Formulation and Solution of 𝑛th-Order Derivative Fuzzy Integrodifferential Equation Using New Iterative Method with a Reliable Algorithm

Abstract

The nth-order derivative fuzzy integro-differential equation in parametric form is converted to its crisp form, and then the new iterative method with a reliable algorithm is used to obtain an approximate solution for this crisp form. The analysis is accompanied by numerical examples which confirm efficiency and power of this method in solving fuzzy integro-differential equations.

1. Introduction

In 1975 Zadeh and then Dubois and Prade [1] introduced fuzzy numbers and fuzzy arithmetic. This concept propagated widely to various problems, for example, fuzzy linear systems [2, 3], fuzzy differential equations [4, 5], fuzzy integral equations [6–8], and fuzzy integro-differential equations [9–12]. Additional topics can be found in [13, 14]. Recently, several numerical methods were suggested to solve integro-differential equations, for example, Sine-Cosine wavelet used by Tavassoli Kajani et al. to obtain a solution of linear integro-differential equations [15] and variational iteration method used by Abbasbandy and Hashemi to formulate and solve fuzzy integro-differential equation [12], by Saberi-Nadjafi and Tamamgar for solving system of integro-differential equations [16], and by Shang and Han for solving 𝑛th-order integro-differential equations [17]. Some other worthwhile works can be found in [18, 19]. Recently, Daftardar-Gejji and Jafari [20, 21] proposed the new iterative method. This method has proven useful for solving a variety of linear and nonlinear equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer and fractional order, and system of equations as well. The new iterative method is simple to understand and easy to implement using computer packages and yields better results [21] than the existing Adomian decomposition method [22], homotopy perturbation method [23], or variational iteration method [24]. For more details, see [25–40].

In the present work we apply the new iterative method with a reliable algorithm to solve the 𝑛th-order derivative fuzzy integro-differential equation in its crisp form.

2. Preliminaries

In this section we set up the basic definitions of fuzzy numbers and fuzzy functions.

Definition 2.1 (see [2, 3, 12]). A fuzzy number in parametric form is an ordered pair of functions (𝑢(𝑟),𝑢(𝑟)), 0≤𝑟≤1 which satisfy the following requirements:(1)𝑢(𝑟) is a bounded left continuous nondecreasing function over [0,1],(2)𝑢(𝑟) is a bounded left continuous nonincreasing function over [0,1],(3)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1.

Remark 2.2 (see [6, 12]). Let 𝑢(𝑟)=(𝑢(𝑟),𝑢(𝑟)), 0≤𝑟≤1 be a fuzzy number, and we can take 𝑢^𝑐(𝑟)=12𝑢(𝑟)+𝑢(𝑟),𝑢^𝑑(𝑟)=12𝑢(𝑟)−𝑢(𝑟). (2.1) It is clear that 𝑢^𝑑(𝑟)≥0, 𝑢(𝑟)=𝑢^𝑐(𝑟)−𝑢^𝑑(𝑟) and 𝑢(𝑟)=𝑢^𝑐(𝑟)+𝑢^𝑑(𝑟), and also a fuzzy number 𝑢∈𝐸 is said symmetric if 𝑢^𝑐(𝑟) is independent of 𝑟 for all 0≤𝑟≤1.
Let 𝑓∶[𝑎,𝑏]→𝐸 for each partition 𝑃={𝑡₀,…,𝑡_𝑛} of [𝑎,𝑏] and for arbitrary 𝜉_𝑖∈𝑡_𝑖−1−𝑡_𝑖], 1≤𝑖≤𝑛suppose that 𝑅_𝑃=∑^𝑛_𝑖=1𝑓(𝜉_𝑖)(𝑡_𝑖−𝑡_𝑖−1) and Δ=max{|𝑡_𝑖−𝑡_𝑖−1|, 𝑖=1,…,𝑛}. The definite integral of 𝑓(𝑡) over [𝑎,𝑏] is ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑡=lim_Δ→0𝑅_𝑃 provided that this limit exists in the metric 𝐷. If the fuzzy function 𝑓(𝑡) is continuous in the metric 𝐷, its definite integral exists. Also we have (∫^𝑏_𝑎𝑓(𝑡,𝑟)𝑑𝑡)=∫^𝑏_𝑎𝑓(𝑡,𝑟)𝑑𝑡 and (∫^𝑏_𝑎𝑓(𝑡,𝑟)𝑑𝑡)=∫^𝑏_𝑎𝑓(𝑡,𝑟)𝑑𝑡.

Definition 2.3 (see [33]). Let 𝑓∶(𝑎,𝑏)→𝑅_𝐹 and 𝑡₀∈(𝑎,𝑏). We say that 𝑓^′ is Hukuhara differentiable at 𝑡₀ if there exists an element 𝑓^′(𝑡₀)∈𝑅_𝐹, such that for all ℎ>0 sufficiently small, ∃𝑓(𝑡₀+ℎ)⊝_𝐻𝑓(𝑡₀),𝑓(𝑡₀)⊝_𝐻𝑓(𝑡₀−ℎ) and lim_ℎ→0𝑓𝑡₀+ℎ⊝_𝐻𝑓𝑡₀ℎ=lim_ℎ→0𝑓𝑡₀⊝_𝐻𝑓𝑡₀−ℎℎ=𝑓^′𝑡₀, (2.2) where ⊝_𝐻 is Hukuhara difference, 𝑅_𝐹 is the set of fuzzy numbers, and the limit is in the metric 𝐷.
In parametric form, if 𝑓=(𝑓,𝑓), then 𝑓^′=(𝑓^′,𝑓^′), where 𝑓^′,𝑓^′ are Hukuhara differentiable of 𝑓,𝑓, respectively.
From (2.2), the 𝑛th-order Hukuhara differentiable, 𝑓^(𝑛), of 𝑓 at 𝑡₀ can be defined as in the following definition.

Definition 2.4. Let 𝑓^(𝑛−1)∶(𝑎,𝑏)→𝑅_𝐹 and 𝑡₀∈(𝑎,𝑏) where 𝑓^(𝑛−1) is (𝑛−1)th-orderHukuhara differentiable of 𝑓 at 𝑡₀ for all 𝑛>1. We say that 𝑓^(𝑛) is 𝑛th-order Hukuhara differentiable at 𝑡₀ if there exists an element 𝑓^(𝑛)(𝑡₀)∈𝑅_𝐹 such that for all ℎ>0 sufficiently small, ∃𝑓^(𝑛−1)(𝑡₀+ℎ)⊝_𝐻𝑓^(𝑛−1)(𝑡₀),𝑓^(𝑛−1) (𝑡₀) ⊝_𝐻𝑓^(𝑛−1)(𝑡₀−ℎ) andlim_ℎ→0𝑓^(𝑛−1)𝑡₀+ℎ⊝_𝐻𝑓^(𝑛−1)𝑡₀ℎ=lim_ℎ→0𝑓^(𝑛−1)𝑡₀⊝_𝐻𝑓^(𝑛−1)𝑡₀−ℎℎ=𝑓^(𝑛)𝑡₀, (2.3a)
𝑛=1,2,…. In parametric form, as above, if =(𝑓,𝑓), then 𝑓^(𝑛)=(𝑓^(𝑛),𝑓^(𝑛)), where 𝑓^(𝑛),𝑓^(𝑛) are 𝑛th-order Hukuhara differentiable of 𝑓,𝑓, respectively. From (2.3a), in case 𝑛=1, we obtain (2.2).

3. Fuzzy Integro-Differential Equation

Consider the 𝑛th-order derivative integro-differential equation [12] as follows:𝑢^(𝑛)(𝑡)+𝑢(𝑡)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠)𝑑𝑠=𝑔(𝑡), (3.1a) where 𝑚, 𝑛∈ℕ, 𝑚<𝑛, 𝑡∈[0,𝑇], 𝜆∈ℝ, 𝑔(𝑡) is a known function and the kernel 𝑘(𝑠,𝑡)≥0 with the initial conditions as follows: 𝑢^(𝑘)(0)=ℎ_𝑘,𝑘=0,1,…,𝑛−1,ℎ_𝑘∈ℝ, (3.1b)

in the fuzzy case; that is, 𝑢 and 𝑔 be fuzzy functions. Let 𝑢(𝑡,𝑟)=𝑢(𝑡,𝑟),𝑢(𝑡,𝑟),𝑔(𝑡,𝑟)=𝑔(𝑡,𝑟),𝑔(),𝑢𝑡,𝑟_(𝑛)(𝑡,𝑟)=𝑢^(𝑛)(𝑡,𝑟),𝑢^(𝑛)(𝑡,𝑟),𝑔^(𝑛)(𝑡,𝑟)=𝑔^(𝑛)(𝑡,𝑟),𝑔^(𝑛)(𝑡,𝑟), (3.2) where all derivatives are, with respect to 𝑡, fuzzy functions. Therefore, related fuzzy integro-differential equation of (3.1a) can be written as follows:𝑢^(𝑛)(𝑡,𝑟)+𝑢(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),𝑢^(𝑛)(𝑡,𝑟)+𝑢(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟), (3.3a)(3.3b)

and its two crisp equations can be written as follows:𝑢^𝑐(𝑛)(𝑡,𝑟)+𝑢^𝑐(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑐(𝑚)(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑑(𝑛)(𝑡,𝑟)+𝑢^𝑑(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑑(𝑚)(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟). (3.4a)(3.4b)

4. New Iterative Method

For simplicity, we present a review of the new iterative method [20, 21, 34–39, 41], and then we introduce a suitable algorithm of this method for solving the 𝑛th-order derivative fuzzy integro-differential equations.

Consider the following general functional equation: 𝑢=𝑓+𝑁(𝑢), (4.1) where 𝑁 is a nonlinear operator from a Banach space 𝐵→𝐵, and 𝑓 is a known function.

We are looking for a solution 𝑢 of (4.1) having the series form: 𝑢=^∞_𝑖=0𝑢_𝑖. (4.2) The nonlinear operator 𝑁 can be decomposed as follows: 𝑁^∞_𝑖=0𝑢_𝑖=𝑁𝑢₀+^∞_𝑖=0𝑁^𝑖_𝑗=0𝑢_𝑗−𝑁^𝑖−1_𝑗=0𝑢_𝑗. (4.3) From (4.2) and (4.3), (4.1) is equivalent to ^∞_𝑖=0𝑢_𝑖=𝑓+𝑁𝑢₀+^∞_𝑖=1𝑁^𝑖_𝑗=0𝑢_𝑗−𝑁^𝑖−1_𝑗=0𝑢_𝑗. (4.4) We define the recurrence relation: 𝑢₀𝑢=𝑓,₁=𝑁𝑢₀,𝑢_𝑛+1=𝑁𝑢₀+𝑢₁+⋯+𝑢_𝑛−𝑁𝑢₀+𝑢₁+⋯+𝑢_𝑛−1,𝑛=1,2,…. (4.5) Then 𝑢₁+⋯+𝑢_𝑛+1=𝑁𝑢₀+⋯+𝑢_𝑛,𝑛=1,2,…,_∞𝑖=0𝑢_𝑖=𝑓+^∞_𝑖=1𝑢_𝑖. (4.6) The 𝑛-term approximate solution of (4.1) is given by 𝑢=𝑢₀+𝑢₁+⋯+𝑢_𝑛−1.

4.1. Reliable Algorithm

After the above presentation of the new iterative method, we introduce a reliable algorithm of this method for solving any 𝑛th-order derivative fuzzy integro-differential equation. Consider the 𝑛th-order derivative integro-differential equation (3.1a) and (3.1b) defined as𝑢^(𝑛)(𝑡)+𝑢(𝑡)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠)𝑑𝑠=𝑔(𝑡), 4.7a with the following initial conditions: 𝑢^(𝑘)(0)=ℎ_𝑘,𝑘=0,1,…,𝑛−1,ℎ_𝑘∈ℝ. 4.7b

The initial value problem (4.7a) and (4.7b) is equivalent to the following integral equation: 𝑢(𝑡)=𝑢₀(𝑡)−𝐼^𝑛_𝑡𝑢(𝑡)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠)𝑑𝑠=𝑢₀(𝑡)−𝑁(𝑢), (4.8) where 𝑢₀(𝑡) is the solution of the 𝑛th-order differential equation:𝑑^𝑛𝑢₀𝑑𝑡^𝑛=𝑔(𝑡),𝑢₀^(𝑘)(0)=ℎ_𝑘,𝑘=0,1,2,…,𝑛−1,ℎ_𝑘∈ℝ,𝑁(𝑢)=𝐼^𝑛_𝑡𝑢(𝑡)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^(𝑚)(𝑠)𝑑𝑠, (4.9a)(4.9b)

and 𝐼^𝑛_𝑡 is an integral operator of order 𝑛. Also, the two crisp equations (3.4a) and (3.4b) are equivalent to the two integral equations:𝑢^𝑐(𝑡,𝑟)=𝑢^𝑐₀(𝑡,𝑟)−𝐼^𝑛_𝑡𝑢^𝑐(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑐(𝑚)(𝑠,𝑟)𝑑𝑠=𝑢^𝑐₀(𝑡,𝑟)−𝑁𝑢^𝑐,𝑢^𝑑(𝑡,𝑟)=𝑢^𝑑₀(𝑡,𝑟)−𝐼^𝑛_𝑡𝑢^𝑑(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑑(𝑚)(𝑠,𝑟)𝑑𝑠=𝑢^𝑑₀(𝑡,𝑟)−𝑁𝑢^𝑑, (4.10a)(4.10b) where 𝑢^𝑐₀(𝑡,𝑟) and 𝑢^𝑑₀(𝑡,𝑟) are the solutions of the 𝑛th-order differential equations: 𝑑^𝑛𝑢^𝑐₀𝑑𝑡^𝑛=𝑔^𝑐(𝑡),𝑢₀^𝑐(𝑘)(0)=ℎ^𝑐_𝑘,𝑘=0,1,2,…,𝑛−1,ℎ^𝑐_𝑘∈ℝ,𝑑^𝑛𝑢^𝑑₀𝑑𝑡^𝑛=𝑔^𝑑(𝑡),𝑢₀^𝑑(𝑘)(0)=ℎ^𝑑_𝑘,𝑘=0,1,2,…,𝑛−1,ℎ^𝑑_𝑘∈ℝ. (4.10c)(4.10d)𝑁(𝑢^𝑐) and 𝑁(𝑢^𝑑) are the following two integral equations: 𝑁𝑢^𝑐=𝐼^𝑛_𝑡𝑢^𝑐(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑐(𝑚)(𝑠,𝑟)𝑑𝑠,𝑁𝑢^𝑐=𝐼^𝑛_𝑡𝑢^𝑑(𝑡,𝑟)+𝜆^𝑇₀𝑘(𝑠,𝑡)𝑢^𝑐(𝑚)(𝑠,𝑟)𝑑𝑠. (4.10e)(4.10f)

We get the solution of (4.8) or the two (4.10a), (4.10b) by employing the recurrence relation (4.5).

4.2. Convergence of the New Iterative Method

Now, we introduce the condition of convergence of the new iterative method, which is proposed by Daftardar-Gejji and Jafari in (2006) [20], also called (DJM) [39]. From (4.3), the nonlinear operator 𝑁 is decomposed as follows [39]: 𝑁(𝑢)=𝑁𝑢₀+𝑁𝑢₀+𝑢₁−𝑁𝑢₀+𝑁𝑢₀+𝑢₁+𝑢₂−𝑁𝑢₀+𝑢₁+⋯. (4.11) Let 𝐺₀=𝑁(𝑢₀) and 𝐺_𝑛=𝑁^𝑛_𝑖=0𝑢_𝑖−𝑁^𝑛−1_𝑖=0𝑢_𝑖,𝑛=1,2,…. (4.12) Then 𝑁(𝑢)=∑^∞_𝑖=0𝐺_𝑖.

Set 𝑢₀=𝑓,𝑢_𝑛=𝐺_𝑛−1,𝑛=1,2,…. (4.13)(4.14) Then 𝑢=^∞_𝑖=0𝑢_𝑖 (4.15) is a solution of the general functional equation (4.1). Also, the recurrence relation (4.5) becomes 𝑢₀𝑢=𝑓,_𝑛=𝐺_𝑛−1,𝑛=1,2,…. (4.16) Using Taylor series expansion for 𝐺_𝑖, 𝑖=1,2,…,𝑛, we have 𝐺₁=𝑁𝑢₀+𝑢₁−𝑁𝑢₀=𝑁𝑢₀+𝑁^′𝑢₀𝑢₁+𝑁^′′𝑢₀𝑢²₁𝑢2!+⋯−𝑁₀=^∞_𝑘=1𝑁^(𝑘)𝑢₀𝑢^𝑘₁,𝐺𝑘!₂=𝑁𝑢₀+𝑢₁+𝑢₂−𝑁𝑢₀+𝑢₁=𝑁^′𝑢₀+𝑢₁𝑢₂+𝑁^′′𝑢₀+𝑢₁𝑢²₂2!+⋯=^∞_𝑗=1^∞_𝑖=0𝑁^(𝑖+𝑗)𝑢₀𝑢^𝑖₁𝑢𝑖!^𝑗2,𝐺𝑗!₃=^∞_𝑖3^∞_=1𝑖2^∞_=0𝑖1=0𝑁^{(𝑖1+𝑖2+𝑖3)}𝑢₀𝑢^𝑖3₃𝑖₃!𝑢^𝑖2₂𝑖₂!𝑢^𝑖1₁𝑖₁!. (4.17) In general: 𝐺_𝑛=^∞_𝑖𝑛^∞_{=1𝑖𝑛−1=0}⋯^∞_𝑖1=0𝑁^{(∑𝑛𝑘=1𝑖𝑘)}𝑢₀^𝑛_𝑗=1𝑢^𝑖𝑗_𝑗𝑖_𝑗!. (4.18) In the following theorem we state and prove the condition of convergence of the method.

Theorem 4.1. If N is 𝐶^(∞) in a neighborhood of 𝑢₀ and𝑁^(𝑛)𝑢₀=sup𝑁^(𝑛)𝑢₀ℎ₁,…,ℎ_𝑛∶ℎ_𝑖≤1,1≤𝑖≤𝑛≤𝐿, (4.19a) for any 𝑛 and for some real 𝐿>0 and ‖𝑢_𝑖‖≤𝑀<1/𝑒, 𝑖=1,2,…, then the series ∑^∞_𝑛=0𝐺_𝑛 is absolutely convergent, and, moreover, 𝐺_𝑛≤𝐿𝑀^𝑛𝑒^𝑛−1(𝑒−1),𝑛=1,2,…. (4.19b)

Proof. In view of (4.18), 𝐺_𝑛≤𝐿𝑀^𝑛∞_𝑖𝑛^∞_{=1𝑖𝑛−1=0}⋯^∞_𝑖1=0^𝑛_𝑗=1𝑢^𝑖𝑗_𝑗𝑖_𝑗!=𝐿𝑀^𝑛𝑒^𝑛−1(𝑒−1). (4.19c) Thus, the series ∑^∞_𝑛=1‖𝐺_𝑛‖ is dominated by the convergent series 𝐿𝑀(𝑒−1) ∑^∞_𝑛=1(𝑀𝑒)^𝑛−1, where 𝑀<1/𝑒. Hence, ∑^∞_𝑛=0𝐺_𝑛 is absolutely convergent, due to the comparison test.
For more details, see [39].

5. Numerical Examples

Example 5.1. Consider the following fuzzy integro-differential equation: 𝑢^′(𝑡,𝑟)+𝜆¹₀𝑘(𝑠,𝑡)𝑢(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),0≤𝑡,𝑟≤1, (5.1) where 𝑢(𝑡,𝑟)=(𝑢(𝑡,𝑟),𝑢(𝑡,𝑟))=(−2𝑟𝑡,4+𝑟𝑡) is the exact solution for (5.1), 𝑔(𝑡,𝑟)=(𝑔(𝑡,𝑟),𝑔(𝑡,𝑟))=(−2𝑟−2𝑟𝑡/3, 𝑟+2𝑡+𝑟𝑡/3, 𝑘(𝑠,𝑡)=𝑠𝑡, and 𝜆=1, with the initial conditions 𝑢(0,𝑟)=0, 𝑢(0,𝑟)=4. Taking 𝑔^𝑐(𝑡,𝑟)=(1/2)(𝑔(𝑡,𝑟)+𝑔(𝑡,𝑟))=−𝑟/2+𝑡−𝑟𝑡/6, 𝑔^𝑑(𝑡,𝑟)=(1/2)(𝑔(𝑡,𝑟)−𝑔(𝑡,𝑟))=3𝑟/2+𝑡+𝑟𝑡/2. The exact solution for related crisp equations is 𝑢^𝑐(𝑡,𝑟)=2−𝑟𝑡/2, 𝑢^𝑑(𝑡,𝑟)=2+3𝑟𝑡/2. At first, we identify 𝑢^𝑐(𝑡,𝑟) and 𝑢^𝑑(𝑡,𝑟). From (3.4a) and (3.4b), (5.1), where 𝑛=1, we have𝑢^𝑐′(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑐(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑐(0,𝑟)=2,𝑢^𝑑′(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟),𝑢^𝑑(0,𝑟)=2. (5.2a)(5.2b)
From (4.10c), (4.10d), we obtain 𝑢^𝑐₀(𝑡,𝑟)=2−𝑟𝑡2+𝑡²2−𝑟𝑡²12,𝑢^𝑑₀(𝑡,𝑟)=2+3𝑟𝑡2+𝑡²2+𝑟𝑡²4. (5.3) Therefore, from (4.10a), (4.10b) the fuzzy integro-differential equations (5.2a) and (5.2b) are equivalent to the following integral equations: 𝑢^𝑐(𝑡,𝑟)=2−𝑟𝑡2+𝑡²2−𝑟𝑡²12−𝐼_𝑡¹₀𝑠𝑡𝑢^𝑐(),𝑢𝑠,𝑟𝑑𝑠_𝑑(𝑡,𝑟)=2+3𝑟𝑡2+𝑡²2+𝑟𝑡²4−𝐼_𝑡¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠. (5.4) Let 𝑁(𝑢^𝑐)=−𝐼_𝑡[∫¹₀𝑠𝑡𝑢^𝑐(𝑠,𝑟)𝑑𝑠],𝑁(𝑢^𝑑)=−𝐼_𝑡[∫¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.1): 𝑢^𝑐₀(𝑡,𝑟)=2−𝑟𝑡2+𝑡²2−𝑟𝑡²12,𝑢^𝑑₀(𝑡,𝑟)=2+3𝑟𝑡2+𝑡²2+𝑟𝑡²4,𝑢^𝑐₁(𝑡,𝑟)=𝑁𝑢^𝑐₀=−9𝑡²+169𝑟𝑡²96,𝑢^𝑑₁(𝑡,𝑟)=𝑁𝑢^𝑑₀=−9𝑡²−169𝑟𝑡²,𝑢32_𝑐2(𝑡,𝑟)=9𝑡²−1289𝑟𝑡²768,𝑢^𝑑₂(𝑡,𝑟)=9𝑡²+1289𝑟𝑡²,⋮𝑢256_𝑐5(𝑡,𝑟)=−9𝑡²+655369𝑟𝑡²393216,𝑢^𝑑₅(𝑡,𝑟)=−9𝑡²−655369𝑟𝑡²,131072 (5.5) and so on. The 6-term approximate solution is 𝑢^𝑐(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑐_𝑛=2−𝑟𝑡2−𝑡²+65536𝑟𝑡²,𝑢393216_𝑑(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑑_𝑛=2+3𝑟𝑡2−𝑡²−65536𝑟𝑡².131072 (5.6) It is clear that the iterations converge to the exact solution of the two crisp equations 𝑢^𝑐,𝑢^𝑑 as the number of iteration converges to ∞, that  is, lim_𝑛→∞𝑢^𝑐_𝑛(𝑡,𝑟)=2−𝑟𝑡/2=𝑢^𝑐_Exact, and lim_𝑛→∞𝑢^𝑑_𝑛(𝑡,𝑟)=2+3𝑟𝑡/2=𝑢^𝑑_Exact. From the relations between 𝑢,𝑢,𝑢^𝑐,𝑢^𝑑, and 𝑢 in Section 2, it follows immediately that this solution is the solution of the fuzzy integro-differential equation (5.1).

Example 5.2. Consider the following fuzzy integro-differential equation: 𝑢^′′(𝑡,𝑟)+𝜆¹₀𝑘(𝑠,𝑡)𝑢(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),0≤𝑡,𝑟≤1. (5.7) In this example 𝑢(𝑡,𝑟)=(𝑢(𝑡,𝑟),𝑢(𝑡,𝑟))=(2𝑟𝑡,4𝑟𝑡) is the exact solution for (5.7), 𝑔(𝑡,𝑟)=(𝑔(𝑡,𝑟),𝑔(𝑡,𝑟))=2𝑟𝑡/3,4𝑟𝑡/3, 𝑘(𝑠,𝑡)=𝑠𝑡, 𝜆=1, and the initial conditions are 𝑢(0,𝑟)=0, 𝑢^′(0,𝑟)=2𝑟, 𝑢(0,𝑟)=0, 𝑢^′(0,𝑟)=4𝑟. Also, the exact solution for related crisp equations is 𝑢^𝑐(𝑡,𝑟)=3𝑟𝑡, 𝑢^𝑑(𝑡,𝑟)=𝑟𝑡 with 𝑔^𝑐(𝑡,𝑟)=𝑟𝑡, 𝑔^𝑑(𝑡,𝑟)=𝑟𝑡/3. At first, we identify 𝑢^𝑐(𝑡,𝑟), 𝑢^𝑑(𝑡,𝑟). From (3.4a) and (3.4b), (5.7), where 𝑛=2, we have𝑢^𝑐′′(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑐(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑐(0,𝑟)=0,𝑢^𝑐′(0,𝑟)=3𝑟,𝑢^𝑑′′(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟),𝑢^𝑑(0,𝑟)=0,𝑢^𝑑′(0,𝑟)=𝑟. (5.8a)(5.8b)
From (4.10c), (4.10d), we obtain 𝑢^𝑐₀(𝑡,𝑟)=3𝑟𝑡+𝑟𝑡³6,𝑢^𝑑₀(𝑡,𝑟)=𝑟𝑡+𝑟𝑡³.18 (5.9) Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.8a) and (5.8b) are equivalent to the following integral equations: 𝑢^𝑐(𝑡,𝑟)=3𝑟𝑡+𝑟𝑡³6−𝐼²_𝑡¹₀𝑠𝑡𝑢^𝑐(),𝑢𝑠,𝑟𝑑𝑠_𝑑(𝑡,𝑟)=𝑟𝑡+𝑟𝑡³18−𝐼²_𝑡¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠. (5.10) Let 𝑁(𝑢^𝑐)=−𝐼²_𝑡[∫¹₀𝑠𝑡𝑢^𝑐(𝑠,𝑟)𝑑𝑠], 𝑁(𝑢^𝑑)=−𝐼²_𝑡[∫¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.7): 𝑢^𝑐₀(𝑡,𝑟)=3𝑟𝑡+𝑟𝑡³6,𝑢^𝑑₀(𝑡,𝑟)=𝑟𝑡+𝑟𝑡³,𝑢18_𝑐1(𝑡,𝑟)=𝑁𝑢^𝑐₀=−31𝑟𝑡³180,𝑢^𝑑₁(𝑡,𝑟)=𝑁𝑢^𝑑₀=−31𝑟𝑡³,𝑢540_𝑐2(𝑡,𝑟)=31𝑟𝑡³5400,𝑢^𝑑₂(𝑡,𝑟)=31𝑟𝑡²,⋮𝑢16200_𝑐5(𝑡,𝑟)=−31𝑟𝑡³145800000,𝑢^𝑑₅(𝑡,𝑟)=−29𝑟𝑡²,437400000 (5.11) and so on. The 6-term approximate solution is 𝑢^𝑐(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑐_𝑛=3𝑟𝑡−𝑟𝑡³145800000,𝑢^𝑑(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑑_𝑛=𝑟𝑡−𝑟𝑡³.437400000 (5.12) It is clear that the iterations converge to the exact solution of 𝑢^𝑐,𝑢^𝑑 as 𝑛→∞, that is, lim_𝑛→∞𝑢^𝑐_𝑛(𝑡,𝑟)=3𝑟𝑡=𝑢^𝑐_Exact, lim_𝑛→∞𝑢^𝑑_𝑛(𝑡,𝑟)=𝑟𝑡=𝑢^𝑑_Exact. As the above example the solution of (5.7) follows immediately.

Example 5.3. The third example is the fuzzy integro-differential equation: 𝑢^′′(𝑡,𝑟)+𝜆¹₀𝑘(𝑠,𝑡)𝑢^′(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),0≤𝑡,𝑟≤1, (5.13) where 𝑢(𝑡,𝑟)=(𝑢(𝑡,𝑟),𝑢(𝑡,𝑟))=(𝑟𝑡,4−3𝑟𝑡) is the exact solution for (5.13), 𝑔(𝑡,𝑟)=(𝑔(𝑡,𝑟),𝑔(𝑡,𝑟))=(−𝑟𝑡/2,3𝑟𝑡/2), 𝑘(𝑠,𝑡)=𝑠𝑡, 𝜆=−1, and the initial conditions are 𝑢(0,𝑟)=0, 𝑢^′(0,𝑟)=𝑟, 𝑢(0,𝑟)=4, 𝑢^′(0,𝑟)=−3𝑟. The exact solution for related crisp equations is 𝑢^𝑐(𝑡,𝑟)=2−𝑟𝑡, 𝑢^𝑑(𝑡,𝑟)=2−2𝑟𝑡 with 𝑔^𝑐(𝑡,𝑟)=𝑟𝑡/2, 𝑔^𝑑(𝑡,𝑟)=𝑟𝑡. At first, we identify 𝑢^𝑐(𝑡,𝑟), 𝑢^𝑑(𝑡,𝑟), and from (3.4a) and (3.4b), (5.13), where 𝑛=2, we have𝑢^𝑐′′(𝑡,𝑟)−¹₀𝑠𝑡𝑢^𝑐′(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑐(0,𝑟)=2,𝑢^𝑐′(0,𝑟)=−𝑟,𝑢^𝑑′′(𝑡,𝑟)−¹₀𝑠𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟),𝑢^𝑑(0,𝑟)=2,𝑢^𝑑′(0,𝑟)=−2𝑟. (5.14a)(5.14b)
From (4.10c), (4.10d), we obtain 𝑢^𝑐₀(𝑡,𝑟)=2−𝑟𝑡+𝑟𝑡³12,𝑢^𝑑₀(𝑡,𝑟)=2−2𝑟𝑡+𝑟𝑡³6. (5.15) Therefore, from (4.10a) and (4.10b), the fuzzy integro-differential equations (5.14a) and (5.14b) are equivalent to the integral equations: 𝑢^𝑐(𝑡,𝑟)=2−𝑟𝑡+𝑟𝑡³12+𝐼²_𝑡¹₀𝑠𝑡𝑢^𝑐′(),𝑢𝑠,𝑟𝑑𝑠_𝑑(𝑡,𝑟)=2−2𝑟𝑡+𝑟𝑡³6+𝐼²_𝑡¹₀𝑠𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠. (5.16) Let 𝑁(𝑢^𝑐)=𝐼²_𝑡[∫¹₀𝑠𝑡𝑢^𝑐′(𝑠,𝑟)𝑑𝑠], 𝑁(𝑢^𝑑)=𝐼²_𝑡[∫¹₀𝑠𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.13): 𝑢^𝑐₀(𝑡,𝑟)=2−𝑟𝑡+𝑟𝑡³12,𝑢^𝑑₀(𝑡,𝑟)=2−2𝑟𝑡+𝑟𝑡³6,𝑢^𝑐₁(𝑡,𝑟)=𝑁𝑢^𝑐₀=−7𝑟𝑡³96,𝑢^𝑑₁(𝑡,𝑟)=𝑁𝑢^𝑑₀=−7𝑟𝑡³,𝑢48_𝑐2(𝑡,𝑟)=−7𝑟𝑡³768,𝑢^𝑑₂(𝑡,𝑟)=−7𝑟𝑡³,⋮𝑢384_𝑐5(𝑡,𝑟)=−7𝑟𝑡³393216,𝑢^𝑑₅(𝑡,𝑟)=−7𝑟𝑡³,196608 (5.17) and so on. The 6-term approximate solution is 𝑢^𝑐(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑐_𝑛=2−𝑟𝑡+𝑟𝑡³393216,𝑢^𝑑(𝑡,𝑟)=⁵_𝑛=0𝑢^𝑑_𝑛=2−2𝑟𝑡+𝑟𝑡³.196608 (5.18) It is clear that the iterations converge to the exact solution of 𝑢^𝑐,𝑢^𝑑 as 𝑛→∞, that is, lim_𝑛→∞𝑢^𝑐_𝑛(𝑡,𝑟)=2−𝑟𝑡=𝑢^𝑐_Exact, lim_𝑛→∞𝑢^𝑑_𝑛(𝑡,𝑟)=2−2𝑟𝑡=𝑢^𝑑_Exact. Therefore the solution of (5.13) follows immediately.

Example 5.4. Let us consider the following fuzzy integro-differential equation: 𝑢^′′(𝑡,𝑟)+𝑢(𝑡,𝑟)+𝜆¹₀𝑘(𝑠,𝑡)𝑢(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),0≤𝑡,𝑟≤1, (5.19) with the exact solution 𝑢(𝑡,𝑟)=(𝑢(𝑡,𝑟),𝑢(𝑡,𝑟))=(3𝑟𝑡,8−𝑟𝑡), 𝑔(𝑡,𝑟)=(𝑔,𝑔)=(2𝑟𝑡,8−4𝑡−(2𝑟𝑡/3)), 𝑘(𝑠,𝑡)=𝑠𝑡, 𝜆=−1, and the initial conditions 𝑢(0,𝑟)=0, 𝑢^′(0,𝑟)=3𝑟, 𝑢(0,𝑟)=8, 𝑢^′(0,𝑟)=−𝑟. The exact solution for related crisp equations is 𝑢^𝑐(𝑡,𝑟)=4+𝑟𝑡, 𝑢^𝑑(𝑡,𝑟)=4−2𝑟𝑡 with 𝑔^𝑐(𝑡,𝑟)=4−2𝑡+2𝑟𝑡/3, 𝑔^𝑑(𝑡,𝑟)=4−2𝑡−4𝑟𝑡/3. From (3.4a), (3.4b), and (5.19), where 𝑛=2, the two crisp equations are𝑢^𝑐′′(𝑡,𝑟)+𝑢^𝑐(𝑡,𝑟)−¹₀𝑠𝑡𝑢^𝑐(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑐(0,𝑟)=4,𝑢^𝑐′(0,𝑟)=𝑟,𝑢^𝑑′′(𝑡,𝑟)+𝑢^𝑑(𝑡,𝑟)−¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟),𝑢^𝑑(0,𝑟)=4,𝑢^𝑑′(0,𝑟)=−2𝑟. (5.20a)(5.20b)
From (4.10c), (4.10d) we obtain 𝑢^𝑐₀(𝑡,𝑟)=4+𝑟𝑡+2𝑡²−𝑡³3+𝑟𝑡³9,𝑢^𝑑₀(𝑡,𝑟)=4−2𝑟𝑡+2𝑡²−𝑡³3−2𝑟𝑡³9. (5.21) Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.20a) and (5.20b) are equivalent to the following integral equations: 𝑢^𝑐(𝑡,𝑟)=4+𝑟𝑡+2𝑡²−𝑡³3+𝑟𝑡³9+𝐼²_𝑡−𝑢^𝑐(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑐(),𝑢𝑠,𝑟𝑑𝑠_𝑑(𝑡,𝑟)=4−2𝑟𝑡+2𝑡²−𝑡³3−2𝑟𝑡³9+𝐼²_𝑡−𝑢^𝑑(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠. (5.22) Let 𝑁𝑢^𝑐=𝐼²_𝑡−𝑢^𝑐(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑐(),𝑁𝑢𝑠,𝑟𝑑𝑠_𝑑=𝐼²_𝑡−𝑢^𝑑(𝑡,𝑟)+¹₀𝑠𝑡𝑢^𝑑(𝑠,𝑟)𝑑𝑠. (5.23) Therefore, from (4.5), we can obtain the following first approximate solutions for (5.19): 𝑢^𝑐₀(𝑡,𝑟)=4+𝑟𝑡+2𝑡²−𝑡³3+𝑟𝑡³9,𝑢^𝑑₀(𝑡,𝑟)=4−2𝑟𝑡+2𝑡²−𝑡³3−2𝑟𝑡³9,𝑢^𝑐₁(𝑡,𝑟)=−2𝑡²+73𝑡³−18029𝑟𝑡³−𝑡2706⁴+𝑡⁵−60𝑟𝑡⁵,𝑢180_𝑑1(𝑡,𝑟)=−2𝑡²+73𝑡³+18029𝑟𝑡³−𝑡1356⁴+𝑡⁵+60𝑟𝑡⁵,𝑢90_𝑐2(𝑡,𝑟)=−311𝑡³−4200421𝑟𝑡³+𝑡1134006⁴−73𝑡⁵+360029𝑟𝑡⁵+𝑡5400⁶−𝑡180⁷+2520𝑟𝑡⁷,𝑢7560_𝑑2(𝑡,𝑟)=−311𝑡³+4200421𝑟𝑡³+𝑡567006⁴−73𝑡⁵−360029𝑟𝑡⁵+𝑡2700⁶−𝑡180⁷−2520𝑟𝑡⁷,⋮3780 (5.24)
and so on. In the same manner the rest of components can be obtained. The 5-term approximate solution is 𝑢^𝑐(𝑡,𝑟)=⁴_𝑛=0𝑢^𝑐_𝑛=4+𝑟𝑡+2.8739𝑡³−31434480.1019𝑟𝑡³+94303440.0523𝑡⁵+272160.0023𝑟𝑡⁵−204120.023𝑡⁷−105840.001𝑟𝑡⁷−952560.13𝑡⁹−1088640.01𝑟𝑡⁹+1632960.01𝑡¹⁰−90720.01𝑡¹¹+1995840.01𝑟𝑡¹¹,𝑢598752_𝑑(𝑡,𝑟)=⁴_𝑛=0𝑢^𝑑_𝑛=4−2𝑟𝑡+2.8739𝑡³+31434480.1019𝑟𝑡³+47151720.0523𝑡⁵−272160.0023𝑟𝑡⁵−102060.023𝑡⁷+105840.001𝑟𝑡⁷−476280.13𝑡⁹+1088640.01𝑟𝑡⁹+816480.01𝑡¹⁰−90720.01𝑡¹¹−1995840.01𝑟𝑡¹¹.299376 (5.25)
Now, let us define the absolute value of the 𝑛th-term error by |𝑒_𝑛(𝑡,𝑟)|=|𝑢^∗(𝑡,𝑟)−𝑢_𝑛(𝑡,𝑟)|, where 𝑢^∗(𝑡,𝑟) is the exact solution and 𝑢_𝑛(𝑡,𝑟) is the 𝑛th-term approximate solution respectively. It is clear from previous above solutions that the smallest value of the absolute 𝑛th-term error is at 𝑡=𝑟=0, that is, |𝑒_𝑛(0,0)|=(|𝑒^𝑐_𝑛(0,0)|,|𝑒^𝑑_𝑛(0,0)|)=(0,0), 𝑛=0,1,…,4 and the largest value is at 𝑡=𝑟=1 as shown in Table 1. Therefore the iterations converge to the exact solution of 𝑢^𝑐,𝑢^𝑑, that is, lim_𝑛→∞𝑢^𝑐_𝑛(𝑡,𝑟)=4+𝑟𝑡=𝑢^𝑐_Exact, lim_𝑛→∞𝑢^𝑑_𝑛(𝑡,𝑟)=4−2𝑟𝑡=𝑢^𝑑_Exact, and the solution of (5.19) follows immediately.

Table 1 

Example 5.5. The final example is the following fuzzy integro-differential equation: 𝑢^′′(𝑡,𝑟)+𝑢(𝑡,𝑟)+𝜆¹₀𝑘(𝑠,𝑡)𝑢^′(𝑠,𝑟)𝑑𝑠=𝑔(𝑡,𝑟),0≤𝑡,𝑟≤1, (5.26) with the exact solution 𝑢(𝑡,𝑟)=(𝑢(𝑡,𝑟), 𝑢(𝑡,𝑟))=(3𝑟𝑡,4+𝑟𝑡), 𝑔(𝑡,𝑟)=(𝑔,𝑔)=(2𝑟𝑡,4+2𝑟𝑡/3, 𝑘(𝑠,𝑡)=𝑠²𝑡, 𝜆=−1 and the initial values 𝑢(0,𝑟)=0, 𝑢^′(0,𝑟)=3𝑟, 𝑢(0,𝑟)=4, 𝑢^′(0,𝑟)=𝑟. The exact solution for related crisp equations is 𝑢^𝑐(𝑡,𝑟)=2+2𝑟𝑡, 𝑢^𝑑(𝑡,𝑟)=2−𝑟𝑡 with 𝑔^𝑐(𝑡,𝑟)=2+4𝑟𝑡/3, 𝑔^𝑑(𝑡,𝑟)=2−2𝑟𝑡/3. As above, where 𝑛=2, the two crisp equations are𝑢^𝑐′′(𝑡,𝑟)+𝑢^𝑐(𝑡,𝑟)−¹₀𝑠²𝑡𝑢^𝑐′(𝑠,𝑟)𝑑𝑠=𝑔^𝑐(𝑡,𝑟),𝑢^𝑐(0,𝑟)=2,𝑢^𝑐′(0,𝑟)=2𝑟,𝑢^𝑑′′(𝑡,𝑟)+𝑢^𝑑(𝑡,𝑟)−¹₀𝑠²𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠=𝑔^𝑑(𝑡,𝑟),𝑢^𝑑(0,𝑟)=2,𝑢^𝑑′(0,𝑟)=−𝑟. (5.27a)(5.27b)
As the above examples, we obtain 𝑢^𝑐₀(𝑡,𝑟)=2+2𝑟𝑡+𝑡²+2𝑟𝑡³9,𝑢^𝑑₀(𝑡,𝑟)=2−𝑟𝑡+𝑡²−𝑟𝑡³9. (5.28) Therefore, the fuzzy integro-differential equations (5.27a) and (5.27b) are equivalent to the following integral equations: 𝑢^𝑐(𝑡,𝑟)=2+2𝑟𝑡+𝑡²+2𝑟𝑡³9+𝐼²_𝑡−𝑢^𝑐(𝑡,𝑟)+¹₀𝑠²𝑡𝑢^𝑐′(),𝑢𝑠,𝑟𝑑𝑠_𝑑(𝑡,𝑟)=2−𝑟𝑡+𝑡²−𝑟𝑡³9+𝐼²_𝑡−𝑢^𝑑(𝑡,𝑟)+¹₀𝑠²𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠. (5.29) Let 𝑁𝑢^𝑐=𝐼²_𝑡−𝑢^𝑐(𝑡,𝑟)+¹₀𝑠²𝑡𝑢^𝑐′(),𝑁𝑢𝑠,𝑟𝑑𝑠_𝑑=𝐼²_𝑡−𝑢^𝑑(𝑡,𝑟)+¹₀𝑠²𝑡𝑢^𝑑′(𝑠,𝑟)𝑑𝑠. (5.30) Therefore, we obtain 𝑢^𝑐₀(𝑡,𝑟)=2+2𝑟𝑡+𝑡²+2𝑟𝑡³9,𝑢^𝑑₀(𝑡,𝑟)=2−𝑟𝑡+𝑡²−𝑟𝑡³9,𝑢^𝑐₁(𝑡,𝑟)=−𝑡²+𝑡³−12𝑟𝑡³5−𝑡⁴−12𝑟𝑡⁵,𝑢90_𝑑1(𝑡,𝑟)=−𝑡²+𝑡³+12𝑟𝑡³−𝑡10⁴+12𝑟𝑡⁵,𝑢180_𝑐2(𝑡,𝑟)=−91𝑡³−1080403𝑟𝑡³+𝑡18900⁴−𝑡12⁵+240𝑟𝑡⁵+𝑡100⁶+360𝑟𝑡⁷,𝑢3780_𝑑2(𝑡,𝑟)=−91𝑡³+1080403𝑟𝑡³+𝑡37800⁴−𝑡12⁵−240𝑟𝑡⁵+𝑡200⁶−360𝑟𝑡⁷,⋮7560 (5.31)
and so on. The 5-term approximate solution is 𝑢^𝑐(𝑡,𝑟)=⁴_𝑛=0𝑢^𝑐_𝑛=2+2𝑟𝑡−0.337𝑡³−272165.0753𝑟𝑡³+15717240.073𝑡⁵+60480.0041𝑟𝑡⁵−102060.01𝑡⁷+90720.017𝑟𝑡⁷−158760.1𝑡⁹−725760.01𝑟𝑡⁹+272160.01𝑡¹⁰+181440.01𝑡¹¹,𝑢299376_𝑑(𝑡,𝑟)=⁴_𝑛=0𝑢^𝑑_𝑛=2−𝑟𝑡−0.337𝑡³+272165.0753𝑟𝑡³+31434480.073𝑡⁵−60480.0041𝑟𝑡⁵−204120.01𝑡⁷−90720.017𝑟𝑡⁷−317520.1𝑡⁹+725760.01𝑟𝑡⁹+544320.01𝑡¹⁰−181440.01𝑡¹¹.598752 (5.32)
As the previous example, it is clear from the obtained results that the smallest value of the absolute 𝑛th-term error is |𝑒_𝑛(0,0)|=(|𝑒^𝑐_𝑛(0,0)|,|𝑒^𝑑_𝑛(0,0)|)=(0,0), 𝑛=0,1,…,4, and the largest value is at 𝑡=𝑟=1 as shown in Table 2. Therefore, the iterations converge to the exact solution of 𝑢^𝑐,𝑢^𝑑, that is, lim_𝑛→∞𝑢^𝑐_𝑛(𝑡,𝑟)=2+2𝑟𝑡=𝑢^𝑐_Exact, lim_𝑛→∞𝑢^𝑑_𝑛(𝑡,𝑟)=2−𝑟𝑡=𝑢^𝑑_Exact. Therefore, the solution of (5.26) follows immediately.

Table 2 

6. Conclusion

In this work, the new iterative method used with a reliable algorithm to solve the 𝑛th-order derivative fuzzy integro-differential equations in crisp form and the cases of first- and second-order derivatives are taken into account. The obtained results concluded that the approximate solutions are in high agreement with corresponding exact solutions, which means that this method is suitable and effective to solve fuzzy integro-differential equations. Moreover, the solutions of the higher order fuzzy integro-differential equations can be calculated, as a future prospects, in a similar manner.