Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

Abstract

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type. The CAS wavelets operational matrix of fractional order integration is derived. A truncated CAS wavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited 2^𝑘(2𝑀+1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.

1. Introduction

A lot of scientific and engineering problems involving fractional phenomenon are already very large and still growing. One of the main advantages of the fractional phenomenon is that the fractional derivatives and fractional integrals provide an excellent approach to the different kinds of physical fields, such as dispersive transports in amorphous semiconductors, tracer transfer in underground water, and seepage in soil or rocks [1–4]. In recent years, more and more researchers are finding that a variety of dynamical problems exhibit fractional order behavior. This indicates that variable order calculus is an effective mathematical framework to describe the complex dynamical problems [5–7]. Many of the numerical methods using various fractional derivative operators and integral operators for solving fractional differential equations have been proposed. Podlubny [8] used the Laplace transform method to solve the fractional partial differential equations with constant coefficients. Odibat and Momani [9] applied generalized differential transform method to solve the numerical solution of linear partial differential equations of fractional order. Zhang [10] discussed a practical implicit method to solve a class of initial boundary value space-time fractional convection-diffusion equations with variable coefficients. Zhuang et al. [11] proposed explicit and implicit Euler method for the variable order fractional advection-diffusion equation.

Bratu’s problem is also discussed in all kinds of applications, such as chemical reaction theory, the fuel ignition model of the thermal combustion theory, and nanotechnology [12–15]. Both mathematicians and physicists have devoted a lot of effort to Bratu’s problem. In [16], Syam and Hamdan presented the Laplace Adomian decomposition method for solving Bratu’s problem. Wazwaz [17] proposed the Adomian decomposition method for solving Bratu’s problem. Aksoy and Pakdemirli [18] had solved Bratu-type equation of new perturbation iteration solutions.

In this paper, we consider the following fractional integrodifferential equations of Bratu type by means of CAS wavelets: 𝐷^𝛼𝑢(𝑥)+𝜆^𝑥₀𝑘(𝑥,𝑡)𝑒^𝑢(𝑡)𝑑𝑡+𝑔(𝑥)=0,𝑛−1<𝛼≤𝑛,0≤𝑥,𝑡≤1, (1) with initial condition 𝑢^(𝑗)(0)=𝑏_𝑗,𝑗=0,1,…,𝑛−1. (2)

2. Definitions and Properties of Fractional Operator

Here we just recall the most typical definitions which are easy to use in physics. The Caputo fractional differential operator 𝐷^𝛼 of order 𝛼>0 is defined as [8] 𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝜏)^{𝑛−𝛼−1}𝑓^(𝑛)(𝜏)𝑑𝜏. (3)

The Riemann-Liouville fractional integration of order 𝛼>0 is defined as 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,𝑛−1<𝛼≤𝑛, (4) where 𝑛 is a positive integer.

It has the following two basic properties for 𝑛−1<𝛼≤𝑛: 𝐷^𝛼𝐼^𝛼𝑓(𝑡)(𝑡),𝐼=𝑓_𝛼𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)0⁺𝑡^𝑘𝑘!,𝑡>0. (5)

3. CAS Wavelets and Some of Their Properties

3.1. CAS Wavelets

The CAS wavelets 𝜓_𝑛𝑚(𝑡)=𝜓(𝑘,𝑛,𝑚,𝑡) have four arguments, 𝑛=1,2,…,2^𝑘, where 𝑘 is any nonnegative integer, 𝑚 is any integer, and 𝑡 is the normalized time. The orthonormal CAS wavelets are defined on the interval [0,1) by [19] 𝜓_𝑛𝑚(𝑡)=2^𝑘/2CAS_𝑚2^𝑘𝑡−𝑛+1,𝑛−12^𝑘𝑛≤𝑡<2^𝑘,0,otherwise, (6) where CAS_𝑚(𝑡)=cos(2𝑚𝜋𝑡)+sin(2𝑚𝜋𝑡).

3.2. Function Approximation

A function 𝑢(𝑡)∈𝐿²[0,1) may be expanded as 𝑢(𝑡)=^∞_{𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡), (7) where 𝑐_𝑛𝑚=⟨𝑢(𝑡),𝜓_𝑛𝑚(𝑡)⟩, in which ⟨⟩ denotes the inner product. The series (7) is truncated as 𝑢(𝑡)≈𝑢_𝑘,𝑀(𝑡)=^2𝑘𝑀_{𝑛=1𝑚=−𝑀}𝑐_𝑛𝑚𝜓_𝑛𝑚=𝐶^𝑇Ψ(𝑡), (8) where 𝐶 and Ψ(𝑡) are two vectors given by 𝐶=𝑐_1(−𝑀),𝑐_1(−𝑀+1),…𝑐_1(𝑀),𝑐_2(−𝑀)𝑐,…,_2(𝑀),…,𝑐₂^𝑘_(−𝑀),…𝑐₂^𝑘_(𝑀)^𝑇=𝑐₁,𝑐₂,…,𝑐₂^𝑘_(2𝑀+1)^𝑇,Ψ(𝑡)=𝜓_1(−𝑀),𝜓_1(−𝑀+1),…,𝜓_1(𝑀),𝜓_2(−𝑀)𝜓,…,_2(𝑀),𝜓_{2𝑘(−𝑀)},…,𝜓_2𝑘(𝑀)^𝑇=𝜓₁,𝜓₂,…,𝜓_{2𝑘(2𝑀+1)}^𝑇. (9)

3.3. Convergence of the CAS Wavelet Bases

In this section, we indicate that the CAS wavelets expansion of a function 𝑢(𝑡), with bounded second derivative, converges uniformly to 𝑢(𝑡).

Lemma 1. If the CAS wavelet expansion of a continuous function 𝑢(𝑡) converges uniformly, then the CAS wavelet expansion converges to the function 𝑢(𝑡).

Proof. Let 𝑣(𝑡)=^∞_{𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡), (10) where 𝑐_𝑛𝑚=⟨𝑢(𝑡),𝜓_𝑛𝑚(𝑡)⟩. Multiplying both sides of (11) by 𝜓_𝑝𝑞(𝑡), 𝑝, and 𝑞 are fixed and then integrating termwise, justified by uniformly convergence, on [0,1], we have ¹₀𝑣(𝑡)𝜓_𝑝𝑞(𝑡)=𝑑𝑡_{10∞𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡)𝜓_𝑝𝑞(𝑡)=𝑑𝑡_{∞𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚¹₀𝜓_𝑛𝑚(𝑡)𝜓_𝑝𝑞(𝑡)𝑑𝑡=𝑐_𝑝𝑞. (11)
Thus ⟨𝑣(𝑡),𝜓_𝑛𝑚(𝑡)⟩=𝑐_𝑛𝑚 for 𝑛,𝑚∈𝑍. Consequently 𝑢,𝑣 have same Fourier expansions with the CAS wavelets basis and therefore 𝑢(𝑡)=𝑣(𝑡).

Theorem 2. A function 𝑢(𝑡)∈𝐿²[0,1], with bounded second derivative, say |𝑢^′′(𝑡)|≤𝛾, can be expanded as an infinite sum of the CAS wavelets and the series converges uniformly to 𝑢(𝑡); that is 𝑢(𝑡)=∑^∞_𝑛=1∑_𝑚∈Ζ𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡). Furthermore, we have 𝑢_𝑘,𝑀(𝑡)(𝑡≤𝛾−𝑢)𝜋^2∞_𝑛=2𝑘^∞_𝑚=𝑀+11𝑛^5/2𝑚²,𝑡∈[0,1]. (12)

Proof. From (6), we obtain 𝑐_𝑛𝑚=¹₀𝑢(𝑡)𝜓_𝑛𝑚(𝑡)=𝑑𝑡_{𝑛/2𝑘()𝑛−1/2𝑘}2^𝑘/2𝑢(𝑡)CAS_𝑚2^𝑘𝑡−𝑛+1𝑑𝑡. (13)
Substituting 2^𝑘𝑡−𝑛+1=𝑥 in (13) yields 𝑐_𝑛𝑚=12^𝑘/21₀𝑢𝑥+𝑛−12^𝑘CAS_𝑚(𝑥)=1𝑑𝑥2^𝑘/21₀𝑢𝑥+𝑛−12^𝑘𝑑sin(2𝑚𝜋𝑥)−cos(2𝑚𝜋𝑥)=12𝑚𝜋2^𝑘/2𝑢𝑥+𝑛−12^𝑘sin(2𝑚𝜋𝑥)−cos(2𝑚𝜋𝑥)2𝑚𝜋¹₀−12^3𝑘/2(2𝑚𝜋)¹⁰𝑢^′𝑥+𝑛−12^𝑘×(()(=1sin2𝑚𝜋𝑥−cos2𝑚𝜋𝑥))𝑑𝑥2^3𝑘/2(2𝑚𝜋)¹⁰𝑢^′𝑥+𝑛−12^𝑘𝑑CAS_𝑚(𝑥)=12𝑚𝜋2^3𝑘/2(2𝑚𝜋)²𝑢^′𝑥+𝑛−12^𝑘CAS_𝑚(𝑥)¹₀−12^5𝑘/2(2𝑚𝜋)²¹⁰𝑢^′′𝑥+𝑛−12^𝑘CAS_𝑚(𝑥)𝑑𝑥. (14)
Thus, we get 𝑐_𝑛𝑚²=12^5𝑘/2(2𝑚𝜋)²¹⁰𝑢^′′𝑥+𝑛−12^𝑘CAS_𝑚(𝑥)𝑑𝑥²≤12^5𝑘/2(2𝑚𝜋)²²¹⁰𝑢^′′𝑥+𝑛−12^𝑘2×𝑑𝑥₁₀CAS_𝑚(𝑥)²≤𝛾𝑑𝑥2^5𝑘/2(2𝑚𝜋)²²¹⁰CAS_𝑚(𝑥)²𝑑𝑥, (15) from orthonormality of CAS wavelets, we know that ∫¹₀|CAS_𝑚(𝑥)|²𝑑𝑥=1, since 𝑛≤2^𝑘, we have |𝑐_𝑛𝑚|≤𝛾/(4𝜋²𝑛^5/2𝑚²). Hence the series ∑^∞_𝑛=1∑_𝑚∈𝑍𝑐_𝑛𝑚 is absolutely convergent. On the other hand, we can obtain ^∞_{𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡≤_{∞𝑛=1𝑚∈𝑍})𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡)≤2^∞_{𝑛=1𝑚∈𝑍}𝑐_𝑛𝑚<∞. (16)
Using Lemma 1, the series ∑^∞_𝑛=1∑_𝑚∈𝑍𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡) converges to 𝑢(𝑡).
Moreover, we conclude that 𝑢_𝑘,𝑀(𝑡)−𝑢(𝑡)≤2^∞_𝑛=2𝑘^∞_{+1𝑚=𝑀+1}𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡)≤4^∞_𝑛=2𝑘^∞_{+1𝑚=𝑀+1}𝑐_𝑛𝑚≤𝛾𝜋^2∞_𝑛=2𝑘^∞_𝑚=𝑀+11𝑛^5/2𝑚². (17)
This completes the proof.

3.4. Operational Matrix of the Fractional Integration

In this part, we may simply introduce the operational matrix of fractional integration of CAS wavelets; more detailed introduction can be found in [19].

Take the points 𝑡_𝑖=𝑖/2^𝑘(2𝑀+1), 𝑖=0,1,2,…,2^𝑘(2𝑀+1)−1, then we define Φ=Ψ𝑡₀Ψ𝑡₁⋯Ψ𝑡₂^𝑘_{(2𝑀+1)−1}. (18)

If 𝐽^𝛼 is fractional integration operator of CAS wavelets, we can get 𝐽^𝛼Ψ(𝑡)=𝑃^𝛼Ψ(𝑡), (19) where 𝑃^𝛼 is called operational matrix of fractional integration of CAS wavelets.

Apart from the CAS wavelets, we consider another basis set of block pulse functions. The set of these functions, over the interval [0,1), is defined as 𝑏_𝑖(𝑡)=𝑖1,𝑙≤𝑡<𝑖+1𝑙,𝑖=0,1,2,…,𝑙−10,otherwise, (20) with a positive integer value for 𝑙, we suppose 𝑙=2^𝑘(2𝑀+1) in this paper.

Let 𝐵_𝑙(𝑡)=[𝑏₀(𝑡),𝑏₁(𝑡),…,𝑏_𝑙−1(𝑡)]^𝑇, then the block pulse functions operational matrix of fractional integration 𝐹^𝛼 is given by [20] 𝐽^𝛼𝐵_𝑙(𝑡)=𝐹^𝛼𝐵_𝑙(𝑡), (21) where 𝐹^𝛼=1𝑙^𝛼1Γ(𝛼+2)1𝜉₁𝜉₂⋯𝜉_𝑙−101𝜉₁⋯𝜉_𝑙−2001⋯𝜉_𝑙−3,𝜉⋮⋮⋮⋮⋮000⋯1_𝑘=(𝑘+1)^𝛼+1−2𝑘^𝛼+1+(𝑘−1)^𝛼+1. (22)

There is a relation between the block pulse functions and CAS wavelets: Ψ(𝑡)=Φ𝐵_𝑙(𝑡). (23)

Therefore we can derive 𝑃^𝛼 easily by using (19), (21), and (23): 𝑃^𝛼=Φ𝐹^𝛼Φ⁻¹. (24)

4. Numerical Solution of (1)-(2)

Consider the nonlinear fractional integrodifferential equations of Bratu type 𝐷^𝛼𝑢(𝑥)+𝜆^𝑥₀𝑘(𝑥,𝑡)𝑒^𝑢(𝑡)(𝑥)𝑢𝑑𝑡+𝑔=0,_(𝑗)(0)=𝑏_𝑗,0≤𝑥,𝑡≤1,𝑛−1<𝛼≤𝑛, (25)

where 𝜆 is arbitrary parameter, 𝑘(𝑥,𝑡)∈𝐿²([0,1]×[0,1]), and 𝑔(𝑥) is a known function.

Let 𝐷^𝛼𝑢(𝑥)≈𝐶^𝑇Ψ(𝑥), 𝑔(𝑥)≈𝐺^𝑇Ψ(𝑥), 𝑘(𝑥,𝑡)≈Ψ^𝑇(𝑥)𝐾Ψ(𝑡), and 𝑒^𝑢(𝑡)≈∑^𝑙_𝑖=0[𝑢(𝑡)]^𝑖/𝑖!, then 𝑢(𝑥)≈𝐶^𝑇𝑃^𝛼Ψ(𝑥)+^𝑛−1_𝑘=0𝑢^(𝑘)(0)𝑥^𝑘𝑘!=𝐶^𝑇𝑃^𝛼Ψ(𝑥)+𝑄Ψ(𝑥)=𝐶^𝑇𝑃^𝛼+𝑄Ψ(𝑥), (26) where coefficient 𝑄 is known and can be obtained by using the initial conditions.

Substituting (23) into (26), we obtain 𝑢(𝑥)≈𝐶^𝑇𝑃^𝛼+𝑄Ψ(𝑥)=𝐶^𝑇𝑃^𝛼+𝑄Φ𝐵_𝑙(𝑥). (27)

Define (𝐶^𝑇𝑃^𝛼+𝑄)Φ=[𝑎₁,𝑎₂,…,𝑎_𝑚]=𝐴, then 𝑢(𝑥)≈𝐴𝐵_𝑙(𝑥).

Applying the properties of BPFs, we have [𝑢(𝑥)]^𝑖≈𝑎^𝑖₁,𝑎^𝑖₂,…,𝑎^𝑖_𝑚𝐵_𝑙(𝑥)=𝐴_𝑖𝐵_𝑙(𝑥). (28)

By substituting the above expanded forms into (1), we get 𝐷^𝛼𝑢(𝑥)+𝜆^𝑥₀𝑘(𝑥,𝑡)𝑒^𝑢(𝑡)𝑑𝑡+𝑔(𝑥)≈𝐶^𝑇Ψ(𝑥)+𝜆^𝑥₀Ψ^𝑇(𝑥)𝐾Ψ(𝑡)^𝑙_𝑗=0[𝑢(𝑡)]^𝑗𝑗!𝑑𝑡+𝐺^𝑇Ψ(𝑥)≈𝐶^𝑇Ψ(𝑥)+𝜆Ψ^𝑇(𝑥)𝐾×^𝑥₀Φ𝐵_𝑙(𝑡)^𝑙_𝑗=0𝐵^𝑇_𝑙(𝑡)𝐴^𝑇_𝑗𝑗!𝑑𝑡+𝐺^𝑇Ψ(𝑥)≈𝐶^𝑇Ψ(𝑥)+𝜆Ψ^𝑇(𝑥)𝐾Φ^𝑙_𝑗=01𝑗!^𝑥₀𝐴^𝑇_𝑗𝐵_𝑙(𝑡)𝑑𝑡+𝐺^𝑇Ψ(𝑥)≈𝐶^𝑇Ψ(𝑥)+𝜆Ψ^𝑇(𝑥)𝐾Φ^𝑙_𝑗=01𝐴𝑗!^𝑇𝑗𝐹¹𝐵_𝑙(𝑥)+𝐺^𝑇Ψ(𝑥)=0, (29) where 𝐴^𝑇_𝑗 is the product operational matrix of 𝐴^𝑇_𝑗.

Putting the collocation points {𝑥_𝑖}^{2𝑘(2𝑀+1)−1}_𝑖=0 into (29), (29) will be 𝐶^𝑇Ψ𝑥_𝑖+𝜆Ψ^𝑇𝑥_𝑖𝐾Φ^𝑙_𝑗=01𝐴𝑗!^𝑇𝑗𝐹¹𝐵_𝑙𝑥_𝑖+𝐺^𝑇Ψ𝑥_𝑖=0. (30)

Solving the nonlinear algebraic equations (30) by using Newton iteration method, we obtain the vector 𝐶^𝑇, and then we get the approximate solution 𝑢(𝑥)=(𝐶^𝑇𝑃^𝛼+𝑄)Ψ(𝑥).

5. Existence of Uniqueness

Theorem 3 (uniqueness theorem). Equation (1) has a unique solution whenever 0<𝛽<1, where 𝛽=𝜆𝑀𝐿Γ(𝛼)𝛼(),𝛼+1𝑀=sup_{0≤𝑥,𝑡≤1}|𝑘(𝑥,𝑡)|<∞. (31)

Proof. Suppose 𝑡∈[0,1], then 𝑢(𝑡) is bounded. Therefore the nonlinear term 𝑒^𝑢(𝑡) in (1) is Lipschitz continuous with |𝑒^𝑢−𝑒^𝑢∗|≤𝐿|𝑢−𝑢_∗|,??𝐿>0.
Let 𝑢 and 𝑢^∗ be two different solutions of (1), then we can get 𝐷^𝛼𝑢(𝑥)=−𝜆^𝑥₀𝑘(𝑥,𝑡)𝑒^𝑢(𝑡)(𝑥),𝐷𝑑𝑡−𝑔_𝛼𝑢_∗(𝑥)=−𝜆^𝑥₀𝑘(𝑥,𝑡)𝑒^{𝑢∗(𝑡)}𝑑𝑡−𝑔(𝑥). (32)
Using Riemann-Liouville fractional integration, we have 𝐼^𝛼𝐷^𝛼𝑢(𝑥)1=−Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑔(𝜉)−𝜆𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^𝑢(𝑡)𝐼𝑑𝑡𝑑𝜉,_𝛼𝐷^𝛼𝑢_∗(𝑥)1=−Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑔(𝜉)−𝜆𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^{𝑢∗(𝑡)}𝑑𝑡𝑑𝜉. (33)
Because 𝐼^𝛼𝐷^𝛼𝑥(𝑡)=𝑥(𝑡)−∑^𝑛−1_𝑘=0(𝑡^𝑘/𝑘!)𝑥^(𝑘)(0⁺), so (33) can transform as 𝑢(𝑥)−^𝑛−1_𝑗=0𝑥^𝑗𝑢𝑗!^(𝑗)(0)1=−Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑔(𝜉)−𝜆𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^𝑢(𝑡)𝑢𝑑𝑡𝑑𝜉,_∗(𝑥)−^𝑛−1_𝑗=0𝑥^𝑗𝑢𝑗!^∗(𝑗)(0)1=−Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑔(𝜉)−𝜆𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^{𝑢∗(𝑡)}𝑑𝑡𝑑𝜉. (34)
Then we have 𝑢−𝑢_∗=−𝜆Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^𝑢(𝑡)+𝜆𝑑𝑡𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^{𝑢∗(𝑡)}=−𝜆𝑑𝑡𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^𝑢(𝑡)−𝑒^{𝑢∗(𝑡)}≤𝜆𝑑𝑡𝑑𝜉Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑘(𝜉,𝑡)𝑒^𝑢(𝑡)−𝑒^{𝑢∗(𝑡)}≤𝑑𝑡𝑑𝜉𝜆𝑀𝐿Γ(𝛼)^𝑥₀^𝜉₀(𝑥−𝜉)^𝛼−1𝑢−𝑢_∗=𝑑𝑡𝑑𝜉𝜆𝑀𝐿Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑢−𝑢_∗=𝜉𝑑𝜉𝜆𝑀𝐿Γ(𝛼)𝑢−𝑢_∗^𝑥₀𝜉(𝑥−𝜉)^𝛼−1=𝑑𝜉𝜆𝑀𝐿Γ(𝛼)𝑢−𝑢_∗𝑥^𝛼+1𝛼−𝑥^𝛼+1=𝛼+1𝜆𝑀𝐿Γ(𝛼)𝑢−𝑢_∗𝑥^𝛼+11𝛼()≤𝛼+1𝜆𝑀𝐿Γ(𝛼)𝛼(𝛼+1)𝑢−𝑢_∗. (35)
Therefore |𝑢−𝑢_∗|(1−(𝜆𝑀𝐿/Γ(𝛼)𝛼(𝛼+1)))≤0.
This implies that |𝑢−𝑢_∗|(1−𝛽)≤0, where 𝛽=𝜆𝑀𝐿/Γ(𝛼)𝛼(𝛼+1).
As 0<𝛽<1, |𝑢−𝑢_∗|=0; implying 𝑢=𝑢_∗, we can prove (1) has the uniqueness solution.

6. Numerical Examples

To show the efficiency and the accuracy of the proposed method, we consider the following two examples.

Example 1. Consider the following equation: 𝐷^0.5𝑢(𝑥)+^𝑥₀𝑥1𝑡−2𝑒^𝑢(𝑡)𝑑𝑡+𝑔(𝑥)=0, (36) with this condition 𝑢(0)=0 and 𝑔(𝑥)=−(Γ(3)/Γ(2.5))𝑥^1.5+(Γ(2)/Γ(1.5))𝑥^0.5−(𝑥/2)(𝑒^𝑥2−𝑥−1). The exact solution of this problem is 𝑢(𝑥)=𝑥²−𝑥. Table 1 shows the absolute errors of the approximate solutions and the exact solution. The comparison between the numerical solutions and the exact solution for different 𝑘 is shown in Figures 1, 2, and 3.

Table 1 

Absolute errors for different values of 𝑘, 𝑀.

Figure 1 

The comparison between numerical and exact solution for 𝑘=1, 𝑀=1.

Figure 2 

The comparison between numerical and exact solution for 𝑘=2, 𝑀=1.

Figure 3 

The comparison between numerical and exact solution for 𝑘=3, 𝑀=1.

Taking a closer look at Table 1 and Figures 1–3, with 𝑘 increasing, we find that the approximate solutions converge to the exact solution.

Example 2. Consider this equation 𝐷^𝛼𝑢(𝑥)+3^𝑥₀(𝑥−𝑡)𝑒^𝑢(𝑡)𝑑𝑡+𝑔(𝑥)=0,0<𝑥,𝑡<1, (37) such that the initial conditions 𝑢(0)=0, 𝑔(𝑥)=−(1/2)𝑥³−(1/𝑥). The exact solution of the problem for 𝛼=1 is given by 𝑢(𝑥)=ln𝑥. The comparison of numerical results for 𝛼=0.5, 𝛼=0.75, and 𝛼=1 and the exact solution for 𝛼=1 are shown in Figure 4. From Figure 4, we can see clearly that the numerical solutions are in very good agreement with the exact solution when 𝛼=1.

Figure 4 

Numerical solution and exact solution of 𝛼=1.

It is evident from Figure 4 that, as 𝛼 close to 1, the numerical solution of the CAS wavelets converges to the exact solution; that is, the solution of fractional integrodifferential equation approaches to the solution of integer order integrodifferential equation.

7. Conclusion

In this paper, a numerical method is presented by numerical solutions of fractional integrodifferential equations of Bratu type. Taking full advantage of the definition of Caputo type fractional derivative and the properties of CAS wavelet, we transform the initial problem into a nonlinear algebraic system equation. By solving the nonlinear system, numerical solutions are obtained. The convergence analysis of CAS wavelets and the uniqueness theorem of this equation are proposed. The numerical results show that the approximation is in very good coincidence with the exact solution.

Acknowledgment

This paper was supported by the International Science & Technology Cooperation Program of China (no. 2012DFG61930).