Convergence of numerical methods for fuzzy differential equations

Abstract

In this paper, semi-implicit and implicit Euler schemes for homogeneous fuzzy differential equations with perturbation term reflected by Liu process are introduced. As to the application with implicit term in numerical scheme, we must make the result gradually explicit by iterative method. In order to obtain numerical method with higher accuracy than fuzzy Euler method, fuzzy trapezoidal scheme is derived. Fuzzy trapezoidal scheme is an implicit formation, which is complicated and cumbersome in computational processing. For the sake of this problem, fuzzy Euler-trapezoidal method is proposed to simplify the algorithm. Furthermore, the convergence properties are investigated for numerical methods. At last, local convergence is proved better than global convergence.

Keywords

Credibility fuzzy differential equations Liu process numerical methods convergence

1 Introduction

Due to the rapid development of economy, science and technology, unexpected and different types of fuzzy events appear in every field of society. Although the fuzzy set theory can be used to process the fuzzy event, it will be limited in practical application since it does not have self-duality. Scholars try hard to overcome this problem. Since Liu and Liu [10] put forward credibility measure in 2002, a lot of research results have been obtained in this field under the framework of credibility theory. Liu process can clearly explain the fuzzy events, but it can not solve the actual fuzzy problems. On the basis of Liu differential and Liu integral, the actual fuzzy event is abstracted into a fuzzy differential equation (FDE) model, which is transformed into the analytic solution of the equation. On the one hand, although theoretically some analytic solutions of FDEs exist and are unique, it is difficult to find their explicit expressions; on the other hand, there are few forms of FDEs whose analytic solution can be obtained. To study the approximate solution to meet the practical needs, it is especially important to construct effective numerical methods in combination with the modern computer technology, according to different actual data. Considering numerical method can solve the problem of calculation, it is necessary to ensure the feasibility of the method. It is well known that a significant basis for measuring numerical methods is convergence. This paper aims to investigate different numerical methods and their convergence.

1.1 The development of credibility theory

Until 1965, Zadeh [27] found a mathematical tool, membership function, the concept of fuzzy set was put forward with the aid of the tool. In 1978, Zadeh [28] came up with possibility measure, which was used to measure a fuzzy event. Then possibility measure was adopted and applied to academic research, but it was discovered a fatal drawback: it is important for self-duality in theory and application, while possibility measure does not possess this property. In order to make up for this defect, Liu and Liu [10] established a measure satisfying self-duality in 2002, which is called credibility measure. Subsequently, Liu [12] raised credibility theory in 2004, that was further supplemented and improved by Liu [13] in 2007. The discovery of credibility theory has opened up a new way in the field of research of fuzzy system. It is not just confined to theory, it also wants to deal with practical fuzzy problems through the idea of mathematical modeling. With the help of credibility theory, we can transform abstract problem into concrete model. Recognizing that there are many fuzzy phenomena that change with time, Liu [14] gave the concept of fuzzy process to solve this problem.

In addition, Liu [14] defined a special fuzzy process with independent stationary increment, which is called Liu process.

Then Dai [5] proved that Liu process conforms to the lipschitz continuity condition. Based on Liu process, Liu [14] has designed a lot of related mathematical concepts and calculation tools like Liu integral, Liu differential. Corresponding to geometric Wiener process, geometric Liu process was used to study the stock model in fuzzy environment. Therefore, Liu [14] built a fuzzy stock model. On the basis of this model, the European option pricing was subsequently proposed by Qin and Li [19]. Inspired by fuzzy vectors, the one-dimensional concept of Liu process and mathematical tools were generalized to the multi-dimensional case by You, Huo and wang [25]. Thus, the multi-dimensional Liu process, the multi-dimensional Liu integral, and the multi-dimensional Liu differential were obtained. In a consequence, what we’re studying is based on these theories.

We admit that fuzzy differential equation driven by Liu process is a kind of special uncertain differential equations. However, credibility measure has better properties than uncertain measure. Therefore, in this paper, we only discuss fuzzy differential equation driven by Liu process.

Though the form of fuzzy differential equation we studied is similar to uncertain differential equation, the measure and theory we used different from those of uncertain differential equation.

The measure used in our fuzzy differential equation satisfies the following four axioms, which is called credibility measure.

Axiom 1 Cr {Θ}=1.

Axiom 2 Cr {A} ≤ Cr {B} whenever A ⊂ B.

Axiom 3 Cr {A} + Cr {A^c} =1 for any event A.

Axiom 4 $Cr {⋃_{i} A_{i}} = sup_{i} Cr {A_{i}}$ for any events {A_i} with $sup_{i} Cr {A_{i}} \leq 0.5$ .

While the measure used in uncertain differential equation, called uncertain measure, satisfies the following four axioms:

Axiom 1’ M {Γ}=1.

Axiom 2’ M {Λ₁} ≤ M {Λ₂} whenever Λ₁ ⊂ Λ₂.

Axiom 3’ M {Λ} + M {Λ^c} =1 for any event Λ.

Axiom 4’ For every countable sequence of events {Λ_i}, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

It is obvious that credibility measure is a kind of uncertain measure, while since Axiom 4 and Axiom 4’ are different for these two measure, credibility measure has better properties than uncertain measure.

Based on different axiom systems, the theory involved are different, we use credibility theory, while solving uncertain differential equation needs uncertainty theory. The main distinct of two theories is the different measure, and the key difference between credibility measure and uncertain measure is reflected in Axiom 4 and Axiom 4’, that is to say, the definition methods of union are different.

1.2 Three types of fuzzy differential equations

Before proposing Liu process, FDEs were mainly divided into two kinds: One was fuzzy treatment of the coefficients of classical differential equations, the other was fuzzy treatment of the initial conditions of classical differential equations. With respect to these two kinds of FDEs, Kaleva [7, 8] and Ding [6] have gained the existence and uniqueness of the solution. Le [9], Mizukoshi [15] have profoundly researched the stability of FDEs, respectively. As for numerical methods of these FDEs, good results have been achieved. For example, Ahmad and Hasan [2], Abbasbandy and Allah [1], Omar and Hasan [18] have put forward Euler method and Taylor formula. Under the concept of generalized differential, Nematollah [17] obtained analytical solution and numerical solution via differential transformation method; Mosleh [16] presented a method for approximating linear FDEs. Under the generalized hukuhara derivative, Van [20] used LTM method to calculate the analytical solution of linear second-order FDEs; Allahviranloo, Gouyandeh and Armand [3] presented an Euler numerical method for solving FDEs based on Taylor expansion. This Euler method is different from fuzzy Euler method mentioned in our paper because (i) the theoretical systems are different: the Euler method in [3] was proposed under the premise of the theoretical background of fuzzy set, while the fuzzy Euler method in our paper used credibility theory; (ii) the meanings of differentials are distinct: the former was based on the concept of generalized hukuhara derivative, however the latter was discussed in the sense of Liu differential, which is corresponding to Ito differential in stochastic environments; (iii) the FDEs dealt with are not alike: the FDE studied in [3] is the FDE mentioned above, yet the FDE in our paper is the third type of FDE, which is fuzzy differential equations driven by Liu process. It just corresponds to Ito-type stochastic differential equation. This types of FDEs contain more fuzziness than the FDEs mentioned above.

After the appearance of Liu process, the third type of FDE, FDE driven by Liu process, defined by Liu [14], is outlined as follows, $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t},$ where X_t is unknown fuzzy process and f, g are given functions, C_t is a standard Liu process. It is a new class of FDEs with perturbation term driven by Liu process. So its ambiguity is broader. Its fuzziness not only includes the fuzzy processing mode of the coefficients and the initial conditions, but also its own unique fuzzy process, which is embodied in its driving process. Many experts and scholars have shifted their research centers of gravity to the study of the properties and solutions of FDEs driven by Liu process. You, Wang and Huo [24] deduced the existence and uniqueness of the solution of homogeneous FDEs driven by Liu process. At the same time, some special types of solutions of FDEs were derived. The existence and uniqueness of solutions for more general types of FDEs were deduced by Chen and Qin [4].

In spite of the existence and uniqueness theorem of solutions, only a few special FDEs can find specific analytical solutions. The solution needs to be approximated by approximate numerical method. Therefore, the discussion of numerical methods is extremely significant. Inspired by this, You and Hao [22] deduced the Euler numerical method for homogeneous FDEs driven by Liu process in 2018. As for this equation, You and Hao [23] also discussed a numerical solution based on fuzzy Taylor expansion. In addition, You and Hao [26] extended the Euler method to the form of multi-dimensional homogeneous FDEs driven by multi-dimensional Liu process, and yielded the multi-dimensional Euler numerical solution. Based on these, this paper will introduce fuzzy Euler-trapezoidal method, which is the improvement of the numerical method in [22]. Compared with the previous fuzzy numerical methods, our paper not only aims to derive the local error and convergence order of the method, but also discusses the global error and its convergence order.

1.3 Structural arrangement

On the premise of homogeneous FDEs driven by Liu process, the structure of this article is as follows. In Section 2, a brief introduction is given to some of the fundamental knowledge points that will be used. In Section 3, fuzzy semi-implicit and implicit Euler scheme are given, fuzzy trapezoidal scheme is introduced with smaller error than Euler method. To simplify algorithm, we present fuzzy Euler-trapezoidal method. In Section 4, the local error and convergence order of the proposed numerical methods are discussed in detail. The core of the discussion in Section 5 is the global error and convergence order of those methods. In Section 6, a brief conclusion is given.

2 Preliminaries

In this section, some of the fundamental knowledge points are given which will be used in next content.

Assuming T is an index set, the credibility space is represented by $(Θ, P, Cr)$ . Supposing ξ is a function that maps from credibility space $(Θ, P, Cr)$ to the real number set R, which is called fuzzy variable. The essence of fuzzy process is a function that maps from the space $T \times (Θ, P, Cr)$ to the real number set R, which is denoted by X (t, θ). For a given t_*, the meaning of the function X (t_*, θ) is a fuzzy variable. Accordingly, if θ_* is given, the meaning of the function X (t, θ_*) is the sample track of a fuzzy process. If function X (t, θ) is a continuous function of t with regard to almost all θ, then the function is called sample continuous. To make it easy to write, we say fuzzy process X (t, θ) with abbreviated symbol X_t.

Definition 2.1. (Liu [11]) The fuzzy variable sequence {ξ_i} is said to be convergent a.s. to ξ if and only if there exists an event A with Cr {A} =1 such that $lim_{i \to \infty} ∣ ξ_{i} (θ) - ξ (θ) ∣ = 0,$ for every θ ∈ A .

Definition 2.2. (Liu [14]) A fuzzy process C_t is said to be a Liu process if

when t = 0, the initial value of C_t is 0;

C_t has stationary and independent increments;

for every fixed interval [s, s + t], the increment of C_s is a normally distributed fuzzy variable with expected value et and variance σ²t².

Note that when e = 0 and σ = 1, Liu process is said to be a standard Liu process.

Theorem 2.1. (Dai [5]) Let C_t be a Liu process. For any given θ ∈ Θ with Cr {θ} >0, the path C_t (θ) is Lipschitz continuous, i.e. there exists a Lipschitz constant K(θ) satisfying |C_t - C_s| ≤ K (θ) |t - s| .

Definition 2.3. (Liu Integral, Liu [14]) Let {t_n} be point sets inserted on the interval [a, b] arbitrarily. Denoted by a = t₁ < t₂ < ⋯ < t_n = b, the mesh is recorded as $Δ = max_{1 \leq i \leq n} ∣ t_{i + 1} - t_{i} ∣ .$ Then the fuzzy integral of fuzzy process X_t with regard to C_t is $\int_{a}^{b} X_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{n} X_{t_{i}} (C_{t_{i + 1}} - C_{t_{i}}),$ provided that the limit exists almost surely and is a fuzzy variable, where X_t is a fuzzy process and C_t is a standard Liu process.

Definition 2.4. (Fuzzy Differential Equation driven by Liu process, Liu [14]) Suppose C_t is a standard Liu process, and f, g are some given functions. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ is called a fuzzy differential equation driven by Liu process, where X_t is unknown fuzzy process.

Especially, for homogeneous FDEs driven by Liu process $d X_{t} = f (X_{t}) d t + g (X_{t}) d C_{t}, X_{t_{0}} = X_{0},$ (2.1) where f, g are given functions with continuous derivative, You and Hao [22] gave the following two conditions for f and g:

Linear growth condition: There is a constant L₁ > 0 such that $| f (x) |^{2} \lor | g (x) |^{2} \leq L_{1} (1 + | x |^{2}), 0 \leq t \leq T .$ where f (x) and g (x) are functions in (2.1).

Global Lipschitz condition: There is a constant L₂ > 0 such that $| f (x) - f (y) |^{2} \lor | g (x) - g (y) |^{2} \leq L_{2} | x - y |^{2}, 0 \leq t \leq T .$ where f (x) and g (x) are functions in (2.1), T > 0.

Theorem 2.2. (You and Hao [22]) Suppose that f (x) and g (x) in (2.1) are given functions that satisfy the Linear growth condition, thus for ∀ 0 ≤ s ≤ t ≤ T, there exists a constant M₂ (θ) such that (X_t - X_s) ² ≤ M₂ (θ) (t - s) ², where M₂ (θ) is related to s, t and initial value, X_t, X_s are the solutions of FDE (2.1) at time t and s, respectively.

Definition 2.5. (You and Hao [22]) Local error of solution of FDE (2.1) is $δ_{n + 1} = ∣ X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}} ∣,$ where X_{t
_n+1} is the analytic solution of the equation (2.1) when t = t_n+1, and ${\tilde{X}}_{t_{n + 1}}$ is the numerical solution of FDE (2.1), which obtained by using ${\tilde{X}}_{t_{n}} = X_{t_{n}}$ to calculate one step of the numerical method.

Definition 2.6. (You and Hao [23]) If there is a constant Q₁ (θ) related to the Lipschitz constant K (θ) , such that ∣δ_n+1 ∣ ≤ Q₁ (θ) h^{p
₁}, then the numerical method is called p₁ order locally convergent, where δ_n+1 is the local error of solution of the Equation (2.1).

3 Numerical methods

In this section, we will design some numerical methods to solve FDEs driven by Liu process.

At present, the numerical methods for solving fuzzy differential equations driven by Liu process are Euler scheme, Taylor scheme. However, fuzzy Euler scheme is simpler than fuzzy Taylor scheme, the accuracy of solution is low and the stability is weak. This defect provides us with new research ideas. Therefore, in order to solve fuzzy differential equations driven by Liu process most accurately and make the numerical solution has better stability, more numerical methods with different schemes are needed.

The form of the homogeneous FDE driven by Liu process discussed in this article is (2.1), i.e. $d X_{t} = f (X_{t}) d t + g (X_{t}) d C_{t}, X_{t_{0}} = X_{0} .$ To write the differential form (2.1) into the integral form on the interval [t_n, t_n+1], we have $X_{t_{n + 1}} = X_{t_{n}} + \int_{t_{n}}^{t_{n + 1}} f (X_{s}) d s + \int_{t_{n}}^{t_{n + 1}} g (X_{s}) d C_{s} .$ (*) For simplicity, write h = Δt_n = t_n+1 - t_n, ΔC_{t
_n} = C_{t
_n+1} - C_{t
_n} . Now if the second term $\int_{t_{n}}^{t_{n + 1}} f (X_{s}) d s$ in the right of above equation is approximated by hf (X_{t
_n}) , and replaces $\int_{t_{n}}^{t_{n + 1}} g (X_{s}) d C_{s}$ with g (X_{t
_n}) ΔC_{t
_n} approximately, then obtain fuzzy Euler scheme ${\tilde{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}},$ (3.1) and we can deduce that its local convergence order is 2 (refer to [22]). Here ${\tilde{X}}_{t_{n}}$ is the approximation of X_{t
_n}, ${\tilde{X}}_{t_{n + 1}}$ is the approximation of X_{t
_n+1}. Note that (3.1) is also called fuzzy explicit Euler scheme.

Different from the method of getting fuzzy explicit Euler scheme, utilizing h [λf (X_{t
_n+1}) + (1 - λ) f (X_{t
_n})] in place of $\int_{t_{n}}^{t_{n + 1}} f (X_{s}) d s,$ we get another formula $\begin{matrix} {\tilde{X}}_{t_{n + 1}} & = {\tilde{X}}_{t_{n}} + h [λ f ({\tilde{X}}_{t_{n + 1}}) + (1 - λ) f ({\tilde{X}}_{t_{n}})] \\ + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}}, \end{matrix}$ (3.2) where λ ∈ [0, 1] is a fixed constant. We refer to (3.2) as fuzzy semi-implicit Euler scheme.

It is obvious that fuzzy semi-implicit Euler scheme is fuzzy explicit Euler scheme if λ = 0. Fuzzy semi-implicit Euler scheme entitles the backward fuzzy Euler scheme when λ = 1 and is called fuzzy trapezoidal Euler scheme when $λ = \frac{1}{2}$ .

Similarly, substituting $\int_{t_{n}}^{t_{n + 1}} f (X_{s}) d s$ , $\int_{t_{n}}^{t_{n + 1}}$ g (X_s) dC_s with $\int_{t_{n}}^{t_{n + 1}} f (X_{t_{n + 1}}) d s, \int_{t_{n}}^{t_{n + 1}} g (X_{t_{n + 1}}) d C_{s}$ in (*), respectively, we get fuzzy implicit Euler scheme ${\tilde{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n + 1}}) + g ({\tilde{X}}_{t_{n + 1}}) Δ C_{t_{n}} .$ (3.3)

Compared with fuzzy explicit scheme, fuzzy implicit scheme has better stability. However, ${\tilde{X}}_{t_{n + 1}}$ in implicit scheme can not be obtained by simple arithmetic operation. The approximate value of ${\tilde{X}}_{t_{n + 1}}$ can be gained generally by iterative method and the essence of the iterative process is gradually explicit. Next we will show the solving process of fuzzy semi-implicit Euler scheme.

According to the fuzzy explicit Euler scheme, we obtain $X_{t_{n + 1}}^{(0)} = {\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}},$ substituting ${\tilde{X}}_{t_{n + 1}}$ with $X_{t_{n + 1}}^{(0)}$ on the right hand of (3.2), we have $\begin{matrix} X_{t_{n + 1}}^{(1)} & = {\tilde{X}}_{t_{n}} + h [λ f (X_{t_{n + 1}}^{(0)}) + (1 - λ) f ({\tilde{X}}_{t_{n}})] \\ + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}} . \end{matrix}$ Replacing ${\tilde{X}}_{t_{n + 1}}$ by $X_{t_{n + 1}}^{(1)}$ in (3.2), then $\begin{matrix} X_{t_{n + 1}}^{(2)} & = {\tilde{X}}_{t_{n}} + h [λ f (X_{t_{n + 1}}^{(1)}) + (1 - λ) f ({\tilde{X}}_{t_{n}})] \\ + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}} . \end{matrix}$ Repeating the iterative processes above, we get $\begin{matrix} X_{t_{n + 1}}^{(k + 1)} = {\tilde{X}}_{t_{n}} & + h [λ f (X_{t_{n + 1}}^{(k)}) + (1 - λ) f ({\tilde{X}}_{t_{n}})] \\ + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}}, k = 0, 1, 2 \dots . \end{matrix}$ (3.4)

Applying Global Lipschitz condition, and minusing (3.2) from (3.4), we obtain $\begin{matrix} ∣ X_{t_{n + 1}}^{(k + 1)} - {\tilde{X}}_{t_{n + 1}} ∣ = h λ ∣ f (X_{t_{n + 1}}^{(k)}) - f ({\tilde{X}}_{t_{n + 1}}) ∣ \\ \leq h λ \sqrt{L_{2}} ∣ X_{t_{n + 1}}^{(k)} - {\tilde{X}}_{t_{n + 1}} ∣ . \end{matrix}$

It can be seen that the iterative solution $X_{t_{n + 1}}^{(k + 1)}$ of (3.4) converges to ${\tilde{X}}_{t_{n + 1}}$ only if $h λ \sqrt{L_{2}} < 1$ .

In order to obtain the formula with smaller error and higher precision than fuzzy Euler scheme, we try to use trapezoidal quadrature formula in place of the right-hand integral in (*), then obtain fuzzy trapezoidal scheme $\begin{matrix} {\tilde{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} & + \frac{h}{2} [f ({\tilde{X}}_{t_{n}}) + f ({\tilde{X}}_{t_{n + 1}})] \\ + \frac{1}{2} [g ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n + 1}})] Δ C_{t_{n}} . \end{matrix}$ (3.5)

The corresponding numerical solving method mentioned above is referred to as fuzzy trapezoidal scheme, the iterative formula of fuzzy trapezoidal scheme is as follows, ${\begin{matrix} X_{t_{n + 1}}^{(0)} = {\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}} \\ X_{t_{n + 1}}^{(k + 1)} = {\tilde{X}}_{t_{n}} + \frac{h}{2} [f ({\tilde{X}}_{t_{n}}) + f (X_{t_{n + 1}}^{(k)})] + \frac{1}{2} [g ({\tilde{X}}_{t_{n}}) \\ + g (X_{t_{n + 1}}^{(k)})] Δ C_{t_{n}}, k = 0, 1, 2 \dots . \end{matrix}$ (3.6)

In order to analyze the convergence of iterative process, the formula (3.5) is subtracted from (3.6), then the result is $\begin{matrix} X_{t_{n + 1}}^{(k + 1)} - {\tilde{X}}_{t_{n + 1}} & = \frac{h}{2} [f (X_{t_{n + 1}}^{(k)}) - f ({\tilde{X}}_{t_{n + 1}})] \\ + \frac{1}{2} [g (X_{t_{n + 1}}^{(k)}) - g ({\tilde{X}}_{t_{n + 1}})] Δ C_{t_{n}} . \end{matrix}$

By using Global Lipschitz condition and Liu process’s Lipschitz condition, we obtain $\begin{array}{l} ∣ X_{t_{n + 1}}^{(k + 1)} - {\tilde{X}}_{t_{n + 1}} ∣ \leq \frac{h \sqrt{L_{2}} (1 + K (θ))}{2} \\ ∣ X_{t_{n + 1}}^{(k)} - {\tilde{X}}_{t_{n + 1}} ∣, \end{array}$ where $\sqrt{L_{2}}$ and K (θ) are Lipschitz constants. If the selected step h is small enough to make $\frac{h \sqrt{L_{2}} (1 + K (θ))}{2} < 1$ , then $X_{t_{n + 1}}^{(k + 1)} \to {\tilde{X}}_{t_{n + 1}}$ , when k → ∞ , this shows that the iterative process (3.6) is convergent.

On the one hand, fuzzy trapezoidal scheme improves the approximation accuracy, and on the other hand, it makes the calculation extremely complex and difficult to address. When the iterative formula (3.6) is applied to the actual calculation, the value of the function f (X_t) , g (X_t) must be recalculated for each iteration, and the iterations have to be repeated several times. Thus it is hard to achieve the desired results effectively thanks to the enormous amount of calculation. In response to this challenge, the core of the solution is not only to make the algorithm simple and easy to operate, but also to guarantee that the calculation result is highly approximate to the exact solution. A natural idea is to find an explicit scheme to derive ${\bar{X}}_{t_{n + 1}}$ as the estimated value of implicit scheme, and then use implicit scheme to correct the estimated value, ${\tilde{X}}_{t_{n + 1}}$ obtained as the corrected value. This method is called estimate-correction scheme. On the basis of the idea, using fuzzy Euler scheme as estimate, and fuzzy trapezoidal scheme as correction, so that we put forward fuzzy Euler-trapezoidal method, it takes the following form ${\begin{matrix} {\tilde{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} + \frac{h}{2} [f ({\tilde{X}}_{t_{n}}) + f ({\bar{X}}_{t_{n + 1}})] + \frac{1}{2} [g ({\tilde{X}}_{t_{n}}) \\ + g ({\bar{X}}_{t_{n + 1}})] Δ C_{t_{n}} \\ {\bar{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}}, \end{matrix}$ (3.7) where $f ({\bar{X}}_{t_{n + 1}}) = f ({\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}})$ , $g ({\bar{X}}_{t_{n + 1}}) = g ({\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}}) + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}}) .$

From [22] we know that the fuzzy Taylor scheme is $\begin{matrix} {\tilde{X}}_{t_{n + 1}} = {\tilde{X}}_{t_{n}} + \frac{h}{2} [F_{1} ({\tilde{X}}_{t_{n}}) + F_{2} ({\tilde{X}}_{t_{n}})] + G_{1} ({\tilde{X}}_{t_{n}}) \\ Δ C_{t_{n}} + \frac{1}{2 h} [G_{1} ({\tilde{X}}_{t_{n}}) + G_{2} ({\tilde{X}}_{t_{n}})] Δ C_{t_{n}}^{2} \end{matrix}$ (3.8) and the local convergence order of fuzzy Taylor scheme is 2. Here $F_{1} ({\tilde{X}}_{t_{n}}) = f ({\tilde{X}}_{t_{n}}), F_{2} ({\tilde{X}}_{t_{n}}) = f ({\tilde{X}}_{t_{n}} + hf ({\tilde{X}}_{t_{n}})),$ $G_{1} ({\tilde{X}}_{t_{n}}) = g ({\tilde{X}}_{t_{n}}), G_{2} ({\tilde{X}}_{t_{n}}) = g ({\tilde{X}}_{t_{n}} + g ({\tilde{X}}_{t_{n}}) Δ C_{t_{n}}) .$

4 Local error and convergence order of one step method

To study the properties of numerical methods, the most important is to discuss the local truncation error and convergence order of numerical methods.

Theorem 4.1. Suppose f and g are given functions that satisfy the Linear growth condition and Global Lipschitz condition in (2.1). Then fuzzy Euler-trapezoidal method is locally convergent almost everywhere, and the convergence order is 2.

Proof: It follows from Theorem 2.2 that the local error of fuzzy Euler-trapezoidal method for FDE (2.1) with $\begin{array}{l} δ_{n + 1} = ∣ X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}} ∣ \\ = | \int_{t_{n}}^{t_{n + 1}} f (X_{s}) d s + \int_{t_{n}}^{t_{n + 1}} g (X_{s}) d C_{s} - \frac{h}{2} [f (X_{t_{n}}) + f ({\bar{X}}_{t_{n + 1}})] - \frac{1}{2} [g (X_{t_{n}}) + g ({\bar{X}}_{t_{n + 1}})] Δ C_{t_{n}} | \\ = | \int_{t_{n}}^{t_{n + 1}} [f (X_{s}) - f (X_{t_{n}})] d s + \int_{t_{n}}^{t_{n + 1}} [g (X_{s}) - g (X_{t_{n}})] d C_{s} + \frac{1}{2} [f (X_{t_{n}}) - f ({\bar{X}}_{t_{n + 1}})] Δ t_{n} + \frac{1}{2} [g (X_{t_{n}}) - g ({\bar{X}}_{t_{n + 1}})] Δ C_{t_{n}} | \\ \leq | \int_{t_{n}}^{t_{n + 1}} [f (X_{s}) - f (X_{t_{n}})] d s ∣ + ∣ \int_{t_{n}}^{t_{n + 1}} [g (X_{s}) - g (X_{t_{n}})] d C_{s} | + \frac{1}{2} ∣ f (X_{t_{n}}) - f (X_{t_{n}} + f (X_{t_{n}}) Δ t_{n} + g (X_{t_{n}}) Δ C_{t_{n}}) ∣ Δ t_{n} + \frac{1}{2} ∣ g (X_{t_{n}}) - g (X_{t_{n}} + f (X_{t_{n}}) Δ t_{n} + g (X_{t_{n}}) Δ C_{t_{n}}) ∣ Δ C_{t_{n}} \\ \leq \sqrt{L_{2}} \int_{t_{n}}^{t_{n + 1}} ∣ X_{s} - X_{t_{n}} ∣ d s + \sqrt{L_{2}} \int_{t_{n}}^{t_{n + 1}} ∣ X_{s} - X_{t_{n}} ∣ d C_{s} + \frac{1}{2} \sqrt{L_{2}} [∣ f (X_{t_{n}}) ∣ Δ t_{n}^{2} + ∣ g (X_{t_{n}}) ∣ Δ t_{n} Δ C_{t_{n}}] + \frac{1}{2} \sqrt{L_{2}} [∣ f (X_{t_{n}}) ∣ Δ t_{n} Δ C_{t_{n}} + ∣ g (X_{t_{n}}) ∣ Δ C_{t_{n}}^{2}], \end{array}$ Applying Theorem 2.2, we have $\begin{array}{l} δ_{n + 1} \leq \sqrt{L_{2} M_{2} (θ)} Δ t_{n} \int_{t_{n}}^{t_{n + 1}} d s + \sqrt{L_{2} M_{2} (θ)} Δ t_{n} \\ \int_{t_{n}}^{t_{n + 1}} d C_{s} + \frac{1}{2} \sqrt{L_{2}} ∣ f (X_{t_{n}}) ∣ Δ t_{n}^{2} + \frac{1}{2} \sqrt{L_{2}} \\ ∣ g (X_{t_{n}}) ∣ Δ C_{t_{n}}^{2} + \frac{1}{2} \sqrt{L_{2}} (∣ f (X_{t_{n}}) ∣ \\ + ∣ g (X_{t_{n}}) ∣) Δ t_{n} Δ C_{t_{n}} . \end{array}$ According to the Lipschitz continuity of Liu process, $\begin{array}{l} Δ C_{t_{n}}^{2} = {(C_{t_{n + 1}} - C_{t_{n}})}^{2} \leq {[K (θ) ∣ t_{n + 1} - t_{n} ∣]}^{2} \\ = K^{2} (θ) Δ t_{n}^{2}, \end{array}$ which is equivalent to ∣ΔC_{t
_n} ∣ ≤ K (θ) Δt_n . Therefore, $\int_{t_{n}}^{t_{n + 1}} d C_{s} = ∣ \lim_{Δ \to 0} \sum_{i = 1}^{k} (C_{s_{i + 1}} - C_{s_{i}}) ∣ \leq ∣ K (θ) ∣ ∣ \int_{t_{n}}^{t_{n + 1}} d s ∣ .$ Thus we have $\begin{array}{l} \lim_{Δ t_{n} \to 0} δ_{t_{n + 1}} \leq \lim_{Δ t_{n} \to 0} [\sqrt{L_{2} M_{2} (θ)} \int_{t_{n}}^{t_{n + 1}} d s \\ + \sqrt{L_{2} M_{2} (θ)} K (θ) \int_{t_{n}}^{t_{n + 1}} d s + \frac{1}{2} \sqrt{L_{2}} ∣ f (X_{t_{n}}) ∣ \\ Δ t_{n} + \frac{1}{2} \sqrt{L_{2}} K^{2} (θ) ∣ g (X_{t_{n}}) ∣ Δ t_{n} + \frac{1}{2} \sqrt{L_{2}} K (θ) \\ (∣ f (X_{t_{n}}) ∣ + ∣ g (X_{t_{n}}) ∣) Δ t_{n}] Δ t_{n} \\ \leq \lim_{Δ t_{n} \to 0} [\sqrt{L_{2} M_{2} (θ)} (1 + K (θ)) + \frac{1}{2} \sqrt{L_{2}} ∣ f (X_{t_{n}}) ∣ \\ + \frac{1}{2} \sqrt{L_{2}} K^{2} (θ) ∣ g (X_{t_{n}}) ∣ + \frac{1}{2} \sqrt{L_{2}} K (θ) (∣ f (X_{t_{n}}) ∣ \\ + ∣ g (X_{t_{n}}) ∣)] Δ t_{n}^{2} = \lim_{Δ t_{n} \to 0} Q_{1} (θ) Δ t_{n}^{2} = 0, \end{array}$ where $\begin{array}{l} Q_{1} (θ) = \sqrt{L_{2} M_{2} (θ)} (1 + K (θ)) + \frac{1}{2} \sqrt{L_{2}} ∣ f (X_{t_{n}}) ∣ \\ + \frac{1}{2} \sqrt{L_{2}} K^{2} (θ) ∣ g (X_{t_{n}}) ∣ + \frac{1}{2} \sqrt{L_{2}} K (θ) \\ (∣ f (X_{t_{n}}) ∣ + ∣ g (X_{t_{n}}) ∣) \end{array}$ So fuzzy Euler-trapezoidal method is locally convergent almost surely, and the convergence order is 2.

Theorem 4.2. Suppose f and g are given functions that satisfy the Linear growth condition and Global Lipschitz condition in (2.1). Then fuzzy semi-implicit, implicit Euler scheme and trapezoidal scheme are locally convergent almost everywhere, and the convergence order is 2.

Proof: The proof processes are similar to those of Theorem 4.1.

5 Global error and convergence order

The former section discusses the local error and the convergence order of numerical methods. While the task of this section is to study the global error and the convergence order of the mentioned numerical methods.

Definition 5.1. Global error of solution of FDE (2.1) is $ɛ_{n + 1} = ∣ X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}}^{(n + 1)} ∣,$ where X_{t
_n+1} is the analytic solution of the equation (2.1) when t = t_n+1, and ${\tilde{X}}_{t_{n + 1}}^{(n + 1)}$ is the numerical solution of FDE (2.1) obtained by calculating n + 1 steps from initial value of the numerical method.

Remark 5.1. Note that a fuzzy variable is a function that maps from credibility space $(Θ, P, Cr)$ to the real number set R, and a fuzzy process X (t, θ) is a function that maps from the space $T \times (Θ, P, Cr)$ to the real number set R. Thus for a fixed t_*, X (t_*, θ) is a function for θ, that is to say, X (t_*, θ) is a fuzzy variable. Since $| X (t_{*}, θ) - \tilde{X} (t_{*}, θ) |$ is the absolute value of the subtraction of the two functions of θ, the result should be a real function, in other word, it is a fuzzy variable. Hence, $ɛ_{n + 1} = ∣ X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}}^{(n + 1)} ∣$ is a fuzzy variable, i.e. it is a real function of θ. Therefore, the result should be a real function rather than a set. This is the reason why the error is introduced as similar to real case in this paper.

Definition 5.2. If there is a constant Q₂ (θ) related to the Lipschitz constant K (θ) , such that $ɛ_{n + 1} \leq Q_{2} (θ) h^{p_{2}},$ then the numerical method is called p₂ order globally convergent, where ɛ_n+1 is the global error of the equation (2.1).

Theorem 5.1. Suppose f and g in (2.1) are given functions that satisfy the Linear growth condition and Global Lipschitz condition. Then fuzzy Euler-trapezoidal method is globally convergent almost surely, and the convergence order is 1.

Proof: The global error of numerical solutions for FDEs (2.1) is: $\begin{array}{l} ε_{n + 1} = | X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}}^{(n + 1)} | \\ = | X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}} + {\tilde{X}}_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}}^{(n + 1)} | \\ \leq | X_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}} | + | {\tilde{X}}_{t_{n + 1}} - {\tilde{X}}_{t_{n + 1}}^{(n + 1)} | \\ \leq δ_{n + 1} + | X_{t_{n}} + \frac{h}{2} [f (X_{t_{n}}) + f ({\bar{X}}_{t_{n + 1}})] + \frac{1}{2} [g (X_{t_{n}}) \\ + g ({\bar{X}}_{t_{n + 1}})] Δ C_{t_{n}} - {\tilde{X}}_{t_{n}}^{(n)} - \frac{h}{2} [f ({\tilde{X}}_{t_{n}}^{(n)}) + f ({\bar{X}}_{t_{n + 1}}^{(n + 1)})] \\ - \frac{1}{2} [g ({\tilde{X}}_{t_{n}}^{(n)}) + g ({\bar{X}}_{t_{n + 1}}^{(n + 1)})] Δ C_{t_{n}} | \leq δ_{n + 1} + | X_{t_{n}} - {\tilde{X}}_{t_{n}}^{(n)} | \\ + \frac{h}{2} | f (X_{t_{n}}) - f ({\tilde{X}}_{t_{n}}^{(n)}) | + \frac{1}{2} | g (X_{t_{n}}) - g ({\tilde{X}}_{t_{n}}^{(n)}) | Δ C_{t_{n}} \\ + \frac{h}{2} | f (X_{t_{n}} + h f (X_{t_{n}}) + g (X_{t_{n}}) Δ C_{t_{n}}) - f ({\tilde{X}}_{t_{n}}^{(n)} \\ + h f ({\tilde{X}}_{t_{n}}^{(n)}) + g ({\tilde{X}}_{t_{n}}^{(n)}) Δ C_{t_{n}}) | + \frac{1}{2} | g (X_{t_{n}} + h f (X_{t_{n}}) \\ + g (X_{t_{n}}) Δ C_{t_{n}}) - g ({\tilde{X}}_{t_{n}}^{(n)} + h f ({\tilde{X}}_{t_{n}}^{(n)}) + g ({\tilde{X}}_{t_{n}}^{(n)}) Δ C_{t_{n}}) | Δ C_{t_{n}} \\ \leq δ_{n + 1} + ε_{n} + \frac{h \sqrt{L_{2}}}{2} ε_{n} + \frac{h K (θ) \sqrt{L_{2}}}{2} ε_{n} + \frac{h \sqrt{L_{2}}}{2} \\ (1 + h \sqrt{L_{2}} + h K (θ) \sqrt{L_{2}}) ε_{n} + \frac{h K (θ) \sqrt{L_{2}}}{2} (1 + h \sqrt{L_{2}} \\ + h K (θ) \sqrt{L_{2}}) ε_{n} \leq (1 + h \sqrt{L_{2}} + h K (θ) \sqrt{L_{2}} + \frac{h^{2} L_{2}}{2} \\ + K (θ) h^{2} L_{2} + \frac{K^{2} (θ) h^{2} L_{2}}{2}) ε_{n} + δ_{n + 1} . \end{array}$ Let $Z_{φ} = \sqrt{L_{2}} (1 + K (θ)) + \frac{{hL}_{2}}{2} + K (θ) {hL}_{2} + \frac{K^{2} (θ) {hL}_{2}}{2} .$ The above formula is written as $ɛ_{n + 1} \leq (1 + {hZ}_{φ}) ɛ_{n} + δ_{n + 1} \leq (1 + {hZ}_{φ}) ɛ_{n} + Q_{1} (θ) h^{2},$ thus the following inequality can be obtained repeatedly and recursively $ɛ_{n + 1} \leq (1 + {hZ}_{φ})^{n + 1} | ɛ_{0} | + \frac{Q_{1} h}{Z_{φ}} [(1 + {hZ}_{φ})^{n} - 1] .$ For x ≥ -1, we have 0 ≤ (1 + x) ⁿ ≤ e^nx . Given t_n - t₀ = nh ≤ T, then $(1 + {hZ}_{φ})^{n} \leq (e^{{hZ}_{φ}})^{n} \leq e^{{TZ}_{φ}} .$ Since there is no initial error, ɛ₀ = 0, the global error is $\begin{array}{l} \lim_{h \to 0} ε_{n + 1} \leq \lim_{h \to 0} \frac{Q_{1} h}{Z_{φ}} [{(1 + h Z_{φ})}^{n} - 1] \\ \leq \lim_{h \to 0} \frac{e^{T Z_{φ}} - 1}{Z_{φ}} Q_{1} (θ) h \\ \leq \lim_{h \to 0} \frac{e^{T (\sqrt{L_{2}} (1 + K (θ)) + \frac{T L_{2}}{2} + K (θ) T L_{2} + \frac{K^{2} (θ) T L_{2}}{2}) - 1}}{\sqrt{L_{2}} (1 + K (θ))} \\ Q_{1} (θ) h = 0, \end{array}$ where $Q_{2} (θ) = \frac{e^{T (\sqrt{L_{2}} (1 + K (θ)) + \frac{{TL}_{2}}{2} + K (θ) {TL}_{2} + \frac{K^{2} (θ) {TL}_{2}}{2}) - 1}}{\sqrt{L_{2}} (1 + K (θ))} Q_{1} (θ) .$ According to the definition of global convergence, fuzzy Euler-trapezoidal method is globally convergent almost everywhere, and the convergence order is 1.

Remark 5.2. The equation in our paper is different from the equation discussed in real case and the proof processes in Section 4 and 5 are different from those in real case. The equation in our paper is fuzzy differential equation driven by Liu process, i.e. $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t},$ where C_t is a standard Liu process, and f, g are some given functions. When solving fuzzy differential equation driven by Liu process, the fuzzy integral $\int_{0}^{t} g (X_{s}) d C_{s}$ is involved, which is called Liu integral. The definition of Liu integral is described in Definition 2.3.

The most significant differences are reflected in the following four points:

f (X_t), g (X_t) and C_t in our paper all have fuzziness;

dC_t is not the derivative of C_t since C_t has no derivative for arbitrary t, it is just a differential form of corresponding Liu integral (a kind of fuzzy integral corresponding to Ito integral). Thus dC_t cannot be converted to some function times dt;

Though dC_t is not derivative, C_t is a Lipschitz continuous function of t;

(X_t - X_s) ² ≤ M (θ) (t - s) ², where M (θ) is related to t, s.

It is just because (3) and (4), we can deal with the Liu integral term similar to $\int_{t_{n}}^{t_{n + 1}} ∣ X_{s} - X_{t_{n}} ∣ d C_{s}$ , deduce our result by amplified method, otherwise, the results can not obtained like the proof of the theorem in real case since we can not deal with Liu integral term by using the method in real case.

Theorem 5.2. Suppose f and g in (2.1) are given functions that satisfy the Linear growth condition and Global Lipschitz condition. Then fuzzy explicit, semi-implicit, implicit Euler scheme, fuzzy trapezoidal scheme and fuzzy Taylor scheme are globally convergent almost surely, and the convergence order is 1.

Proof: The proof of this theorem is similar to that of Theorem 5.1.

Theorem 4.1 and Theorem 5.1 can be illustrated by the following examples.

Example 5.1 Consider dX_t = 10X_tdt - 2X_tdC_t. Since f (x) =10x and g (x) =2x satisfy Linear growth condition $| f (x) |^{2} \lor | g (x) |^{2} \leq 100 (1 + | x |^{2})$ and Global Lipschitz condition $| f (x) - f (y) |^{2} \lor | g (x) - g (y) |^{2} \leq 100 | x - y |^{2},$ it follows from Theorem 4.1 that the fuzzy Euler-trapezoidal is locally convergent almost surely, and the convergence order is 2. Furthermore, from Theorem 5.1 we know the fuzzy Euler-trapezoidal is globally convergent almost surely, and the convergence order is 1.

Example 5.2 Consider dX_t = 5 sin(X_t) dt + 7 sin(- X_t) dC_t. Because f (x) =5 sin(x) and g (x) =7 sin(- x) satisfy Linear growth condition $| f (x) |^{2} \lor | g (x) |^{2} \leq 49 (1 + | x |^{2})$ and Global Lipschitz condition $| f (x) - f (y) |^{2} \lor | g (x) - g (y) |^{2} \leq 49 | x - y |^{2},$ the numerical solution of the fuzzy Euler-trapezoidal is locally convergent almost surely, and the convergence order is 2; the numerical solution of the fuzzy Euler-trapezoidal is globally convergent almost surely, and the convergence order is 1, according to Theorem 4.1 and Theorem 5.1, respectively.

6 Conclusions

In order to improve the approximation effect of the fuzzy Euler scheme, fuzzy semi-implicit, fuzzy implicit Euler scheme, fuzzy trapezoidal scheme and fuzzy Euler-trapezoidal of fuzzy differential equation driven by Liu process are presented in this paper, and two convergence orders of numerical methods are given: the local convergence order is 2 and the global convergence order is 1.

Footnotes

Acknowledgments

This work was supported by Natural Science Foundation of China Grant No. 61773150 and Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.

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