1. Introduction

In the past decades, fractional differential equations have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering ^{[2, 10, 12, 13]}. Most fractional differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. The fractional spline function of a polynomial form (see ^{[5, 6, 7, 8]}) is a new approach to provide an analytical approximation to linear and nonlinear problems, and it is particularly valuable as a tool for scientists and applied mathematicians, because they provide immediate and visible symbolic terms of numerical approximate solutions to both linear and nonlinear differential equations.

In this work we construct a new fractional spline which interpolates the (1/2-th derivative for the first case, 1/2, 3/2 -th derivatives for the second case, and 1/2, 3/2, 5/2-th derivatives for the last case of a given function at the knots and its value at the beginning of the interval considered. We obtain a direct simple formula for the proposed fractional spline. error bounds for the function is derived in the sense of the Hermite interpolation. To illustrate the efficiency and the error analysis two numerical examples are considered.

2. Preliminaries

In this section, we recall some relevant definitions. There are many ways to define fractional integral and derivative. In this paper we will use Riemann-Liouville fractional integral and Caputo fractional derivative.

Let α be a positive real number and f(x) be a function defined on the right side of a, then Definition 1. ^{[1, 2, 10]} The Riemann-Liouville fractional integral of order α > 0 is defined by

where is the gamma function.

Definition 2. ^{[1, 2, 10]} The Caputo fractional derivative of order α > 0 is defined by

3. Description of the Fractional Splines

We construct here a class of interpolating fractional splines of degree jα, for j = 2, 4, 6 and α = 0.5. error estimates for these splines are also represented. Since all cases considered are similar, details are given only for the first case of 2α.

Let 0 = x₀ < x₁ < … < x_n-1 < x_n = 1 be a uniform partition of . Set the stepsize h = x_i+1 - x_i (i = 0(1)n) and note that

If g is a real-valued function in [0, 1], then g_i stands for g(x_i) (i = 0(1)n). Since all cases considered are similar, details are given only for the first case. We have the following cases:

We suppose that s^(1/2)(x) ∈ C² and s(x) in each subinterval [x_i, x_i+1] has a form:

(1)

where a_i, b_i and c_i are constants to be determined.

Theorem 1. suppose that s^(1/2)(x) ∈ C² and s(x) in each subinterval [x_i, x_i+1] has the form (1). Given the real numbers s(1=2) i = f(1=2) i (i = 0(1)n) and f₀, there exist a unique s(x) such that

(2)

The fractional spline which satisfies (2) in [x_i, x_i+1] is of the form:

(3)

where

(4)

and x = x_i + th, t∈, with a similar expression for s(x) in [x_i-1, x_i].

The coefficient s_i in (3) are given by the recurrence formula:

(5)

Proof. Indeed we can express any p(t) in in the following form:

To determine , we write the above equality for we get

Solving this we obtain

Now for a fixed , set . In the subinterval the fractional spline satisfying (2) is:

We have a similar expression for in . From the continuity condition of we arrive the above recurrence formula (5). This completes the proof.

3.2. Error Bounds for the Fractional Spline of Degree Case

In this section, the error estimates are presented for the above interpolating fractional spline in using one of the best theorem of the Hermite interpolation (Theorem 2). Note that denotes the norm.

Theorem 2. ^{[4, 8, 9]} Let be given. Let be the unique Hermite interpolation polynomial of degree that matches g and its first derivatives at 0 and h. Then

(6)

where

(7)

The bounds in (6) are best possible for only.

Theorem 3. Suppose that be the fractional spline defined in section 3.1, and that , then for any we have

(8)

Proof. Since we have is the Hermite interpolation polynomial of degree 1 matching at . So for any we have using (6) with

From which, we get

This gives

since, and then the last equation becomes

and since , following ^[11], p. 20, we have

which leads to

Thus we have proved the theorem.

3.3. Spline of Degree Case (Existence and Uniqueness)

We suppose that and in each subinterval has a form:

(9)

From which the following theorem can be obtained:

Theorem 4. Suppose that and in each subinterval has a form (1). Given the real numbers and , there exist a unique such that

(10)

The fractional spline which satisfies (10) in is of the form:

(11)

where

(12)

and , with a similar expression for in .

The coefficient in (11) are given by the recurrence formula:

(13)

Proof. In this case we can express any p(t) in [0,1] in the following form:

and to determine the coefficients , we write the above equality for .

By the same technique of theorem 1 we obtain the desired result and consequently the proof is completed.

3.4. Error Bounds for the Fractional Spline of Degree Case

Here we will derive the error estimates are presented for the fractional spline that we have mentioned in Section 3.3, the error bounds have shown in the below theorem and its proof is similar subsequence of theorem 3.

Theorem 5. Suppose that be the fractional spline defined in section 3.3, and that then for any we have

(14)

Proof. Because is the Hermite interpolation polynomial of degree 2 matching at . So for any we have using (6) with

and following ^[11], we have

which leads to

Which proves the theorem.

3.5. Spline of Degree Case (Existence and Uniqueness)

We suppose that and in each subinterval has a form:

(15)

Which deduces the following theorem:

Theorem 6. Let be the fractional spline defined in Section 3.5. Given the real numbers and , there exist a unique such that

(16)

The fractional spline which satisfies (16) in is of the form:

(17)

where

(18)

and , with a similar expression for in .

The coefficient in (17) are given by the recurrence formula:

(19)

Proof. In this case we can express any in in the following form:

and to determine the coeficients , we write the above equality for . By the same technique of theorem 1 we obtain the desired result and consequently the proof is completed.

Error estimates for the fractional spline that we have mentioned in Section 3.5 are explained by the following theorem:

Theorem 5. Suppose that be the fractional spline defined in section 3.3, and that then for any we have

(20)

Proof. Because is the Hermite interpolation polynomial of degree 3 matching at . So for any we have using (6) with

and following ^[11], we have

This gives

Which proves the theorem.

4. Algorithms

The following remarks are needed in solving a problem:

1. Note that the above formulation and analysis was done in . However, this does not constitute a serious restriction since the same techniques can be carried out for the general interval . This is achieved using the linear transformation

(21)

Form to .

2. Use the equations (5), (13) and (19) to compute , respectively, in each cases.

3. Use the equations (3), (11) and (17) to compute at n equally spaced points in each subinterval and in each cases.

5. Illustrations Results

To illustrate our methods as error estimates has been found in theorems (3, 5 and 7) and to compare each of them with the other one, we have solved two examples of fractional equation. We have implemented all of problems’ calculations with MATLAB 12b.

Example 1. Consider the following fractional differential equation

(22)

with

For which, all actual error bounds for each cases are presented in Table 1,

Table 1. The observed maximum errors

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Example 2. Let

(23)

For which, all actual error bounds for each cases are presented in Table 2.

Table 2. The observed maximum errors

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Example 3. Consider the following fractional differential equation

(24)

Numerical and exact solutions are presented in Table 3. we give here the fractional spline of degree 6a for . Also, the exact and numerical solutions are demonstrated in Figure 1. for .

Figure 1. Exact and approximate solutions of Example 3 with h=0.2

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Table 3. Exact, approximate and absolute error

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6. Conclusions

In this paper, the existence and uniqueness of three fractional splines of degree are derived and in each case we have obtained direct simple formulas. These formulas are agree with those obtained for degree of integer, such as in ^[3], where a different approach was used. Also, Error estimates are derived which, together with the numerical results, showed the method to be efficient.

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