## 1. Introduction

We begin by defining the Caputo fractional time-derivative [9,10] of a given function as

which is a fractional derivative of order . In [2], the use of the Laplace transform was applied to a Caputo fractional derivative term to obtain the fractional integral term

(1) |

which was studied numerically and convergent schemes were developed for this integral inspired by the works presented in [9-13], [15-16]. The integral (1) can be expressed as the convolution , where . Our work will examine and derive numerical schemes to discretize integrals of the form (1), which has numerous engineering and physics applications, see [11], [12] and [16]. One of the major advantages of applying the Laplace Transform to a fractional derivative term, as seen in [2], is the ability to preserve the same assumption of regularity as required for fractional derivative discretizations and recovering an extra order of accuracy, where denotes the time step size. Further, we now have the ability to relax the regularity assumption from requiring the objective function under the usual L1-method to instead be any , where and . This is achieved by a usual Taylor series expansion to obtain convergence results for whole number values of , and by utilizing a fractional Taylor series expansion to approximate functions with a fractional order of regularity, see [15]. By requiring more regularity, we are able to obtain a higher order of convergence, as seen in Theorems 3.6 and 3.7. This method naturally generalizes to any convolution type-quadrature where the kernel function is positive, decreasing, and satisfies , as seen in Theorems 3.4 and 3.5.

The remainder of the paper is organized as follows. Section 2 will provide discretizations for fractional integrals of the above form, and a general scheme is established for fractional integrals of other forms based on the integral kernel. We obtain general schemes of orders up to order of accuracy. Section 3 establishes all of the necessary consistency, stability, and convergence results for each of these schemes, in addition to a discussion of the implementation of the schemes. We prove optimal order of convergence of our stable schemes, where the order is at least 3. The instability of schemes of order 6 and above are presented as well. The main convergence results are featured in Theorems 3.4 through 3.6. Section 4 presents two fractional integral equations as numerical examples that validate our findings. Future works will consider the application of these methods to fractional order diffusion processes based on the validity of the schemes and the order of accuracy they can provide.

## 2. Discretized numerical schemes

In order to discretize the Caputo fractional integral (1), we must consider the Taylor expansion for a given function at the arbitrary point , the usual time interval over which the integral is considered. That is, the Taylor expansion centered at the point for on , given , is

(2) |

Define and let and such that . From the above, similar Taylor expansions centered at any given can be constructed for each of the previous points . That is,

(3) | ||||

(4) | ||||

(5) | ||||

(6) | ||||

(7) |

Thus, the -th order approximation of for any at the point requires we solve the linear system of equations:

(8) | ||||

(9) |

Each of the terms are then replaced by their Taylor expansions about the point and then solved, after neglecting the higher order terms of and . For example, a second order approximation of is provided in [2], Theorem 3.2, which can be recovered by solving the equation

This equation can be rewritten as a system of equations

Which yields the solution , . Therefore, we may numerically approximate the integral as seen in [2] using and as solved for above:

which recovers the equation that was studied in greater detail in [2]. We remark that under this construction, we satisfy the condition . This directly implies that the coefficients and presented above are positive. We now provide the values of the coefficients for each scheme up to 4th order accuracy. Higher order schemes can be derived using the generalized system of equations (8) and (9). We remark that in general, for each i.

First order accurate:

Second order accurate:

Third order accurate:

Fourth order accurate:

We present the generalized equation to solve for a n-th order accurate approximation to . The above equations may be rewritten by the matrix equation

(10) |

The first matrix is, in fact, the transpose of the usual Vandermonde matrix [14] where each entry, other than the first row of ones, is a multiple of the time partition . That is, we set , , , … so that

(11) |

where is the Vandermonde matrix with each expressed in terms of as above, and

The following lemma asserts the existence of any general n-th order numerical scheme. Since we have

provided that which directly implies that the matrix is invertible under this condition. The following lemma is immediate from the above.

###### Lemma 2.1.

Let . Then, (11) has a unique solution for each .

We now compute the unique solution based on the previous lemma. From [14], we can establish the generalized inverse of the Vandermonde matrix.

###### Theorem 2.2.

Let . Then, (11) has a unique solution for each , with the solution

(12) |

###### Proof.

###### Remark 2.3.

By utilizing the fractional Taylor series expansion instead for on , as discussed in [15], we may obtain similar results to those outlined in Theorem 2.2. This can further relax the regularity assumption to .

Using the fractional Taylor series expansion, we can assert that we have an order scheme defined by the following:

order accurate:

We now examine the consistency, stability, and convergence of these schemes based on the generalized scheme

(15) |

## 3. Numerical Consistency, Stability, and Convergence

### 3.1. Numerical Consistency and Stability

We motivate our discussion of stability by examining the results presented in [1]. The quadrature studied in [1] is of the form

(16) |

which provides the error estimate given

a sequence of constants, , such thatWe will directly compare these results to the ones established in the previous section to prove stability and assert convergence. Our goal is to decompose the integrand into a convolution integral , where we may relax the continuity assumptions on the kernel function . This goal is motivated in part from the results obtained in [2], where we seek a generalization for the integral kernel. We begin by recalling some basic definitions for quadrature methods. From (1.15) of [1], a quadrature method is said to be consistent if it satisfies

for the global quadrature of the integral (16). We will relate (16) and a generalization of the results provided in [2].

###### Lemma 3.1.

Let for any prescribed . Let denote the order of the desired approximation to the function , let such that and . Then, for an order scheme, as described in Theorem 2.2, we have

(17) |

###### Proof.

The following remark is a natural extension of the first lemma, which allows for direct comparison to prove stability using the Theorem 3.7 in [1].

###### Remark 3.2.

By expanding the series

and by collecting all of the repeating terms for each , we may condense the double summation into a single summation term

(23) |

where we note that to satisfy the previous lemma. Further, by defining for fixed

(24) |

We arrive at a form identical to the generalized quadrature rule posed in [1]

(25) |

###### Theorem 3.3.

The approximation scheme (25) is consistent for any , where is the order of approximation.

###### Proof.

From the consistency requirement in [1], we must show that the scheme (25) satisfies any time step

(26) |

for any fixed . That is, we have from Remark 3.2

(27) | ||||

(28) | ||||

(29) | ||||

(30) |

From (11), the first equation in the Vandermonde matrix requires , hence,

(31) |

On the other hand, by relabelling the coefficients of (26) and by noting that ,

(32) | ||||

(33) |

By equating (25) and (33), we have

(34) |

Since each is arbitrary under this construction, we select to satisfy . Thus, we have for the scheme (25)

(35) |

hence the scheme (25) is consistent. For simplicity and for implementation, we take for each k to trivially satisfy these conditions since . ∎

We must further satisfy stability requirements in order to prove the convergence of these schemes for any order . From [1], we have the following theorem asserting stability under arbitrary quadrature rules:

###### Theorem.

(3.7 of [1]) The stability polynomial

(36) |

is Schur, if . Assuming each and satisfy , the recurrence for

when for is stable whenever , given .

We remark that under these results, we must simply satisfy the requirement that each in (25) to satisfy a similar stability criterion for this generalized quadrature. This leads immediately to two stability results:

###### Theorem 3.4.

Let be a positive kernel on and let for all k. Then, the approximation scheme (25) is stable for and , where is the order of approximation.

###### Proof.

The case where is immediate since , hence . For , then

(37) | ||||

(38) | ||||

(39) | ||||

(40) | ||||

(41) | ||||

(42) |

Using similar analysis we are able to come to the same conclusion for and , given . Therefore, when , the scheme (25) is stable. ∎

We require additional assumptions on the integral kernel to ensure that the scheme is stable in the case where the order of approximation to (25) is any order .

###### Theorem 3.5.

Let be a positive, nonincreasing kernel on and let for all k. The approximation scheme (25) is stable for any order of accuracy.

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