{ - \frac{2}{3}}}, {b_0} = 4{a_0}{(4{a_4})^{ - \frac{1}{3}}}, {\eta _1} = {(4{a_4})^{\frac{1}{3}}}\eta .$

(3.6)

Equation (3.5) can be expressed as the following:

${({\omega '_{{\eta _1}}})^2} = {\omega ^3} + {b_1}{\omega ^2} + {b_0}\omega .$

(3.7)

Then we can get the integral expression of Eq (3.7)

$\pm ({\eta _1} - {\eta _0}) = \int {\frac{{d\omega }}{{\sqrt {\omega ({\omega ^2} + {b_1}\omega + {b_0})} }}} ,$

(3.8)

where ${\eta _0}$ is the constant of integration.

Here, we get the $F(\omega) = {\omega ^2} + {b_1}\omega + {b_0}$ and $\Delta = b_1^2 - 4{b_0}.$ In order to solve Eq (3.7), we discuss the third order polynomial discrimination system in four cases.

Case 1: $\Delta = 0$ and $\omega \gt 0.$

When ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _1} = - \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.9)

${\omega _2} = - \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.10)

Thus, the classification of all solutions of Eq (3.7) is obtained by the third order polynomial discrimination system. The exact traveling wave solutions of the Eq (1.1) are obtained by combining the above solutions and the conditions (3.6) with Eq (3.1), can be expressed as below:

${p_1}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tanh (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.11)

In Eq (3.11), ${p_1}(x, y, t)$ is a dark soliton solution, it expresses the energy depression on a certain intensity background. Figure 1 depict two-dimensional graph, three-dimensional graph, contour plot and density plot of the solution.

${q_1}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}))$

(3.12)

$\begin{array}{l} {p_2}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \coth (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.13)

Figure 1. Module length graphs of Eq (3.12) when

${r_1} = - 2, {r_2} = 1, {r_3} = - 1, {r_4} = 1, {\lambda _1} = - 1, {\lambda _3} = 3, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

where ${p_1}(x, y, t), {q_1}(x, y, t), {p_2}(x, y, t), {q_2}(x, y, t)$ are hyperbolic function solutions. Specially, ${p_2}(x, y, t)$ is a bright soliton solution.

${q_2}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.14)

When ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _3} = \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.15)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_3}(x, y, t) = \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tan (\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}. \end{array}$

(3.16)

${q_3}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.17)

In Eq (3.16) and Eq (3.17), ${p_3}(x, y, t)$ and ${q_3}(x, y, t)$ are trigonometric function solutions. ${q_3}(x, y, t)$ is a periodic wave solution, and it Shows the periodicity of ${q_3}(x, y, t)$ in Figure 2(a), (b).

Figure 2. Module length graphs of Eq (3.17) when

${r_1} = 2, {r_2} = 1, {r_3} = 1, {r_4} = 1, {\lambda _1} = 1, {\lambda _3} = - 1, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

When ${b_1} = 0,$ the solution of Eq (3.7) is

${\omega _4} = \frac{4}{{{{({\eta _1} - {\eta _0})}^2}}}.$

(3.18)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_4}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{ - \frac{1}{3}}}} \frac{2}{{{{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}}}{e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}},$

(3.19)

${q_4}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{ - \frac{1}{3}}}\frac{4}{{{{({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})}^2}}},$

(3.20)

where ${p_4}(x, y, t)$ is exponential function solution, and ${q_4}(x, y, t)$ is rational function solution.

Case 2: $\Delta = 0$ and ${b_0} = 0.$

When $\omega \gt - {b_1}$ and ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _5} = \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.21)

${\omega _6} = \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.22)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_5}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tanh }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.23)

${q_5}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.24)

$\begin{array}{l} {p_6}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\coth }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.25)

${q_6}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.26)

where ${p_5}(x, y, t), {q_5}(x, y, t), {p_6}(x, y, t)$ and ${q_6}(x, y, t)$ are hyperbolic function solutions.

When $\omega \gt - {b_1}$ and ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _7} = - \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.27)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_7}(x, y, t) = \\ \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tan }^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) + 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.28)

${q_7}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.29)

where ${p_7}(x, y, t)$ and ${q_7}(x, y, t)$ are trigonometric function solutions.

Case 3: $\Delta \gt 0$ and ${b_0} \ne 0.$ Let $u \lt v \lt s,$ there $u, v$ and $s$ are constants satisfying one of them is zero and two others are the root of $F(\omega) = 0.$

When $u \lt \omega \lt v,$ we can get the solution of Eq (3.7) is

${\omega _8} = u + (v - u)s{n^2}(\frac{{\sqrt {s - u} }}{2}({\eta _1} - {\eta _0}), c),$

(3.30)

where ${c^2} = \frac{{v - u}}{{s - u}}.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_8}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.31)

${q_8}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)].$

(3.32)

When $\omega \gt s,$ the solution of Eq (3.7) is

${\omega _9} = \frac{{ - v \cdot s{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c) + s}}{{c{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c)}}.$

(3.33)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_9}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}} {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.34)

${q_9}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}.$

(3.35)

Case 4: $\Delta \lt 0.$

When $\omega \gt 0,$ similarly we get

${\omega _{10}} = \frac{{2\sqrt {{b_0}} }}{{1 + cn(b_0^{\frac{1}{4}}({\eta _1} - {\eta _0}), c)}} - \sqrt {{b_0}} ,$

(3.36)

where ${c^2} = (1 - \frac{{{b_1}\sqrt {{b_0}} }}{2})/2.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_{10}}(x, y, t) = \pm \sqrt {2\sqrt {{a_0}} {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}} ,$

(3.37)

${q_{10}}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}2\sqrt {{a_0}} {(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1],$

(3.38)

where ${p_8}(x, y, t), {q_8}(x, y, t), {p_9}(x, y, t), {q_9}(x, y, t), {p_{10}}(x, y, t)$ and ${q_{10}}(x, y, t)$ are Jacobian elliptic function solutions.

4. Conclusions

In this paper, the complete discrimination system of polynomial method has been applied to construct the single traveling wave solutions of the Fokas system. The Jacobian elliptic function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solutions are obtained. The obtained solutions are very rich, which can help physicists understand the propagation of traveling wave in monomode optical fibers. Furthermore, we have also depicted two-dimensional graphs, three-dimensional graphs, contour plots and density plots of the solutions of Fokas system, which explains the state of solitons from different angles.

Acknowledgments

This work was supported by Scientific Research Funds of Chengdu University (Grant No.2081920034).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	W. Yu, P. Guo, Q. Wang, G. Guan, Y. Huang, X. Yu, Randomized observation periods for compound Poisson risk model with capital injection and barrier dividend, Adv. Differ. Equ., 220 (2021), 220.
[2]	W. Yu, P. Guo, Q. Wang, G. Guan, Q. Yang, Y. Huang, et al., On a periodic capital injection and barrier dividend strategy in the compound Poisson risk model, Mathematics, 8 (2020), 511. doi: 10.3390/math8040511
[3]	A. V. Boikov, The Cramér-Lundberg model with stochastic premium process, Theory Probab. Appl., 47 (2003), 489–493. doi: 10.1137/S0040585X9797987
[4]	C. Labbé, K. P. Sendova, The expected discounted penalty function under a risk model with stochastic income, Appl. Math. Comput., 215 (2009), 1852–1867.
[5]	T. A. Belkina, N. B. Konyukhova, S. V. Kurochkin, Singular boundary value problem for the integro-differential equation in an insurance model with stochastic premiums: Analysis and numerical solution, Comput. Math. Math. Phys., 52 (2012), 1384–1416. doi: 10.1134/S0965542512100077
[6]	H. Albrecher, O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals, Insur. Math. Econ., 35 (2004), 245–254. doi: 10.1016/j.insmatheco.2003.09.009
[7]	G. Wang, K. C Yuen, On a correlated aggregate claims model with thinning-dependence structure, Insur. Math. Econ., 36 (2005), 456–468. doi: 10.1016/j.insmatheco.2005.04.004
[8]	J. Pan, G. Wang, Expected discounted penalty function for a thinning risk model, Chin. J. Appl. Probab. Stat., 25 (2009), 544–552.
[9]	M. Boudreault, H. Cossette, D. Landriault, E. Marceau, On risk model with dependence between interclaim arrivals and claim sizes, Scand. Actuar. J., 5 (2006), 265–285.
[10]	H. Albrecher, C. Constantinescu, S. Loisel, Explicit ruin formulas for models with dependence among risks, Insur. Math. Econ., 48 (2011), 265–270. doi: 10.1016/j.insmatheco.2010.11.007
[11]	M. Vidmar, Ruin under stochastic dependence between premium and claim arrivals, Scand. Actuar. J., 2018 (2018), 505–513. doi: 10.1080/03461238.2017.1391114
[12]	B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of The XVth International Congress of Actuaries, 2 (1957), 433–443.
[13]	C. Yin, K. C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model, Stat. Probab. Lett., 81 (2011), 1841–1846. doi: 10.1016/j.spl.2011.07.022
[14]	H. Albrecher, E. C. Cheung, S. Thonhauser, Randomized observation periods for the compound poisson risk model: Dividends, ASTIN Bull., 41 (2011), 645–672.
[15]	K. Noba, J. L. Pérez, K. Yamazaki, K. Yano, On optimal periodic dividend strategies for Lévy risk processes, Insur. Math. Econ., 80 (2018), 29–44. doi: 10.1016/j.insmatheco.2018.02.004
[16]	Z. Zhang, E. C. Cheung, H. Yang, On the compound poisson risk model with periodic capital injections, ASTIN Bull., 48 (2018), 435–477. doi: 10.1017/asb.2017.22
[17]	B. Avanzi, E. C. Cheung, B. Wong, J. K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insur. Math. Econ., 52 (2013), 98–113. doi: 10.1016/j.insmatheco.2012.10.008
[18]	J. Song, F. Sun, The dual risk model under a mixed ratcheting and periodic dividend strategy, Commun. Stat. Theory Methods, in press.
[19]	D. Peng, D. Liu, Z. Liu, Dividend problems in the dual risk model with exponentially distributed observation time, Stat. Probab. Lett., 83 (2013), 841–849. doi: 10.1016/j.spl.2012.11.025
[20]	M. C. Choi, E. C. Cheung, On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions, Insur. Math. Econ., 59 (2014), 121–132. doi: 10.1016/j.insmatheco.2014.08.009
[21]	B. Avanzi, V. Tu, B. Wong, On optimal periodic dividend strategies in the dual model with diffusion, Insur. Math. Econ., 55 (2014), 210–224. doi: 10.1016/j.insmatheco.2014.01.005
[22]	B. Avanzi, V. Tu, B. Wong, Optimal dividends under Erlang $(2)$ inter-dividend decision times, Insur. Math. Econ., 79 (2018), 225–242. doi: 10.1016/j.insmatheco.2018.01.009
[23]	H. U. Gerber, E. S. Shiu, On the time value of ruin, N. Am. Actuar. J., 2 (1998), 48–72.
[24]	D. C. Dickson, C. Hipp, On the time to ruin for Erlang(2) risk processes, Insur. Math. Econ., 29 (2001), 333–344. doi: 10.1016/S0167-6687(01)00091-9
[25]	S. Li, J. Garrido, On a general class of renewal risk process: Analysis of the Gerber-Shiu function, Adv. Appl. Probab., 37 (2005), 836–856. doi: 10.1239/aap/1127483750
[26]	E. J. Baurdoux, J. C. Pardo, J. L. Pérez, J. F. Renaud, Gerber-Shiu distribution at Parisian ruin for Lévy insurance risk processes, J. Appl. Probab., 53 (2016), 572–584. doi: 10.1017/jpr.2016.21
[27]	E. C. Cheung, D. Landriault, G. E. Willmot, J. K. Woo, Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insur. Math. Econ., 46 (2010), 117–126. doi: 10.1016/j.insmatheco.2009.05.009
[28]	S. Li, Y. Lu, K. P. Sendova, The expected discounted penalty function: From infinite time to finite time, Scand. Actuar. J., 2019 (2019), 336–354. doi: 10.1080/03461238.2018.1560955
[29]	W. Wang, Z. Zhang, Computing the Gerber-Shiu function by frame duality projection, Scand. Actuar. J., 219 (2019), 291–307.
[30]	L. Zhang, The Erlang( $n$ ) risk model with two-sided jumps and a constant dividend barrier, Commun. Stat. Theory Methods, 2020 (2020), 1–19.

This article has been cited by:

1.	Andrew Geoly, Ernest Greene, Masking the Integration of Complementary Shape Cues, 2019, 13, 1662-453X, 10.3389/fnins.2019.00178
2.	Ernest Greene, Comparing methods for scaling shape similarity, 2019, 6, 2373-7972, 54, 10.3934/Neuroscience.2019.2.54
3.	Hannah Nordberg, Michael J Hautus, Ernest Greene, Visual encoding of partial unknown shape boundaries, 2018, 5, 2373-7972, 132, 10.3934/Neuroscience.2018.2.132
4.	Ernest Greene, Hautus Michael J, Evaluating persistence of shape information using a matching protocol, 2018, 5, 2373-7972, 81, 10.3934/Neuroscience.2018.1.81
5.	Ernest Greene, Jack Morrison, Computational Scaling of Shape Similarity That has Potential for Neuromorphic Implementation, 2018, 6, 2169-3536, 38294, 10.1109/ACCESS.2018.2853656
6.	Ernest Greene, New encoding concepts for shape recognition are needed, 2018, 5, 2373-7972, 162, 10.3934/Neuroscience.2018.3.162
7.	Cheng Chen, Kang Jiao, Letao Ling, Zhenhua Wang, Yuan Liu, Jie Zheng, 2023, Chapter 47, 978-981-19-3631-9, 382, 10.1007/978-981-19-3632-6_47
8.	Bridget A. Kelly, Charles Kemp, Daniel R. Little, Duane Hamacher, Simon J. Cropper, Visual Perception Principles in Constellation Creation, 2024, 1756-8757, 10.1111/tops.12720

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2299) PDF downloads(79) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(3)

AIMS Mathematics

On the improved thinning risk model under a periodic dividend barrier strategy

Related Papers:

Abstract

1. Introduction

2. Description of the method

3. Traveling wave solution of Eq (1.1)

4. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

On the improved thinning risk model under a periodic dividend barrier strategy

Related Papers:

Abstract

1. Introduction

2. Description of the method

3. Traveling wave solution of Eq (1.1)

4. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog