# Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Material on Fractional Calculus

## 3. Multi-Step Methods for FDEs

- product-integration (PI) rules,
- fractional linear multi-step methods (FLMMs).

#### 3.1. Product-Integration Rules

#### 3.2. Fractional Linear Multi-Step Methods

`FLMM2.m`implementing three different FLMMs is also presented.

#### 3.3. Implicit vs. Explicit Methods

#### 3.4. Efficient Treatment of the Persistent Memory

## 4. Some Particular Families of FDEs and Systems of FDEs

#### 4.1. Multi-Order Systems of FDEs

#### 4.2. Linear Multi-Term FDEs

## 5. MATLAB Routines for Fractional-Order Problems

`f_fun`, require also the Jacobian ${\mathbf{J}}_{f}(t,y)$ of the right-hand side, namely the function handle

`J_fun`; this is necessary to solve the inner nonlinear equation by means of Newton–Raphson iterations as described in Section 3.3.

`[t, y] = FDE_PI1_Ex(alpha,f_fun,t0,T,y0,h,param)``[t, y] = FDE_PI1_Im(alpha,f_fun,J_fun,t0,T,y0,h,param,tol,itmax)``[t, y] = FDE_PI2_Im(alpha,f_fun,J_fun,t0,T,y0,h,param,tol,itmax)``[t, y] = FDE_PI12_PC(alpha,f_fun,t0,T,y0,h,param,mu,mu_tol)`

`alpha`must be a vector of the same size of the problem, while with a standard system (6),

`alpha`is a scalar value.

`[t, y] = MT_FDE_PI1_Ex(alpha,lambda,f_fun,t0,T,y0,h,param)``[t, y] = MT_FDE_PI1_Im(alpha,lambda,f_fun,J_fun,t0,T,y0,h,param,tol,itmax)``[t, y] = MT_FDE_PI2_Im(alpha,lambda,f_fun,J_fun,t0,T,y0,h,param,tol,itmax)``[t, y] = MT_FDE_PI12_PC(alpha,lambda,f_fun,t0,T,y0,h,param,mu,mu_tol)`

`f_fun`: function defining the right-hand side $f(t,y)$ or $f(t,y,param)$ of the equation;`param`denotes a possible optional parameter (or a set of parameters collected in a single vector);`J_fun`: Jacobian matrix (or derivative in the scalar case) of the right-hand side $f(t,y)$ of the equation (only for implicit methods) with respect to the second variable y; also, the Jacobian ${\mathbf{J}}_{f}(t,y)$ can have some parameters, namely ${\mathbf{J}}_{f}(t,y,param)$;`t0`and`T`: initial and final endpoints of the integration interval;`y0`: matrix of the initial conditions with the number of rows equal to the size of the problem and the number of columns equal to the smallest integer greater than $max\{{\alpha}_{1},\cdots ,{\alpha}_{Q}\}$;`h`: step-size for integration; it must be real and positive;`param`: (optional) vector of possible parameters for the evaluation of the vector field $f(t,y)$ and its Jacobian (if not necessary, this vector can be omitted or an empty vector`[ ]`can be used);`tol`: (optional) fixed tolerance for stopping the Newton–Raphson iterations when solving the internal system of nonlinear equations (only for implicit methods); when not specified, the default value ${10}^{-6}$ is assumed;`itmax`: (optional) maximum number of iterations for the Newton–Raphson method; when not specified, the default value`itmax = 100`is used;`mu`: (optional) number of corrector iterations (only for predictor-corrector methods); when not specified, the default value`mu = 1`is used;`mu_tol`: (optional) tolerance for testing the convergence of corrector iterations when`mu=Inf`; when not specified, the default value`mu_tol = ${10}^{-6}$`is used.

`t`: the vector of nodes on the interval $[{t}_{0},T]$ in which the numerical solution is evaluated;`y`: the matrix whose columns are the values of the solution evaluated in the points of`t`.

## 6. Some Applicative Examples

`alpha = 0.6; lambda = -10 ;``f_fun = @(t,y,lam) lam * y;``J_fun = @(t,y,lam) lam ;``param = lambda ;``t0 = 0 ; T = 5 ; y0 = 1 ; h = 2^(−5) ;`

`[t, y] = FDE_PI1_Ex(alpha,f_fun,t0,T,y0,h,param) ;``[t, y] = FDE_PI1_Im(alpha,f_fun,J_fun,t0,T,y0,h,param) ;``[t, y] = FDE_PI2_Im(alpha,f_fun,J_fun,t0,T,y0,h,param) ;``[t, y] = FDE_PI12_PC(alpha,f_fun,t0,T,y0,h,param).`

`f_fun = @(t,y,al) 40320/gamma(9-al)*t.^(8-al) - ...``3*gamma(5+al/2)/gamma(5-al/2)*t.^(4-al/2)+9/4*gamma(al+1) + ...``(3/2*t.^(al/2)-t.^4).^3 - y.^(3/2) ;``J_fun = @(t,y,al) -3/2.*y.^(1/2) ;``alpha = 0.5 ;``param = alpha ;``t0 = 0 ; T = 1 ;``y0 = 0 ;`

`alpha = [0.8,0.7] ;``A = 1 ; B = 3 ;``param = [ A , B ] ;``f_fun = @(t,y,par) [ ...``par(1) - (par(2)+1)*y(1) + y(1)^2*y(2) ; ...``par(2)*y(1) - y(1)^2*y(2) ] ;``J_fun = @(t,y,par) [ ...``-(par(2)+1) + 2*y(1)*y(2) , y(1)^2 ; ...``par(2) - 2*y(1)*y(2) , -y(1)^2 ] ;``t0 = 0 ; T = 100 ;``y0 = [ 1.2 ; 2.8] ;`

`alpha = [0.5, 0.2, 0.6] ;``f_fun = @(t,y) [ ...``(((y(2)-0.5).*(y(3)-0.3)).^(1/6) + sqrt(t))/sqrt(pi) ; ...``gamma(2.2)*(y(1)-1) ; ...``gamma(2.8)/gamma(2.2)*(y(2)-0.5) ] ;``J_fun = @(t,y) [ ...``0 , (y(2)-0.5).^(-5/6).*(y(3)-0.3).^(1/6)/6/sqrt(pi) , ...``(y(2)-0.5).^(1/6).*(y(3)-0.3).^(-5/6)/6/sqrt(pi) ; ...``gamma(2.2) , 0 , 0 ; ...``0 , gamma(2.8)/gamma(2.2) , 0 ] ;``t0 = 0 ; T = 5 ;``y0 = [ 1 ; 0.500000001 ; 0.300000001 ] ;`

`alpha = [2 3/2 0] ;``lambda = [1 2 1/2] ;``f_fun = @(t,y) t.^2 - y.^(3/2) ;``J_fun = @(t,y) -3/2*y.^(1/2) ;``t0 = 0 ; T = 5 ;``y0 = [0 , 0 ] ;`

`[t, y] = MT_FDE_PI1_Ex(alpha,lambda,f_fun,t0,T,y0,h)``[t, y] = MT_FDE_PI1_Im(alpha,lambda,f_fun,J_fun,t0,T,y0,h)``[t, y] = MT_FDE_PI2_Im(alpha,lambda,f_fun,J_fun,t0,T,y0,h)``[t, y] = MT_FDE_PI12_PC(alpha,lambda,f_fun,t0,T,y0,h) .`

`alpha = [3 2.5 2 1 0.5 0] ;``lambda = [1 1 1 4 1 4] ;``f_fun = @(t,y) 6*cos(t) ;``J_fun = @(t,y) 0 ;``t0 = 0 ; T = 100 ;``y0 = [1 , 1, -1 ] ;`

## 7. Conclusions

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FDE | Fractional differential equation |

ODE | Ordinary differential equation |

RL | Riemann–Liouville |

PI | Product integration |

FLLM | Fractional linear multi-step method |

LMM | Linear multi-step method |

PC | Predictor-corrector |

MOS | Multi-order system |

FFT | Fast Fourier transform |

VIE | Volterra integral equation |

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**Figure 1.**Scheme of the efficient algorithm for the convolution quadrature (9); squares ${S}_{p}$ represent the evaluation of partial sums of length p, and triangles ${T}_{r}$ represent convolution sums of fixed length r.

**Figure 2.**Behavior of the solution of the Brusselator multi-order system (MOS) (32) in the phase plane (

**a**) and in the $(t,x)$ and $(t,z)$ planes (

**b**).

**Table 1.**Some linear multi-step methods (LMMs) with corresponding polynomials $\rho \left(z\right)$ and $\sigma \left(z\right)$ and generating function $\delta \left(\xi \right)$.

Name | Formula | $\mathit{\rho}\left(\mathit{z}\right)$ | $\mathit{\sigma}\left(\mathit{z}\right)$ | $\mathit{\delta}\left(\mathit{\xi}\right)$ |
---|---|---|---|---|

BDF1 | ${y}_{n}={y}_{n-1}+h{f}_{n}$ | $z-1$ | z | $1-\xi $ |

BDF2 | ${y}_{n}-\frac{4}{3}{y}_{n-1}+\frac{1}{3}{y}_{n-2}=\frac{2}{3}h{f}_{n}$ | ${z}^{2}-\frac{4}{3}z+\frac{1}{3}$ | $\frac{2}{3}{z}^{2}$ | $\frac{3}{2}-2\xi +\frac{1}{2}{\xi}^{2}$ |

BDF3 | ${y}_{n}-\frac{18}{11}{y}_{n-1}+\frac{9}{11}{y}_{n-2}-\frac{2}{11}{y}_{n-3}=\frac{6}{11}h{f}_{n}$ | ${z}^{3}-\frac{18}{11}{z}^{2}+\frac{9}{11}z-\frac{2}{11}$ | $\frac{6}{11}{z}^{3}$ | $\frac{11}{6}-3\xi +\frac{3}{2}{\xi}^{2}-\frac{1}{3}{\xi}^{3}$ |

Trapez. | ${y}_{n}={y}_{n-1}+\frac{h}{2}\left({f}_{n-1}+{f}_{n}\right)$ | $z-1$ | $\frac{1}{2}z+\frac{1}{2}$ | $2\frac{1-\xi}{1+\xi}$ |

**Table 2.**MATLAB routines for some fractional-order problems. FDEs: fractional differential equations; PI: product-integration.

Name | Problem | Method |
---|---|---|

FDE_PI1_Ex.m | System of FDEs (6) or (23) | Explicit PI rectangular rule (10) |

FDE_PI1_Im.m | System of FDEs (6) or (23) | Implicit PI rectangular rule (11) |

FDE_PI2_Im.m | System of FDEs (6) or (23) | Implicit PI trapezoidal rule (12) |

FDE_PI12_PC.m | System of FDEs (6) or (23) | Predictor-corrector PI rules (13) |

MT_FDE_PI1_Ex.m | Multi-term FDE (25) | Explicit PI rectangular rule (27) |

MT_FDE_PI1_Im.m | Multi-term FDE (25) | Implicit PI rectangular rule (28) |

MT_FDE_PI2_Im.m | Multi-term FDE (25) | Implicit PI trapezoidal rule (29) |

MT_FDE_PI12_PC.m | Multi-term FDE (25) | Predictor-corrector PI rules |

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-2}$ | 7.52(12) | 6.80(−4) | 5.55(−4) | 5.43(21) | ||||

${2}^{-3}$ | 3.57(17) | **** | 3.31(−4) | 1.036 | 1.81(−4) | 1.614 | 2.57(27) | **** |

${2}^{-4}$ | 8.14(17) | **** | 1.63(−4) | 1.020 | 5.95(−5) | 1.609 | 7.87(21) | **** |

${2}^{-5}$ | 1.57(−1) | **** | 8.11(−5) | 1.011 | 1.95(−5) | 1.606 | 4.22(−4) | **** |

${2}^{-6}$ | 3.99(−5) | **** | 4.04(−5) | 1.006 | 6.43(−6) | 1.604 | 3.96(−5) | **** |

${2}^{-7}$ | 2.00(−5) | 0.997 | 2.01(−5) | 1.003 | 2.12(−6) | 1.602 | 8.90(−6) | 2.153 |

${2}^{-8}$ | 1.00(−5) | 0.998 | 1.01(−5) | 1.002 | 6.98(-7) | 1.602 | 2.43(−6) | 1.873 |

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-4}$ | 8.03(−2) | 7.55(−2) | 3.71(−3) | 3.56(−3) | ||||

${2}^{-5}$ | 3.85(−2) | 1.060 | 3.79(−2) | 0.997 | 1.04(−3) | 1.842 | 6.03(−4) | 2.560 |

${2}^{-6}$ | 1.89(−2) | 1.025 | 1.90(−2) | 0.998 | 2.76(−4) | 1.907 | 2.28(−4) | 1.407 |

${2}^{-7}$ | 9.40(−3) | 1.009 | 9.48(−3) | 1.000 | 7.19(−5) | 1.941 | 1.04(−4) | 1.135 |

${2}^{-8}$ | 4.69(−3) | 1.002 | 4.74(−3) | 1.001 | 1.85(−5) | 1.961 | 4.50(−5) | 1.204 |

${2}^{-9}$ | 2.35(−3) | 1.000 | 2.37(−3) | 1.001 | 4.70(−6) | 1.974 | 1.83(−5) | 1.294 |

${2}^{-10}$ | 1.17(−3) | 0.999 | 1.18(−3) | 1.002 | 1.19(−6) | 1.982 | 7.15(−6) | 1.359 |

**Table 5.**Errors and EOC at $T=100.0$ for the Brusselator system of FDEs (32) with $({\alpha}_{1},{\alpha}_{2})=(0.8,0.7)$, $(A,B)=(1.0,3.0)$ and $({x}_{0},{z}_{0})=(1.2,2.8)$.

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-2}$ | 4.64(−1) | 1.03(0) | 4.90(−2) | 1.16(0) | ||||

${2}^{-3}$ | 2.32(−1) | 0.996 | 5.20(−1) | 0.988 | 7.84(−3) | 2.643 | 2.92(−1) | 1.994 |

${2}^{-4}$ | 1.22(−1) | 0.926 | 2.25(−1) | 1.211 | 2.85(−3) | 1.460 | 5.80(−2) | 2.333 |

${2}^{-5}$ | 6.86(−2) | 0.834 | 9.84(−2) | 1.191 | 7.63(−4) | 1.903 | 1.28(−2) | 2.179 |

${2}^{-6}$ | 3.69(−2) | 0.896 | 4.52(−2) | 1.124 | 1.92(−4) | 1.991 | 3.41(−3) | 1.910 |

${2}^{-7}$ | 1.92(−2) | 0.941 | 2.15(−2) | 1.071 | 4.60(−5) | 2.060 | 1.01(−3) | 1.758 |

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-2}$ | 2.56(−1) | 1.37(−1) | 7.30(−3) | 7.84(−2) | ||||

${2}^{-3}$ | 1.31(−1) | 0.963 | 7.41(−2) | 0.892 | 3.16(−3) | 1.210 | 3.50(−2) | 1.163 |

${2}^{-4}$ | 6.60(−2) | 0.992 | 3.95(−2) | 0.905 | 1.35(−3) | 1.227 | 1.56(−2) | 1.171 |

${2}^{-5}$ | 3.29(−2) | 1.005 | 2.09(−2) | 0.918 | 5.72(−4) | 1.238 | 6.89(−3) | 1.175 |

${2}^{-6}$ | 1.63(−2) | 1.011 | 1.10(−2) | 0.930 | 2.41(−4) | 1.245 | 3.04(−3) | 1.178 |

${2}^{-7}$ | 8.09(−3) | 1.013 | 5.72(−3) | 0.940 | 1.01(−4) | 1.250 | 1.34(−3) | 1.180 |

**Table 7.**Errors and EOC at $T=5.0$ for the multi-term FDE (34) with $(a,b)=(2.0,0.5)$, $y\left(0\right)=0$ and ${y}^{\prime}\left(0\right)=0$.

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-2}$ | 3.52(−2) | 8.17(−2) | 2.72(−4) | 8.53(−2) | ||||

${2}^{-3}$ | 2.16(−2) | 0.704 | 3.94(−2) | 1.053 | 7.03(−5) | 1.950 | 2.36(−2) | 1.855 |

${2}^{-4}$ | 1.22(−2) | 0.826 | 1.88(−2) | 1.067 | 1.75(−5) | 2.005 | 7.21(−3) | 1.709 |

${2}^{-5}$ | 6.58(−3) | 0.888 | 9.00(−3) | 1.063 | 4.30(−6) | 2.026 | 2.36(−3) | 1.609 |

${2}^{-6}$ | 3.47(−3) | 0.925 | 4.34(−3) | 1.052 | 1.04(−6) | 2.046 | 8.00(−4) | 1.564 |

${2}^{-7}$ | 1.80(−3) | 0.948 | 2.11(−3) | 1.041 | 2.46(-7) | 2.082 | 2.75(−4) | 1.540 |

PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | |||||
---|---|---|---|---|---|---|---|---|

h | Error | EOC | Error | EOC | Error | EOC | Error | EOC |

${2}^{-2}$ | 2.23(−2) | 3.07(−2) | 1.69(−3) | 2.20(−2) | ||||

${2}^{-3}$ | 1.03(−2) | 1.120 | 1.34(−2) | 1.199 | 4.04(−4) | 2.062 | 4.35(−3) | 2.335 |

${2}^{-4}$ | 4.33(−3) | 1.244 | 6.16(−3) | 1.119 | 9.84(−5) | 2.036 | 1.24(−3) | 1.808 |

${2}^{-5}$ | 2.29(−3) | 0.918 | 2.92(−3) | 1.079 | 2.42(−5) | 2.024 | 3.98(−4) | 1.642 |

${2}^{-6}$ | 1.20(−3) | 0.934 | 1.40(−3) | 1.055 | 5.97(−6) | 2.018 | 1.34(−4) | 1.575 |

${2}^{-7}$ | 6.18(−4) | 0.959 | 6.84(−4) | 1.036 | 1.50(−6) | 1.993 | 4.58(−5) | 1.544 |

h | N | PI 1 Expl | PI 1 Impl. | PI 2 Impl. | PI P.C. | N (log_{2}N)^{2} |
---|---|---|---|---|---|---|

${2}^{-2}$ | 20,000 | 1.1224 | 1.8754 | 1.9601 | 2.6078 | 4.08 × 10^{6} |

${2}^{-3}$ | 40,000 | 2.0545 | 3.6506 | 3.8378 | 5.1718 | 9.35 × 10^{6} |

${2}^{-4}$ | 80,000 | 4.1367 | 7.3493 | 7.6927 | 10.4410 | 2.12 × 10^{7} |

${2}^{-5}$ | 160,000 | 8.3460 | 14.7751 | 15.4799 | 21.0344 | 4.78 × 10^{7} |

${2}^{-6}$ | 320,000 | 16.9107 | 29.6091 | 30.9253 | 42.3168 | 1.07 × 10^{8} |

${2}^{-7}$ | 640,000 | 34.7736 | 60.1003 | 62.8137 | 86.0035 | 2.38 × 10^{8} |

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**MDPI and ACS Style**

Garrappa, R.
Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. *Mathematics* **2018**, *6*, 16.
https://doi.org/10.3390/math6020016

**AMA Style**

Garrappa R.
Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. *Mathematics*. 2018; 6(2):16.
https://doi.org/10.3390/math6020016

**Chicago/Turabian Style**

Garrappa, Roberto.
2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial" *Mathematics* 6, no. 2: 16.
https://doi.org/10.3390/math6020016