Approximate Solution of 𝑛th-Order Fuzzy Linear Differential Equations

Abstract

The approximate solution of 𝑛th-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

1. Introduction

Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers. Most practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions, and therefore a fuzzy initial problem occurs and needs to be solved. However, most fuzzy initial value problems could not be solved exactly. So it is necessary to consider their approximating methods.

Prior to discussing fuzzy differential equations and their associated numerical algorithms, it is necessary to present an appropriate brief introduction to derivative of the fuzzy-valued function. The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1], and it was followed up by Dubois and Prade [2] who used the extension principle in their approach. Other fuzzy derivative concepts have been proposed by Puri and Ralescu [3] and Goetschel Jr. and Voxman [4] as an extension of the Hukuhara derivative of multivalued functions. In recent years, many works have been produced in the aspects of theories and applications on fuzzy differential equations; see [5–15]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [16, 17] who later applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [18, 19]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [20, 21], Seikkala [22], Ouyang and Wu [23], Kloeden [24], and Wu [25]. A generalization of fuzzy differential equation was given by Aubin [26, 27], Baĭdosov [6], Leland [28], and Colombo and Kř‌ivan [29]. Some numerical methods for solving fuzzy differential equations were introduced in [30–33].

For an 𝑛th-order linear differential equation 𝑦^(𝑛)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)+⋯+𝑎₁(𝑡)𝑦^′+𝑎₀(𝑡)𝑦=𝑔(𝑡) (1) with fuzzy initial conditions 𝑦𝑡₀=𝑏̃₀,𝑦^′𝑡₀=𝑏̃₁,…,𝑦^(𝑛−1)𝑡₀=𝑏̃_𝑛−1, (2) where 𝑔(𝑡),𝑎_𝑖(𝑡),𝑖=0,1,…,𝑛−1,𝑡∈[𝑡₀,𝑇] are continuous functions, Buckley and Feuring [34] presented two analytical methods for solving them. The first was to fuzzify the crisp solution and then check to see if it satisfies the fuzzy differential equations with fuzzy initial conditions. The second method was the reverse of the first one, in that they firstly solved the fuzzy initial value problem and then checked to see if it defines a fuzzy function. In 2008, Allahviranloo et al. [35] utilized the collocation method to transfer (1) and (2) into a crisp 2(𝑛+1)×2(𝑛+1) system of linear equations 𝑆(𝑡)𝑋=𝑌(𝑟) (3) for three specious cases; that is, all coefficient functions 𝑎_𝑘(𝑡) are positive, negative, 𝑎_𝑘(𝑡)(𝑘=0,1,…,𝑛−𝑚), are negative 𝑎_𝑘(𝑡)(𝑘=𝑛−𝑚+1,…,𝑛−1) are positive, respectively. By computing (3), they obtained the approximate solution of (1) and (2). However, their methods will be restricted for the general case.

In this paper, the 𝑛th-order linear differential equation with fuzzy initial conditions is further investigated. It shows that the result obtained in this paper is an extension of Allahviranloo's conclusions. In addition, considering that the case of the order of the fuzzy differential equation and the number of basic functions in assumed solution are not always equal, we obtain the approximate solution of the original equations (1) and (2) by calculating the minimal norm least squares solution of crisp system of linear equations. Three illustrating examples are given, and one of them is compared with Allahviranloo's work and is shown be more accurate. The structure of this paper is organized as follows.

In Section 2, we recall some basic definitions and results about fuzzy numbers and the undetermined fuzzy coefficients method. In Section 3, a class of 𝑛th-order fuzzy linear differential equations is investigated by converting it to a crisp system of linear equations, and some corollaries for the special cases are given. The proposed algorithms are illustrated by solving some examples in Section 4, and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. Fuzzy Number

Definition 1 (see [1]). A fuzzy number is a fuzzy set like 𝑢∶𝑅→𝐼=[0,1] which satisfies the following.(1)𝑢is upper semicontinuous.(2)𝑢 is fuzzy convex; that is, 𝑢(𝜆𝑥+(1−𝜆)𝑦)≥min{𝑢(𝑥),𝑢(𝑦)} for all 𝑥,𝑦∈𝑅,𝜆∈[0,1].(3)𝑢 is normal; that is, there exists 𝑥₀∈𝑅 such that 𝑢(𝑥₀)=1.(4)Supp 𝑢={𝑥∈𝑅∣𝑢(𝑥)>0} is the support of 𝑢, and its closure cl(supp 𝑢) is compact.
Let 𝐸¹ be the set of all fuzzy numbers on 𝑅.

Definition 2 (see [36]). A fuzzy number 𝑢 in parametric form is a pair (𝑢,𝑢) of functions 𝑢(𝑟), 𝑢(𝑟), 0≤𝑟≤1, which satisfies the following requirements.(1)𝑢(𝑟) is a bounded monotonic increasing left continuous function.(2)𝑢(𝑟) is a bounded monotonic decreasing left continuous function.(3)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1.
For arbitrary fuzzy number 𝑥=(𝑥(𝑟),𝑥(𝑟)),𝑦=(𝑦(𝑟),𝑦(𝑟))∈𝐸¹, 0≤𝑟≤1, and real number 𝑘∈𝑅,(1)𝑥=𝑦 if and only if 𝑥(𝑟)=𝑦(𝑟) and 𝑥(𝑟)=𝑦(𝑟);(2)𝑥+𝑦=(𝑥(𝑟)+𝑦(𝑟),𝑥(𝑟)+𝑦(𝑟));(3)𝑥−𝑦=(𝑥(𝑟)−𝑦(𝑟),𝑥(𝑟)−𝑦(𝑟));(4)𝑘𝑥=𝑘𝑥(𝑟),𝑘𝑥(𝑟)𝑘,𝑘≥0,𝑥(𝑟),𝑘𝑥(𝑟),𝑘<0. (4)

Definition 3 (see [23]). For arbitrary 𝑢=(𝑢(𝑟),𝑢(𝑟)),𝑣=(𝑣(𝑟),𝑣(𝑟))∈𝐸¹,0≤𝑟≤1, the quantity 𝐷(𝑢,𝑣)=¹₀𝑢(𝑟)−𝑣(𝑟)²𝑑𝑟+¹₀(𝑢(𝑟)−𝑣(𝑟))²𝑑𝑟^1/2 (5) is the distance between fuzzy numbers 𝑢 and 𝑣.

2.2. The Undetermined Fuzzy Coefficients Method

The undetermined fuzzy coefficients method is to seek an approximate solution as 𝑦_𝑁(𝑡)=^𝑁_𝑘=0𝛼_𝑘𝜙_𝑘(𝑡), (6) where 𝜙_𝑘(𝑡),𝑘=0,1,…,𝑁, are positive basic functions whose all differentiations are positive. We compute the fuzzy coefficients 𝛼_𝑘 in (6) by setting the error to zero as follows: 𝐸=𝐷𝑦^(𝑛)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)+⋯+𝑎₁(𝑡)𝑦^′+𝑎₀(𝑡)𝑔(𝑡)𝑦𝑡𝑦,+𝐷₀,𝑏̃₀+⋯+𝐷𝑦^(𝑛−1)𝑡₀,𝑏̃_𝑛−1. (7)

We substitute (6) in (7) and represent them in parametric forms; then, 𝑦^(𝑛)(𝑡,𝑟)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)(𝑡,𝑟)+⋯+𝑎₁(𝑡)𝑦^′(𝑡,𝑟)+𝑎₀(𝑡)𝑦(𝑡,𝑟)=𝑔(𝑡,𝑟),𝑦𝑡₀,𝑟=𝑏₀(𝑟),⋮𝑦^(𝑛−1)𝑡₀,𝑟=𝑏_𝑛−1(𝑟),𝑦^(𝑛)(𝑡,𝑟)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)(𝑡,𝑟)+⋯+𝑎₁(𝑡)𝑦^′(𝑡,𝑟)+𝑎₀(𝑡)𝑦()=𝑡,𝑟𝑔(𝑡,𝑟),𝑦𝑡₀,𝑟=𝑏₀(𝑟),⋮𝑦^(𝑛−1)𝑡₀,𝑟=𝑏_𝑛−1(𝑟). (8)

Lemma 4 (see [35]). Let basic functions 𝜙_𝑘(𝑡),𝑘=0,1,…,𝑁, and all of their differentiations be positive; without loss of generality, then (𝑦_𝑁)^(𝑖)(𝑡)=𝑦_𝑁^(𝑖)(𝑡) and (𝑦_𝑁)^(𝑖)(𝑡)=𝑦_𝑁^(𝑖)(𝑡),𝑖=0,1,…,𝑛.

3. Basic Results

In order to solve (1) and (2), we need to consider the system of linear equations (8). In this section, we study its general case at first then give some corollaries for some special cases.

3.1. New Models

Theorem 5. Suppose that each 𝑎_𝑖(𝑡),𝑖=0,1,…,𝑛−1, in (1) is either nonnegative or negative over [𝑡₀,𝑇]; then, the 𝑛th-order fuzzy linear differential equations 𝑦^(𝑛)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)+⋯+𝑎₁(𝑡)𝑦^′+𝑎₀(𝑡)(𝑡),𝑦𝑡𝑦=𝑔₀=𝑏̃₀,𝑦^′𝑡₀=𝑏̃₁,…,𝑦^(𝑛−1)𝑡₀=𝑏̃_𝑛−1 (9) can be extended into a system of linear equations 𝛽₀𝛽₁⋯𝛽_𝑁𝜉₀𝜉₁⋯𝜉_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁𝜉00⋯0₀𝜉₁⋯𝜉_𝑁𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (10) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+_{𝑎𝑖(𝑡)≥0}𝑎_𝑖(𝑡)𝜙_𝑘^(𝑖)(𝑡)𝜉,𝑖=0,1,…,𝑛−1,_𝑘=_{𝑎𝑖(𝑡)<0}𝑎_𝑖(𝑡)𝜙_𝑘^(𝑖)(𝑡)𝜎,𝑖=0,1,…,𝑛−1,_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (11)
By setting 𝑡=𝑎,𝑎∈[𝑡₀,𝑇], and by solving (10), the fuzzy approximate solution of the original fuzzy linear differential equations is obtained as follows:𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡),𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡). (12)

Proof. Let (5) be substituted in (1) and (2); we have ^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙^′_𝑘(𝑡)+𝑎₀(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘(𝑡)=𝑔(𝑡),_𝑁𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘𝑡₀=𝑏̃₀(𝑟),_𝑁𝑘=0𝛼_𝑘(𝑟)𝜙^′_𝑘𝑡₀=𝑏̃₁(𝑟),…,_𝑁𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛−1)𝑡₀=𝑏̃_𝑛−1(𝑟). (13)
We express the pervious equations in parametric forms; then, ^𝑁_𝑘=0𝛼_𝑘(𝑟),𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟),𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟),𝛼_𝑘(𝑟)𝜙^′_𝑘(𝑡)+𝑎₀(𝑡)^𝑁_𝑘=0𝛼_𝑘(𝑟),𝛼_𝑘(𝑟)𝜙_𝑘(𝑡)=𝑔(𝑡,𝑟),𝑔(𝑡,𝑟_𝑁𝑘=0),𝛼_𝑘(𝑟)𝜙_𝑘𝑡₀=𝑏₀(𝑟),…,^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛−1)𝑡₀=𝑏_𝑛−1(𝑟),_𝑁𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘𝑡₀=𝑏₀(𝑟),…,^𝑁_𝑘=0𝛼_𝑘(𝑟)𝜙_𝑘^(𝑛−1)𝑡₀=𝑏_𝑛−1(𝑟). (14)
By setting 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+_{𝑎𝑖(𝑡)≥0}𝑎_𝑖(𝑡)𝜙_𝑘^(𝑖)(𝑡)𝜉,𝑖=0,1,…,𝑛−1,_𝑘=_{𝑎𝑖(𝑡)<0}𝑎_𝑖(𝑡)𝜙_𝑘^(𝑖)(𝑡)𝜎,𝑖=0,1,…,𝑛−1,_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁, (15)
thus we have the corresponding systems 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟) as follows: 𝛽₀𝛽₁⋯𝛽_𝑁𝜉₀𝜉₁⋯𝜉_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁𝜉00⋯0₀𝜉₁⋯𝜉_𝑁𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (16) being 2(𝑛+1)×2(𝑁+1) linear systems.
By setting 𝑡=𝑎,𝑎∈[𝑡₀,𝑇], and by solving (10), we obtain the values of parameters 𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟); (17) therefore, we get the fuzzy approximate solution of the original fuzzy equation as follows: 𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡),𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡). (18)

Now, we consider another special case; that is, coefficient functions 𝑎_𝑘(𝑡),𝑘=0,1,…,𝑛−1, are positive and negative alternately.

Corollary 6. Suppose that the coefficients functions 𝑎_𝑛−1(𝑡),𝑎_𝑛−3(𝑡),… are positive and 𝑎_𝑛−2(𝑡),𝑎_𝑛−4(𝑡),… are negative, and 𝑛 is an odd number. Then, the 𝑛th-order fuzzy linear differential equations (1) and (2) can be extended into the following linear equations: ^𝑁_𝑘=0𝛼_𝑘(𝑟)𝛽_𝑘+^𝑁_𝑘=0𝛼_𝑘(𝑟)𝛾_𝑘=𝑔(𝑡,𝑟_𝑁𝑘=0),𝛼_𝑘(𝑟)𝜎_0𝑘=𝑏₀(𝑟),_𝑁𝑘=0⋮𝛼_𝑘(𝑟)𝜎_{𝑛−1,𝑘}=𝑏_𝑛−1(𝑟),_𝑁𝑘=0𝛼_𝑘(𝑟)𝛽_𝑘+^𝑁_𝑘=0𝛼_𝑘(𝑟)𝛾_𝑘=𝑔(𝑡,𝑟_𝑁𝑘=0),𝛼_𝑘(𝑟)𝜎_0𝑘=𝑏₀(𝑟),_𝑁𝑘=0⋮𝛼_𝑘(𝑟)𝜎_{𝑛−1,𝑘}=𝑏_𝑛−1(𝑟), (19) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝜙_𝑘^(𝑛−1)(𝑡)+𝑎_𝑛−3(𝑡)𝜙_𝑘^(𝑛−3)(𝑡)+⋯+𝑎₀(𝑡)𝜙_𝑘(𝑡),𝛾_𝑘=𝑎_𝑛−2(𝑡)𝜙_𝑘^(𝑛−2)(𝑡)+𝑎_𝑛−4(𝑡)𝜙_𝑘^(𝑛−4)(𝑡)+⋯+𝑎₁(𝑡)𝜙^′_𝑘(𝑡),𝜎_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (20)
By setting 𝑋(𝑟)=𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟)^⊤,𝑌(𝑟)=𝑔(𝑡,𝑟),𝑏₀(𝑟),…,𝑏_𝑛−1(𝑟),𝑔(𝑡,𝑟),𝑏₀(𝑟),…,𝑏_𝑛−1(𝑟)^⊤, (21)
(19) is a system of linear equations 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟) such that 𝑆(𝑡)=𝑆₁𝑆₂𝑆₂𝑆₁, (22) where 𝑆₁=𝛽₀𝛽₁⋯𝛽_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁,𝑆₂=𝛾₀𝛾₁⋯𝛾_𝑁.00⋯0⋮⋮⋯⋮00⋯0 (23)
When 𝑛 is an even number, then the 𝑛th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations 𝛽₀𝛽₁⋯𝛽_𝑁𝛾₀𝛾₁⋯𝛾_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁𝛾00⋯0₀𝛾₁⋯𝛾_𝑁𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (24) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝜙_𝑘^(𝑛−1)(𝑡)+𝑎_𝑛−3(𝑡)𝜙_𝑘^(𝑛−3)(𝑡)+⋯+𝑎₁(𝑡)𝜙^′_𝑘(𝑡),𝛾_𝑘=𝑎_𝑛−2(𝑡)𝜙_𝑘^(𝑛−2)(𝑡)+𝑎_𝑛−4(𝑡)𝜙_𝑘^(𝑛−4)(𝑡)+⋯+𝑎₀(𝑡)𝜙_𝑘(𝑡),𝜎_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (25)
Three special cases in Allahviranloo et al.’s paper [35] are viewed as corollaries from Theorem 5.

Corollary 7. Suppose that coefficients functions 𝑎_𝑘(𝑡),𝑘=0,1,2,…,𝑛−1, are nonnegative; then, the 𝑛th-order fuzzy linear differential equations (1) and (2) can be extended into the following linear equations: 𝛽₀𝛽₁⋯𝛽_𝑁𝜎00⋯0₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁00⋯000⋯0𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (26) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝜙_𝑘^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)𝜙^′_𝑘(𝑡)+𝑎₀(𝑡)𝜙_𝑘(𝑡),𝜎_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (27)

Corollary 8. Suppose that coefficients functions 𝑎_𝑘(𝑡),𝑘=0,1,2,…,𝑛−1, are negative; then, the 𝑛th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations 𝛽₀𝛽₁⋯𝛽_𝑁𝛾₀𝛾₁⋯𝛾_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁𝛾00⋯0₀𝛾₁⋯𝛾_𝑁𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (28) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡),𝛾_𝑘=𝑎_𝑛−1(𝑡)𝜙_𝑘^(𝑛−1)(𝑡)+𝑎_𝑛−2(𝑡)𝜙_𝑘^(𝑛−2)(𝑡)+⋯+𝑎₀(𝑡)𝜙_𝑘(𝑡),𝜎_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (29)

Corollary 9. Suppose that coefficients functions 𝑎_𝑛−1(𝑡),𝑎_𝑛−3(𝑡),…,𝑎_𝑛−𝑚(𝑡) are nonnegative and 𝑎_{𝑛−𝑚−1}(𝑡),𝑎_{𝑛−𝑚−2}(𝑡),…,𝑎₀(𝑡) are negative; then, the 𝑛th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations 𝛽₀𝛽₁⋯𝛽_𝑁𝛾₀𝛾₁⋯𝛾_𝑁𝜎₀₀𝜎₀₁⋯𝜎_0𝑁𝜎00⋯0⋮⋮⋯⋮⋮⋮⋯⋮_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁𝛾00⋯0₀𝛾₁⋯𝛾_𝑁𝛽₀𝛽₁⋯𝛽_𝑁00⋯0𝜎₀₀𝜎₀₁⋯𝜎_0𝑁⋮⋮⋯⋮⋮⋮⋯⋮00⋯0𝜎_𝑛−10𝜎_𝑛−11⋯𝜎_𝑛−1𝑁×𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)⋮𝛼_𝑁(𝑟)=𝑔()𝑏𝑡,𝑟₀(𝑟)⋮𝑏_𝑛−1(𝑟)𝑔(𝑡,𝑟)𝑏₀(𝑟)⋮𝑏_𝑛−1(𝑟), (30) where 𝛽_𝑘=𝜙_𝑘^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝜙_𝑘^(𝑛−1)(𝑡)+⋯+𝑎_𝑛−𝑚(𝑡)𝜙_𝑘^{(𝑛−𝑚)}(𝑡),𝛾_𝑘=𝑎_{𝑛−𝑚−1}(𝑡)𝜙_𝑘^{(𝑛−𝑚−1)}(𝑡)+𝑎_{𝑛−𝑚−2}(𝑡)𝜙_𝑘^{(𝑛−𝑚−2)}(𝑡)+⋯+𝑎₀(𝑡)𝜙_𝑘(𝑡),𝜎_𝑗𝑘=𝜙_𝑘^(𝑗)𝑡₀,𝑗=0,1,…,𝑛−1,𝑘=0,1,…,𝑁. (31)

3.2. Method to Solve Linear Equations

The previous crisp linear equations (10), (19), (24), (26), (28), and (30) are all 2(𝑛+1)×2(𝑁+1) linear systems, and they have the same form as follows: 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟). (32)

In the process of solving (32) by setting 𝑡=𝑎,𝑎∈[𝑡₀,𝑇], whether it is consistent or inconsistent, we obtain the minimal norm least squares solution [37] by using the generalized inverse of the coefficient matrix 𝑆(𝑎); that is, 𝑋(𝑟)=𝑆^†(𝑎)𝑌(𝑟). (33)

Thus, we get 𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),…,𝛼_𝑁(𝑟). (34)

Therefore, we obtain the fuzzy approximate solution of the original fuzzy equation as follows: 𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡),𝑦(𝑡,𝑟)=𝛼₀(𝑟)𝜙₀(𝑡)+𝛼₁(𝑟)𝜙₁(𝑡)+⋯+𝛼_𝑁(𝑟)𝜙_𝑁(𝑡). (35)

4. Numerical Examples

Example  1. Consider the following second-order fuzzy linear differential equation: 𝑦^′′−4𝑦^′𝑦(0)=(),𝑦+4𝑦=4𝑡−4,𝑡≥0,2+𝑟,4−𝑟_′(0)=(3+2𝑟,9−2𝑟). (36)

The exact solution of equation is 𝑌(𝑡,𝑟)=(2+𝑟)𝑒^2𝑡+(−1+𝑟)𝑡𝑒^2𝑡+𝑡,𝑌(𝑡,𝑟)=(4−𝑟)𝑒^2𝑡+(1−𝑟)𝑡𝑒^2𝑡+𝑡. (37)

Let 𝜙_𝑘(𝑡)=𝑡^𝑘,𝑘=0,1,2,3; then, 𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³,𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³. (38)

From (10), the extended linear equation 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟) is as follows: 44𝑡2+4𝑡²6𝑡+4𝑡³0−4−8𝑡−12𝑡²10000000010000000−4−8𝑡−12𝑡²44𝑡2+4𝑡²6𝑡+4𝑡³×𝛼0000100000000100₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)=.4𝑡−42+𝑟3+2𝑟4𝑡−44−𝑟9−2𝑟 (39)

By setting 𝑡=1/2, the parameters 𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟) are obtained, and by putting them into (38), we have 𝑦()=()+()𝑡+()𝑡𝑡,𝑟2+𝑟3+2𝑟3.28767123−1.22739726𝑟²+(0.76712328−1.13972602𝑟)𝑡³,𝑦()=()+()𝑡+()𝑡𝑡,𝑟4−𝑟9−2𝑟−0.08767123+1.22739726𝑟²+(−2.36712328+1.13972602𝑟)𝑡³. (40)

Tables 1 and 2 show comparisons between the exact solution and the approximate solution at 𝑡=0.001 for some 𝑟∈[0,1], and all data were calculated by MATLAB 6.x.

Table 1 

Comparisons between the exact solution and the approximate solution.

Table 2 

Comparisons between the exact solution and the approximate solution.

Example  2. Consider the following three-order fuzzy linear differential equation: 𝑡³𝑦^′′′−3𝑡²𝑦^′′+6𝑡𝑦^′𝑦(1)=(),𝑦−6𝑦=0,𝑡≥1,𝑟,2−𝑟_′(1)=(),𝑦−1+𝑟,1−𝑟_′′(1)=(2+2𝑟,6−2𝑟). (41)

The exact solution of equation is 𝑌(𝑡,𝑟)=(3+2𝑟)𝑡+(−5−2𝑟)𝑡²+(2+𝑟)𝑡³,𝑌(𝑡,𝑟)=(7−2𝑟)𝑡+(−9+2𝑟)𝑡²+(4−𝑟)𝑡³. (42)

Let 𝜙_𝑘(𝑡)=𝑡^𝑘,𝑘=0,1,2,3; then, 𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³,𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³. (43)

From (19), the extended linear equation 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟) is as follows: 06𝑡²12𝑡24−6𝑡³−6𝑡²−12𝑡−24111100000123000000260000−6𝑡³−6𝑡²−12𝑡6−240𝑡²12𝑡×𝛼24000011110000012300000026₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)=0𝑟0.−1+𝑟2+2𝑟2−𝑟1−𝑟6−2𝑟 (44)

By setting 𝑡=4/3, the parameters 𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟) are obtained, and by putting them into (43), we have 𝑦()=()+()𝑡+()𝑡𝑡,𝑟−2.1760+2.1760𝑟9.5280−4.5280𝑟−11.5280+4.5280𝑟²+(4.1760−1.1760𝑟)𝑡³,𝑦()=()+()𝑡+()𝑡𝑡,𝑟2.1760−2.1760𝑟0.4720+4.5280𝑟−2.4720−4.5280𝑟²+(1.8240+1.1760𝑟)𝑡³. (45)

Tables 3, 4, 5, and 6 show comparisons between the exact solution and the approximate solution at 𝑡=1.01 and 𝑡=1.001 for some 𝑟∈[0,1].

Table 3 

Comparisons between the exact solution and the approximate solution (𝑡=1.01).

Table 4 

Comparisons between the exact solution and the approximate solution (𝑡=1.01).

Table 5 

Comparisons between the exact solution and the approximate solution (𝑡=1.001).

Table 6 

Comparisons between the exact solution and the approximate solution (𝑡=1.001).

Example  3 (see [35]). Consider the following fuzzy linear differential equation: 𝑦^′′[],𝑦(0)=(),𝑦+𝑦=−𝑡,𝑡∈0,1−0.1+0.1𝑟,0.1−0.1𝑟_′(0)=(0.088+0.1𝑟,0.288−0.1𝑟). (46)

The exact solution of equation is 𝑌(𝑡,𝑟)=(−0.1+0.1𝑟)cos𝑡+(1.088+0.1𝑟)sin𝑡−𝑡,𝑌(𝑡,𝑟)=(0.1−0.1𝑟)cos𝑡+(1.288−0.1𝑟)sin𝑡−𝑡. (47)

Let 𝜙_𝑘(𝑡)=𝑡^𝑘,𝑘=0,1,2,3; then, 𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³,𝑦(𝑡,𝑟)=𝛼₀+𝛼₁𝑡+𝛼₂𝑡²+𝛼₃𝑡³. (48)

From (26), the extended linear equation 𝑆(𝑡)𝑋(𝑟)=𝑌(𝑟) is as follows: 1𝑡2+𝑡²6𝑡+𝑡³0000100000000100000000001𝑡2+𝑡²6𝑡+𝑡³×𝛼0000100000000100₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)𝛼₀(𝑟)𝛼₁(𝑟)𝛼₂(𝑟)𝛼₃(𝑟)=.−𝑡−0.1+0.1𝑟0.088+0.1𝑟−𝑡0.1−0.1𝑟0.288−0.1𝑟 (49)

By setting 𝑡=1/2, the parameters 𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟),𝛼₀(𝑟),𝛼₁(𝑟),𝛼₂(𝑟),𝛼₃(𝑟) are obtained, and by putting them into (48), we have 𝑦()=()+()𝑡+()𝑡𝑡,𝑟−0.100+0.100𝑟0.088+0.100𝑟−0.067371−0.02276𝑟²+(−0.093572−0.03161𝑟)𝑡³,𝑦()=()+()𝑡+()𝑡𝑡,𝑟0.100−0.100𝑟0.288−0.100𝑟−0.11289+0.022760𝑟²+(−0.15679+0.031612𝑟)𝑡³. (50)

Tables 7, 8, 9, and 10 show comparisons between the exact solution and the approximate solution at 𝑡=0.01 and 𝑡=0.001 for some 𝑟∈[0,1].

Table 7 

Comparisons between the exact solution and the approximate solution (𝑡=0.01).

Table 8 

Comparisons between the exact solution and the approximate solution (𝑡=0.01).

Table 9 

Comparisons between the exact solution and the approximate solution (𝑡=0.001).

Table 10 

Comparisons between the exact solution and the approximate solution (𝑡=0.001).

From Tables 7, 8, 9, and 10, we can see that our results are more accurate than those of Example 3.1 [35] in Allahviranloo et al.’s work.

5. Conclusion

In this paper, an approximate method similar to the undetermined fuzzy coefficients method, based on a positive basis for solving fuzzy differential equations, was further discussed. The more general case was considered, and the model system of linear equation was set up according to different case of coefficient functions. Fuzzy approximate solution was obtained by solving the model system. Our work enriched the theory of fuzzy linear differential equations.

Acknowledgments

The work is supported by the Natural Scientific Funds of PR China (71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-11-20).