key: cord-0827857-e9b2rqpd
authors: Roomi, Vahid; Ahmadi, Hamid Reza
title: The Liouville Formula for the Uncertain Homogeneous Linear System and Explicit Solutions of the System
date: 2021-06-14
journal: Differ Equ Dyn Syst
DOI: 10.1007/s12591-021-00573-9
sha: b1beb85dd3d3e23a1738e84f85a87370ae796b0b
doc_id: 827857
cord_uid: e9b2rqpd

This paper presents some new definitions and results about a system of uncertain homogeneous linear differential equations. Introducing the uncertain fundamental system and uncertain fundamental matrix for the uncertain system, the Liouville formula will be proven for the system. Moreover, the explicit solutions of the system will be presented.

When no samples are available to estimate a probability distribution, we have to invite some domain experts to evaluate the belief degree that each event will happen. In order to model such phenomena, uncertainty theory was founded by Liu [7] in 2007, refined by Liu [6] in 2010, and became a branch of mathematics based on the normality axiom, duality axiom, subadditivity axiom, and product axiom. It is a new tool to study subjective uncertainty. The first fundamental concept in uncertainty theory is uncertain measure which is used to indicate the belief degree that an uncertain event may occur. Liu process and uncertain calculus were initialized by Liu (2009) to deal with differentiation and integration of functions of uncertain processes. Furthermore, uncertain differential equations, a type of differential equations driven by the Liu process, was defined by Liu [4] . Uncertainty theory and uncertain differential equations has been studied in many literatures (for example, see [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] and the references cited therein).

In this paper the fundamental matrix for the uncertain homogeneous linear system will be introduced and the Liouville formula for the system will be proven. Liouville formula gives us better information about determinant of uncertain fundamental matrix. Moreover, Definition 2 [5] An uncertain process C t is said to be a Liu process if (i) C 0 = 0 and almost all sample paths are Lipschitz continuous, (ii) C t has stationary and independent increments, (iii) every increment C s+t − C s is a normal uncertain variable with expected value 0 and variance t 2 .

Let C it , i = 1, 2, … , n be independent Liu processes. Then, C t = (C 1t , C 2t , … , C nt ) T is called an n-dimensional Liu process [12] .

Definition 3 [5] Let X t be an uncertain process and C t be a Liu process. For any partition of closed interval [a, b] with a = t 1 < t 2 < ⋯ < t n+1 = b , the mesh is written as Then, Liu integral of X t with respect to C t is defined as provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be integrable.

An uncertain differential equation is a type of differential equation involving uncertain processes. We introduce uncertain differential equation and system of uncertain linear differential equations as follows.

Definition 4 [4] Suppose C t is a Liu process, and f and g are two functions. Then is called an uncertain differential equation. A solution is an uncertain process C t that satisfies (2.1) identically in t.

Definition 5 [8] Let t be a positive real variable and X t = (X 1t , X 2t , … , X nt ) T be an n-dimensional uncertain process whose elements X jt are integrable uncertain processes. Also, A(t) = [a ij (t)] and B(t) = [b ij (t)] are n × n matrices of integrable uncertain real functions and U t = (u 1 (t), u 2 (t), … , u n (t)) T and V t = (v 1 (t), v 2 (t), … , v n (t)) T are n-component vectors of uncertain integrable real functions. Then is called a system of uncertain linear differential equations.

If U(t) and V(t) in (2.2) are identically 0; that is, if (2.2) has the form then the equation is called uncertain homogeneous linear system ( [8] ). Chen and Liu in [1] considered uncertain differential equation (2.1) and presented the following theorem about the existence and uniqueness of solutions of (2.1). 

Stability of uncertain differential equation (2.1) is considered by the authors in [11] as follows.

Recently, Lio and Liu in [3] have obtained a COVID-19 spread model as uncertain differential equation

where X t is the cumulative number of COVID-19 infections in China at time t, C t is Liu process, and t and t are unknown time-varying parameters at this moment. Then, they have inferred the zero-day of COVID-19 spread in China.

be an n × n matrix of uncertain integrable real functions and X t = (X 1t , X 2t ,..., X nt ) T be an n-dimensional uncertain process whose elements X jt and all a ij (t)X jt are integrable uncertain processes. Then, the Liu integral of A(t)X t with respect to C t on [a, b] is defined by In this case, A(t)X t is said to be Liu integrable with respect to C t [8] .

Recently, the authors in [8] considered system (2.2) with an initial condition and proved the following theorem about the existence and uniqueness of solutions of the initial value problem.

|A(t)| ⩽ k(t) , |B(t)| ⩽ k(t) , |U(t)| ⩽ k(t) , and |V(t)| ⩽ k(t) on [a, b]. Then, system (2.2) with initial condition X t 0 = X 0 has a unique solution X(t) on [a, b] in the following sense. dX t = t X t dt + t X t dC t , � b a A(t)X t dC t = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ n ∑ j=1 ∫ a b a 1j (t)X jt dC jt n ∑ j=1 ∫ a b a 2j (t)X jt dC jt ⋮ n ∑ j=1 ∫ a b a nj (t)X jt dC jt ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . X t = X 0 + ∫ t 0 [A(s)X s + U(s)]ds + ∫ t 0 [B(s)X s + V(s)]dC s . X t = X 0 + ∫ t t 0 [A(s)X s + U(s)]ds + ∫ t t 0 [B(s)X s + V(s)]dC s , t ∈ [a, b].

In this section, the fundamental system and fundamental matrix associate with the system of uncertain homogeneous linear differential equations will be introduced. The main result in this section is to prove the Liouville formula for the system.

A set of n linearly independent solutions of (2.3) is called an uncertain fundamental system of (2.3).

Definition 7 An n × n matrix whose columns are linearly independent solutions of (2.3) is called uncertain fundamental matrix of (2.3).

Proof If there exists s 0 ∈ (a, b) such that X s 0 = 0 , then X s and Y s ≡ 0 are two solutions of (2.3). Therefore, by the uniqueness of solutions X s ≡ 0. ◻

Let Y s be an n × n matrix whose columns are solutions of (2

Let Y s be an n × n matrix whose columns are solutions of (2.3) in (a, b) . A necessary and sufficient condition for Y s to be an uncertain fundamental matrix of (2.3) is that there exists s 0 ∈ (a, b) such that det Y s 0 ≠ 0.

Proof Let det Y s 0 ≠ 0 , then by Theorem 5 det Y s ≠ 0 . Now, let y s ≠ 0 be a solution of (2.3) with initial condition y s 1 ≠ 0 . Since det Y s 1 ≠ 0 , there exists C = (c 1 , c 2 , … , c n ) T such that Y s 1 C = y s 1 or ∑ n i=1 c i y is 1 = y s 1 . Thus, by the uniqueness of solutions, ∑ n i=1 c i y si = y s . Therefore, Y s is an uncertain fundamental matrix. Now let det Y s ≡ 0. In this case, det Y s 1 = 0 for all s 1 ∈ (a, b) . Thus, the columns of Y s 1 are linearly dependent. Therefore, there exist c 1 , c 2 , … , c n such that ∑ n i=1 c i y is 1 ≡ 0 . But by Theorem 4, ∑ n i=1 c i y si ≡ 0 . Thus, the columns of Y s are linearly dependent and therefore, Y s is not an uncertain fundamental matrix. Therefore, if Y s be an uncertain fundamental matrix, we must have det Y s ≠ 0 or there exists s 0 ∈ (a, b) such that det Y s 0 ≠ 0 . ◻

The following corollary is a direct conclusion of Theorems 5 and 6.

Corollary 1 Let Y s be an n × n matrix whose columns are solutions of (2.3) in (a, b) . A necessary and sufficient condition for Y s to be an uncertain fundamental matrix of (2.3) is that det Y s ≠ 0. Since det Y s ≠ 0 on (a, b), then Therefore, F s is a constant matrix. Since then F s is nonsingular and the proof is complete. ◻

The authors in [1] proved the following theorem and calculated the exact solution of an uncertain linear equation. We will use this theorem to state our main result in this section.

Theorem 8 [1] Let u 1t , u 2t , v 1t , v 2t be integrable uncertain processes. Then the linear uncertain differential equation

has a solution where Now we state our main result in this section which is an extension of a theorem in the theory of ordinary differential equations, which is known as Liouville formula, to the uncertain homogeneous linear systems. 

Therefore,

In the first determinant on the right, by adding the appropriate multiple of each row to the first row and carrying out similar operations on the other determinants on the right, we obtain

[a 1k (s)y kn s ds + b 1k (s)y kn s dC s ] y 21 

Proof Let X t be a fundamental matrix. Then X t is a solution of (3.1) and therefore satisfies in (3.2) . On the other hand, according to Corollary 1, X t is a fundamental matrix if only if det X t ≠ 0 . According to relation (3.2) , det X t ≠ 0 if only if there exists t 0 ∈ (a, b) such that det X t 0 ≠ 0 . Thus X t is an uncertain fundamental matrix if and only exists t 0 ∈ (a, b) such that det 

Proof Let I i be the i-th column of identity matrix, X it = e D(t) I i for 1 ⩽ i ⩽ n and X t = (X 1t , … , X nt ). Then every column of X t is a solution of (4.1) and X t = e D(t) I = e D(t) . From X t 0 = e D(t 0 ) = e 0 = I and det X t 0 = 1 ≠ 0 it can be concluded that X t = e D(t) is a fundamental matrix. 

. (t + 2C t ) n n! I n = e (t+2C t ) I Therefore, the fundamental matrix of the system is as follows.

In order to calculate the fundamental matrix for t 0 ≠ 0 and C t 0 ≠ 0 , it suffices to put t − t 0 , t 2 − t 0 2 , C t − C t 0 and C t 2 − C t 0 2 instead of t, t 2 , C t and C t 2 respectively.

In this paper, we introduced the uncertain fundamental system and uncertain fundamental matrix for the uncertain homogeneous linear system. The main contribution of this paper was to prove the Liouville formula for the system and calculating the explicit solutions of the system.

Existence and uniqueness theorem for uncertain differential equations

Multi-dimensional uncertain differential equation: existence and uniqueness of solution

Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China. Fuzzy Optim. Decision Mak

Fuzzy process, hybrid process and uncertain process

Some research problems in uncertainty theory

Uncetainty Theory: A Branch of Mathematics for Modelling Human Uncertainty

Uncertainty Theory

Existence and uniqueness of solutions of uncertain linear systems

Multi-dimensional uncertain calculus with Liu process

Uncertain Differential Equations

Some stability theorems of uncertain differential equations

Multi-dimensional canonical process

The authors would like to thank anonymous referees for their valuable comments that improved the manuscript.

Funding No funding was received to assist with the preparation of this manuscript.

Code Availability Not applicable.

In this section, introducing the exponential matrix, the explicit solutions of (2.3) will be presented. Suppose that B(t) is an n × n matrix the elements of which are continuous functions on (a, b), and B 0 = I n×n is the identity matrix. In this case m ∑ s=0 1 s! B s is also an n × n matrix. For the integers p and q we haves! B s is a Cauchy sequence and therefore it converges to an n × n matrix. This matrix is called the exponential matrix of B(t) and is denoted by e B(t) . It is well known that, from linear algebra, for two n × n matrices A and B, if AB = BA , then e A+B = e A e B . Now we present the main result of this section.

Proof First note that for differentiable matrices M and N whose elements are also uncertain differentiable functions, the following relations hold.

Now assume that Then, using (4.2) and the hypothesis on A, B and D, it can be concluded thatTherefore, 

is a fundamental matrix.