A New Neumann Series Method for Solving a Family of Local Fractional Fredholm and Volterra Integral Equations

Abstract

We propose a new Neumann series method to solve a family of local fractional Fredholm and Volterra integral equations. The integral operator, which is used in our investigation, is of the local fractional integral operator type. Two illustrative examples show the accuracy and the reliability of the obtained results.

1. Introduction

Many initial- and boundary-value problems associated with ordinary differential equations (ODEs) and partial differential equations (PDEs) can be transformed into problems of solving the corresponding approximate integral equations. However, some initial- and boundary-value domains are fractal curves, which are everywhere continuous, but nowhere differentiable. As a result, we cannot employ the classical calculus, which requires that the defined functions should be differentiable, in order to process various classes of ordinary differential equations (ODEs) and partial differential equations (PDEs). Applications of fractional calculus, in general, and fractional differential equations [1–10], in particular, as well as various transport phenomena in complex and disordered media and fractional systems, have attracted considerable attention during the past two decades or so [11–22].

Recently, local fractional calculus [23–40], processing local fractional continuous non-differential functions, was successfully applied to model the stress-strain relation in fractal elasticity [26, 27], fractal release equation [32], wave equations on Cantor sets [34], fractal heat equation [34], diffusion equation arising in discontinuous heat transfer in fractal media [35], Laplace equation within local fractional operators [36], Schrödinger equation in fractal time-space [37], damped wave equation and dissipative wave equation in fractal strings [38], heat-conduction equation on Cantor sets without heat generation in fractal media [39], and so on. There are some analytical and numerical methods for solving local fractional ODEs and PDEs, such as fractional complex transform method with local fractional operator [35], local fractional variational iteration method [37], Cantor-type cylindrical-coordinate method [38], local fractional Fourier series method [39], local fractional series expansion method [40], Fourier and Laplace transforms with local fractional operator [39], and reference therein.

The Neumann series method was applied to solve the integral equations [41, 42]. Recently, the fractional Neumann series method was considered in [43, 44]. This paper focuses on a new Neumann series method for solving the local fractional Fredholm and Volterra integral equation being here facts in mind. This paper is structured as follows. Section 2 introduces the notations and the basic concepts. Section 3 is devoted to a new Neumann series method via local fractional integral operator. Two illustrative examples are explained in Section 4. Finally, conclusions are reported in Section 5.

2. Preliminaries

In order to investigate the local fractional continuity of non-differential functions, we suggest the result derived from fractal geometry [34, 39].

Let 𝑓(𝑥) be local fractional continuous on interval (𝑎,𝑏); then we write [34, 35] 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). (1) If 𝑓∶(𝐹,𝑑)→(Ω^′,𝑑^′) is a bi-Lipschitz mapping, then 𝜌^𝑠𝐻^𝑠(𝐹)≤𝐻^𝑠𝑓(𝐹)≤𝜏^𝑠𝐻^𝑠(𝐹), (2) which leads to 𝜌^𝛼𝑥₁−𝑥₂^𝛼≤𝑓𝑥₁−𝑓𝑥₂≤𝜏^𝛼𝑥₁−𝑥₂^𝛼, (3) so that 𝑓𝑥₁−𝑓𝑥₂<𝜀^𝛼, (4) where 𝜌,𝜏>0 and 𝑥₁,𝑥₂∈𝐹.

The result deduced from fractal geometry is related to fractal coarse-grained mass function 𝛾^𝛼[𝐹,𝑎,𝑏], which reads [34] as 𝛾^𝛼[]=𝐻𝐹,𝑎,𝑏^𝛼(𝐹∩(𝑎,𝑏))Γ(),1+𝛼 (5) with 𝐻^𝛼(𝐹∩(𝑎,𝑏))=(𝑏−𝑎)^𝛼, (6) where 𝐻^𝛼 is an 𝛼-dimensional Hausdorff measure.

Notice that we consider that the dimensions of any fractal spaces (e.g., Cantor spaces or the Cantor-like spaces) are a positive numbers. It looks like the Euclidean space because its dimension is also positive number. The detailed results were considered in [34].

For 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏), local fractional integral of 𝑓(𝑥) of order 𝛼 in the interval [𝑎,𝑏] is given by [34, 37, 39] _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^{𝑗=𝑁−1}_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼, (7) where Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗,Δ𝑡=max{Δ𝑡₁,Δ𝑡₂,Δ𝑡_𝑗,…} and [𝑡_𝑗,𝑡_𝑗+1], 𝑗=0,…,𝑁−1,𝑡₀=𝑎,𝑡_𝑁=𝑏, is a partition of the interval [𝑎,𝑏].

For any 𝑥∈(𝑎,𝑏), we have [34] _𝑎𝐼_𝑥^(𝛼)𝑓(𝑥), (8) denoted by 𝑓(𝑥)∈𝐼_𝑥^(𝛼)(𝑎,𝑏). (9) If 𝑓(𝑥)∈𝐼_𝑥^(𝛼)(𝑎,𝑏), then we have [34] 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). (10) For detailed content of fractal geometrical explanation of local fractional integral, we can see [34, 35]. Some properties of local fractional integral operator were suggested in (A.1)–(A.5).

3. A New Neumann Series Method to Deal with the Local Fractional Fredholm and Volterra Integral Equations

In this section, we consider a new Neumann series method to process the local fractional Fredholm and Volterra integral equations.

A new Neumann series method to deal with the local fractional Fredholm integral equation is written in the following form: 𝑢(𝑥)(𝑥)+𝜆=𝑓^𝛼Γ(1+𝛼)^𝑏_𝑎𝐾(𝑥,𝑡)𝑢(𝑡)(𝑑𝑡)^𝛼. (11) It is obtained if we set 𝑢₀(𝑥)=𝑓(𝑥), (12) such that 𝑢₁(𝑥)=𝑢₀(𝑥)+𝜆^𝛼Γ(1+𝛼)^𝑏_𝑎𝐾(𝑥,𝑡)𝑢₀(𝑡)(𝑑𝑡)^𝛼=𝑓(𝑥)+𝜆^𝛼𝜓₁(𝑥), (13) where 𝜓₁(𝑥)=(1/Γ(1+𝛼))∫^𝑏_𝑎𝐾(𝑥,𝑡)𝑓(𝑡)(𝑑𝑡)^𝛼.

The zeroth approximation can be written as 𝑢₂(𝑥)(𝑥)+𝜆=𝑓^𝛼Γ(1+𝛼)^𝑏_𝑎𝐾(𝑥,𝑡)𝑢₁(𝑡)(𝑑𝑡)^𝛼(𝑥)+𝜆=𝑓^𝛼Γ()×1+𝛼_𝑏𝑎𝐾(𝑥,𝑡)𝑓(𝑥)+𝜆^𝛼𝜓₁(𝑥)(𝑑𝑡)^𝛼=𝑓(𝑥)+𝜆^𝛼𝜓₁(𝑥)+𝜆^2𝛼𝜓₂(𝑥), (14) where 𝜓₂(𝑥)=(1/Γ(1+𝛼))∫^𝑏_𝑎𝐾(𝑥,𝑡)𝜓₁(𝑥)(𝑑𝑡)^𝛼.

Proceeding in this manner, the final solution 𝑢(𝑥) can be obtained as 𝑢(𝑥)=𝑓(𝑥)+𝜆^𝛼𝜓₁(𝑥)+𝜆^2𝛼𝜓₂(𝑥)+⋯+𝜆^𝑛𝛼𝜓_𝑛(𝑥)(𝑥)++⋯=𝑓^∞_𝑛=1𝜆^𝑛𝛼𝜓_𝑛(𝑥), (15) where 𝜓_𝑛(𝑥)=(1/Γ(1+𝛼))∫^𝑏_𝑎𝐾(𝑥,𝑡)𝜓_𝑛−1(𝑥)(𝑑𝑡)^𝛼,𝑛≥1.

Now we structure a new Neumann series method to handle the local fractional Volterra integral equation, which reads as 𝑢(𝑥)(𝑥)+𝜆=𝑓^𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)𝑢(𝑡)(𝑑𝑡)^𝛼. (16) The method is applicable provided that 𝑢(𝑥) is a local fractional analysis function; that is, 𝑢(𝑥) have a local fractional Taylor’s expansion around 𝑥=0.

𝑢(𝑥) can be expressed by a local fractional series expansion; which reads as 𝑢(𝑥)=^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼, (17) where the coefficients 𝑎_𝑛 and 𝑥 are constants that are required to be determined.

We have ^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼(𝑥)+𝜆=𝑓^𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼(𝑑𝑡)^𝛼. (18) Thus, using a few terms of the expansion in both sides, we find that 𝑎₀+𝑎₁𝑥^𝛼+𝑎₂𝑥^2𝛼+⋯+𝑎_𝑛𝑥^𝑛𝛼(𝑥)+𝜆+⋯=𝑓^𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)𝑎₀(𝑑𝑡)^𝛼+𝜆^𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)𝑎₁𝑥^𝛼(𝑑𝑡)^𝛼+𝜆^𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)𝑎₂𝑥^2𝛼(𝑑𝑡)^𝛼+𝜆+⋯_𝛼Γ(1+𝛼)^𝑥_𝑎𝐾(𝑥,𝑡)𝑎_𝑛𝑥^𝑛𝛼(𝑑𝑡)^𝛼+⋯. (19) We then write the local fractional Taylor’s expansions for 𝑓(𝑥) and count the first few integrals in (19). After the integration is performed, we equate the coefficients of the same powers of 𝑥^𝛼 in both sides of (19). By this way, we can determine completely the unknown coefficients and produce solution in a local fractional series form.

4. Examples

Example 1. Solve the following local fractional Fredholm integral equation: 𝑢(𝑥)()+1=Γ1+𝛼Γ(1+𝛼)¹₀𝑥^𝛼𝑢(𝑡)(𝑑𝑡)^𝛼. (20) Let us consider the zeroth approximation given by 𝑢₀(𝑡)=Γ(1+𝛼). (21) The first approximation can be computed as follows: 𝑢₁(𝑥)()+1=Γ1+𝛼Γ(1+𝛼)¹₀𝑥^𝛼Γ(1+𝛼)(𝑑𝑡)^𝛼=Γ(1+𝛼)+𝑥^𝛼. (22) Proceeding in this manner, we find the following local fractional series approximation: 𝑢₂(𝑥)()+1=Γ1+𝛼Γ(1+𝛼)¹₀𝑥^𝛼Γ(1+𝛼)+𝑡^𝛼(𝑑𝑡)^𝛼=Γ(1+𝛼)+𝑥^𝛼Γ()1+1+𝛼Γ().1+2𝛼 (23) Similarly, the third approximation reads as follows: 𝑢₃(𝑥)()+1=Γ1+𝛼Γ()×1+𝛼₁₀𝑥^𝛼Γ(1+𝛼)+𝑡^𝛼Γ()1+1+𝛼Γ()()1+2𝛼𝑑𝑡^𝛼=Γ(1+𝛼)+𝑥^𝛼Γ()1+1+𝛼Γ()+Γ1+2𝛼²(1+𝛼)Γ²().1+2𝛼 (24) The fourth approximation yields 𝑢₄(𝑥)()+1=Γ1+𝛼Γ(1+𝛼)×¹₀𝑥^𝛼Γ(1+𝛼)+𝑡^𝛼×Γ()1+1+𝛼Γ()+Γ1+2𝛼²(1+𝛼)Γ²()()1+2𝛼𝑑𝑡^𝛼=Γ(1+𝛼)+𝑥^𝛼Γ()1+1+𝛼Γ()+Γ1+2𝛼₂(1+𝛼)Γ²()+Γ1+2𝛼³(1+𝛼)Γ³().1+2𝛼 (25) In conclusion, we get 𝑢_𝑛(𝑥)=Γ(1+𝛼)+𝑥^𝛼𝑛_𝑖=0Γ(1+𝛼)Γ(1+2𝛼)^𝑛. (26) Hence, 𝑢(𝑥)=lim_𝑛→∞𝑢_𝑛(𝑥)=Γ(1+𝛼)+𝑥^𝛼lim^𝑛_{𝑛→∞𝑖=0}Γ(1+𝛼)Γ(1+2𝛼)^𝑛()+𝑥=Γ1+𝛼^𝛼Γ(1+𝛼)Γ()().1+2𝛼−Γ1+𝛼 (27)

Example 2. Obtain the solution of the following local fractional Volterra equation: 𝑢(𝑥)𝑥=1+^𝛼Γ()+11+𝛼Γ(1+𝛼)^𝑥₀(𝑡−𝑥)^𝛼Γ()𝑢(𝑡)1+𝛼)(𝑑𝑡^𝛼. (28) Suppose that there exists the solution in the following local fractional series form: 𝑢(𝑥)=^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼. (29) Then, upon substituting the local fractional series into the equation, we find that ^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼=𝑥^𝛼Γ()+11+𝛼Γ(1+𝛼)^𝑥₀(𝑡−𝑥)^𝛼Γ(1+𝛼)^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼(𝑑𝑡)^𝛼=𝑥^𝛼Γ()−1+𝛼^∞_𝑛=1Γ(𝑛𝛼+1)𝑎_𝑛𝑥^(𝑛+2)𝛼Γ)).((𝑛+2𝛼+1 (30)Comparing the coefficients of the same powers of 𝑥^𝛼, we get 𝑎₀=1,𝑎₁=1Γ(),𝑎1+𝛼₂𝑎=−⁰Γ(2𝛼+1),𝑎₃Γ()𝑎=−𝛼+1₁Γ(),⋮𝑎3𝛼+1_𝑛Γ))𝑎=−((𝑛−2𝛼+1_𝑛−2Γ(),𝑛𝛼+1 (31) and so on. Thus, the values of the coefficients can be calculated as follows: 𝑎₀=1,𝑎₁=1Γ(),𝑎1+𝛼₂1=−Γ(2𝛼+1),𝑎₃1=−Γ(),𝑎3𝛼+1₄=1Γ(4𝛼+1),𝑎₅=1Γ(),𝑎5𝛼+1₆1=−Γ(6𝛼+1),𝑎₇1=−Γ(),⋮7𝛼+1 (32) Hence, the local fractional series solution is given by 𝑢(𝑥)=^∞_𝑛=1𝑎_𝑛𝑥^𝑛𝛼=𝑥1−^3𝛼Γ()+𝑥3𝛼+1Γ(^5𝛼)−𝑥5𝛼+1Γ(^7𝛼)+𝑥7𝛼+1+⋯_𝛼Γ()−𝑥1+𝛼Γ(^2𝛼)+𝑥2𝛼+1Γ(^4𝛼)−𝑥4𝛼+1_6𝛼Γ(6𝛼+1)+⋯=cos_𝛼𝑥^𝛼+sin_𝛼𝑥^𝛼, (33) which are satisfied with the condition given by [34, 39] 𝐸_𝛼𝑖^𝛼𝑥^𝛼=cos_𝛼𝑥^𝛼+𝑖^𝛼sin_𝛼𝑥^𝛼, (34) where the Mittag-Leffler function defined on fractal set of fractal dimension 𝛼 is suggested by [34, 39] 𝐸_𝛼𝑥^𝛼=^∞_𝑘=0𝑥^𝛼𝑘Γ().1+𝑘𝛼 (35)

5. Conclusions

Local fractional differential and integral operators have proven to be useful tools to deal with everywhere continuous (but nowhere differentiable) functions in fractal areas ranging from fundamental science to engineering. In this paper, it is proven that a new Neumann series method can be used for solving the local fractional Fredholm and Volterra integral equations, and their solutions are fractal functions. The proposed method is efficient and leads to accurate, approximately convergent solutions to local fractional Fredholm and Volterra integral equations. It is demonstrated that the solutions of local fractional Fredholm and Volterra integral equations are fractal functions, which are equipped with local fractional continuities. However, the classical and fractional Neumann series methods [41–44] were only applied to continuous functions.

Appendix

The following properties of local fractional integral operator are valid [34]. (a)For any𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏),0<𝛼≤1, we have local fractional multiple integrals, which are written as [34] _𝑥0𝐼_𝑥^(𝑘𝛼)𝑓(𝑥)=^𝑘times_𝑥0𝐼_𝑥^(𝛼)…_𝑥0𝐼_𝑥^(𝛼)𝑓(𝑥). (A.1)(b)If 𝜓(𝑥,𝑦)∈𝐶_𝛼(𝑎,𝑏)×𝐶_𝛼(𝑐,𝑑), then [34] _𝑎𝐼_𝑏𝑐^(𝛼)𝐼_𝑏^(𝛼)𝜓𝑥,𝑦=_𝑐𝐼_𝑑𝑎^(𝛼)𝐼_𝑏^(𝛼)𝜓𝑥,𝑦. (A.2)(c)The sine and cosine subfunctions can, respectively, be written as follows [34, 39]: sin_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^{𝛼(2𝑘+1)}Γ[(,1+𝛼2𝑘+1)]cos_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^2𝛼𝑘Γ(1+2𝛼𝑘),0<𝛼≤1. (A.3)(d)Suppose that 𝑓(𝑡) is local fractional continuous on the interval [𝑎,𝑏]. Then _𝑎𝐼^(𝛼)_𝑥𝑎𝐼_𝜏^(𝛼)𝑓(𝑡)=_𝑎𝐼_𝑥^(𝛼)(𝑥−𝑡)^𝛼𝑓(𝑡)Γ()([.1+𝛼𝑥∈𝑎,𝑏]) (A.4)(e)We have ₀𝐼^(𝛼)_𝑥0𝐼_𝜏^(𝛼)𝑡^𝑘𝛼Γ()=𝑡𝑘𝛼+1Γ)^(𝑘+2)𝛼).((𝑘+2𝛼+1 (A.5)