Spectral Methods for Fractional Differential Equations Chinenye Assumpta Nnakenyi (
[email protected]) African Institute for Mathematical Sciences (AIMS) Supervised by: Prof J.A.C. Weideman Stellenbosch University, South Africa Dr. N. Hale Stellenbosch University, South Africa
21 May 2015 Submitted in partial fulfillment of a structured masters degree at AIMS South Africa
Abstract This project focuses on the spectral collocation method for fractional differential equations. Orthogonal polynomials such as Jacobi and Legendre are used alongside Chebyshev points in the derivation of the spectral differentiation matrices for fractional differential equations and spectral differentiation matrices for ordinary differential equations. We apply the method to some applications, including a model of the motion of a gas in a fluid, and a model of the motion of a large thin plate immersed in a Newtonian fluid. We find that for some problems we obtain exponential convergence, but for others only algebraic convergence.
Declaration I, the undersigned, hereby declare that the work contained in this research project is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.
Chinenye Assumpta Nnakenyi, 21 May 2015 i
Contents Abstract
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1 Introduction. 1.1 Background and History. 1.2 Aims and Objectives. . . 1.3 Literature Review. . . . . 1.4 Organization of work. . .
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1 1 1 2 2
2 Preliminaries. 2.1 Fractional Calculus. . . . 2.2 Orthogonal Polynomials. 2.3 Polynomial Interpolation. 2.4 Spectral Methods. . . . . 2.5 Chebyshev Points. . . . . 2.6 Orders of convergence. .
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3 . 3 . 4 . 7 . 8 . 9 . 10
(ODEs). . . . . . . . . . . . . . . . . . . . . . . . .
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12 12 14 14 15
4 Spectral Collocation for Fractional Differential Equations (FDEs). 4.1 Fractional Lagrange Interpolants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fractional Differentiation Matrix Dµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fractional Differentiation Matrix D1+µ . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 16 17 19
3 Spectral Collocation for Ordinary Differential Equations 3.1 Differentiation Matrix. . . . . . . . . . . . . . . . . . . 3.2 Higher Derivatives. . . . . . . . . . . . . . . . . . . . 3.3 Solving Boundary Value Problems (BVPs). . . . . . . . 3.4 Example. . . . . . . . . . . . . . . . . . . . . . . . . .
5 Applications. 22 5.1 Model of gas in a fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Bagley–Torvik Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6 Conclusion and Further work. 28 6.1 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Further work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A Differentiation matrices.
29
References
35
ii
1. Introduction. Recently, there has been considerable interest in fractional differential equations (FDEs) in physical systems like viscous fluid flows, fractured or porous media, bioengineering applications (Magin, 2004) and viscoelastic materials (Chatterjee, 2005). Space fractional derivatives represent super-diffusion and appear in modelling epidemics where concentrations can take rare but large jumps (Zhang et al., 2009). Contrary to differential equations of integer order in which derivatives depend only on the local behaviour of functions, FDEs take the global information in a weighted form Ghoreishi and Yazdani (2011). Time fractional derivatives, which represent anomalous diffusion, are used to model processes with a memory like neural synapse responses and DNA sequences (Kilbas et al., 2006).
1.1
Background and History.
Fractional calculus was developed by Leibniz in 1695 alongside with L’Hˆopital as seen in a letter Leibniz wrote to L’Hˆopital suggesting the idea of fractional derivates. This stemmed from a question where dn y L’Hˆopital asked what the derivative is, if n equals 21 (Leibniz, 1849). Later on, mathematicians dxn like Euler, Lagrange, Liouville, Gr¨ unwald and Fourier worked rigorously on the formal foundations of fractional calculus. These fractional integrals include complex orders of differintegrals and left and right derivatives (Kisela, 2008). Spectral methods are techniques that are numerically used to solve certain differential equations in applied mathematics. In spectral methods, we aim at writing the solution of a differential equation as a sum of a basis functions. The coefficients of the sum are chosen to satisfy the solution of the differential equation. Spectral methods were introduced by Lanczos (1938). Lanczos showed the power of Fourier series and Chebyshev polynomials in a number of problems where they had not been used before. Later in the 1970s, Orszag introduced spectral methods again, alongside Kreiss, Oliger and others, for the purpose of solving partial differential equations in fluid mechanics (Trefethen, 1996). Finite difference methods and finite element methods represent functions locally, that is, over a small sub-domain, whereas spectral methods use basis functions that are non-zero globally (over the whole domain). Spectral methods often exhibit exponential convergence. This rate of convergence yields the ability to achieve high precision with a small number of points (spectral accuracy). Computationally, spectral methods can be less expensive than finite difference and finite element methods (Costa, 2004; Trefethen, 1996).
1.2
Aims and Objectives.
The aim of this project is to use the spectral collocation method to solve FDEs as well as ordinary differential equations (ODEs). We also compare the solutions of the FDEs with their ODEs counterpart.
1
Section 1.3. Literature Review.
1.3
Page 2
Literature Review.
The analytic solutions of some FDEs are not known, therefore, approximations and numerical methods are commonly applied (Zhang et al., 2012). Diethelm et al. (2002) presented a predictor-corrector approach for numerical solutions of FDEs. Doha et al. (2012) developed a new Jacobi operational matrix for solving multi-term FDEs. Lubich (1983, 1986) introduced the idea of discretized fractional calculus with a finite difference method. The finite difference method is a local approach whereas fractional derivatives are global differential operators. Although the implementation of a finite difference method is easy, the challenges are limited accuracy and high cost of computing the long-range memory in the discretization of the fractional derivatives at each point (Zayernouri and Karniadakis, 2014). To address this problem, global schemes such as spectral methods are more efficient for discretizing fractional differential equations (Zayernouri and Karniadakis, 2014). Spectral methods are widely used in solving differential equations and variational problems (Canuto et al., 1988). They give very accurate approximations for a smooth solution with relatively few degrees of freedom. Recently, Doha et al. (2011) developed an efficient Chebyshev spectral method for solving multi-term fractional order differential equations. Esmaeili and Shamsi (2011) introduced a direct solution of a special family of fractional initial value problems using a pseudo-spectral method. Li and Xu (2009, 2010) developed a time-fractional diffusion equation in which they achieved exponential convergence. Collocation schemes for fractional equations are relatively easy to implement. They overcome the difficulty in the treatment of nonlinear fractional partial differential equations and multi-term fractional partial differential equations. Khader (2011) presented a Chebyshev collocation method for the discretization of the space-fractional diffusion equation. In this project, we mostly focus on the spectral collocation technique introduced by Zayernouri and Karniadakis (2014).
1.4
Organization of work.
The organization of this project is as follows: In Chapter 2, we recap some basic definitions and important concepts. Spectral collocation for ODEs is introduced in Chapter 3 as well as the derivation of the ordinary differentiation matrix. Thereafter, in Chapter 4, spectral collocation for FDEs is discussed and we derive the fractional differentiation matrix. Chapter 5 is devoted to applications of our method. In Chapter 6, we conclude and give the further work. We give an appendix of the spectral differentiation matrices code in Python software.
2. Preliminaries. In this chapter, we build up our work by looking into fractional calculus and discussing what it entails. After that, we define orthogonal polynomials and give examples that will be important in subsequent chapters. Consequently, we consider polynomial interpolation and introduce the associated Vandermonde matrices. Later on, we give an overview of spectral methods and finally we discuss Chebyshev points and see why they are important.
2.1
Fractional Calculus.
In fractional calculus, we are concerned with real orders of the differential or integral operator. Hence, the name fractional calculus is a misnomer as it encompasses more than just fractions. In this section, we define the gamma function and consider the definition of fractional derivatives in the sense of Riemann–Liouville and Caputo. Thereafter, we see how Riemann–Liouville and Caputo derivatives are related. These definitions are based on Podlubny (1998). 2.1.1 Gamma function. The gamma function as we shall see later is important in the definition of fractional derivatives. For a number x in R, the gamma function is defined as Z ∞ e−t tx−1 dt . (2.1.1) Γ(x) = 0
A fundamental property of gamma function is x ∈ N.
Γ(x + 1) = xΓ(x) = x!
(2.1.2)
This generalizes the factorial x! and allows x to take non-integer values. 2.1.2 Riemann–Liouville fractional derivatives. The Riemann–Liouville fractional derivatives is the most commonly used definition of fractional derivatives. The left-sided and right-sided Riemann–Liouville fractional derivatives of order µ, when 0 < µ < 1, are defined as Z x 1 d f (s) µ (RL D f )(x) = ds, x > −1 (2.1.3) −1 x Γ(1 − µ) dx −1 (x − s)µ and µ (RL x D1
1 f )(x) = Γ(1 − µ)
−d dx
Z x
1
f (s) ds, (s − x)µ
x −1, then a useful property of the Riemann–Liouville fractional derivative is RL q RL p RL p RL q RL p+q (2.1.5) −1 Dx f (x) = −1 Dx −1 Dx f (x) = −1 Dx −1 Dx f (x).
3
Section 2.2. Orthogonal Polynomials.
Page 4
2.1.3 Caputo fractional derivatives. The left-sided and right-sided Caputo fractional derivatives of order µ, when 0 < µ < 1, are defined as Z x 1 f 0 (s) µ (C D f )(x) = ds, x > −1 (2.1.6) −1 x Γ(1 − µ) −1 (x − s)µ and 1 Γ(1 − µ)
µ (C x D1 f )(x) =
Z
1
x
f 0 (s) ds, (s − x)µ
x . . . > x0 = 1 (that is, the order of the nodes are reversed). Alternatively, the diagonal entries of the differentiation matrix D can be computed in a way that represents exactly the derivative of a constant as: Djj = −
n X
Dij .
j=0 j6=i
The equation (3.1.13), is often referred to as the “negative sum trick”.
(3.1.13)
Section 3.2. Higher Derivatives.
3.2
Page 14
Higher Derivatives.
We now consider higher order derivatives for our differentiation matrices. Suppose we have an ODE u00 (x)+u0 (x) + u = g(x) ,
(3.2.1)
u(±1) = 0 ,
(3.2.2)
then, an approximation to u00 (x) by the differentiation matrix is u00 (x) ≈ D2 u(x) = D × D u(x) . Generally, for any nth order derivative, we have un (x) ≈ Dn u(x) = D × D × · · · × D u(x) . {z } | n times
We now proceed to solve (3.2.1)-(3.2.2) which is a boundary value problem.
3.3
Solving Boundary Value Problems (BVPs).
In solving BVPs like (3.2.1) by differentiation matrices, we implement the homogeneous Dirichlet boundary conditions (3.2.2) as follows: • The boundary condition u(−1) is treated with respect to the first row of the differentiation matrix, and • the boundary condition u(1) is treated with respect to the last row of the differentiation matrix. For example, in the case of (3.2.1), we have the approximation to the ODE to be D2 u(x) + Du(x) + u(x) = g(x) , which implies (D2 + D + I) u(x) = g(x) . {z } | A
In matrix form, we have: Now, using the boundary condition 1 0 0 0
A
u(−1) g 1 g2 .. = .. . . . gN −1 gN u(1)
u(−1) = 0 and u(1) = 0, we now have our matrices to be u(−1) 0 0 ··· 0 0 g2 .. . .. A¯ = , . gN −1 ··· 0 0 1 0 u(1)
Section 3.4. Example.
Page 15
where A¯ is A with the first and last rows removed. Since g1 and gN are zeros, we can delete the first and last rows and columns of matrix A. Using what is left, we then solve for the unknown solutions. Note that there are other ways to implement boundary conditions as seen in (Driscoll and Hale, 2015). Let us now demonstrate how differentiation matrices are used to solve ODEs.
3.4
Example.
Let us consider the second order linear BVP u00 (x) + u(x) = e−2x ,
−1 < x < 1 ,
(3.4.1)
u(±1) = 0 .
(3.4.2)
Solving (3.4.1)-(3.4.2) analytically, we obtain the exact solution u(x) =
− cosh(2) cos(x) sin(1) + sinh(2) sin(x) cos(1) + e−2x cos(1) sin(1) . 5 cos(1) sin(1)
(3.4.3)
We want to solve this problem numerically by the approximation D2 u(x) + u(x) = e−2x −2x (D2 + I) u(x) = e|{z} | {z } c
L
L u(x) = c ,
(3.4.4)
where L is an (n + 1) × (n + 1) matrix. The solution u(x) is obtained by solving the linear system (3.4.4). We use Python to obtain the graphical solutions shown in Figure 3.1 (left) for n = 50. Figure 3.1 (right) shows the convergence of the approximation as the number of points is increased. With fifteen points, we get around fourteen digits of accuracy. After this, convergence stagnates because of rounding errors. 0.2
max err = 4.396e-14
10
0.0
Error
10-5
L∞ error
10-6
−0.4
u(x)
e−3n
-3
10-4
−0.2
−0.6
10-7 10-8 10-9
10-10
−0.8
10-11
−1.0
10-12 10-13
Exact Approximate
−1.2 −1.4 −1.0
Log-Linear
10-2
−0.5
0.0
x
0.5
10-14 1.0
10-15
0
10
20
n
30
40
50
Figure 3.1: Left: Solution of (3.4.1)-(3.4.2) using Chebyshev points. Right: Convergence of the approximation as we increase the number of points. We note that the rate of convergence is geometric.
4. Spectral Collocation for Fractional Differential Equations (FDEs). We now consider spectral collocation for FDEs. We start by looking into fractional Lagrange interpolants, thereafter we derive our fractional differentiation matrices Dµ .
4.1
Fractional Lagrange Interpolants.
For our fractional Lagrange interpolants, we start by looking at the work by Zayernouri and Karniadakis (2014). They develop a fractional collocation scheme for the problem as follows: τ 0 Dt u(x, t)
= Lv u(x, t),
x ∈ [−1, 1],
t ∈ [0, T ],
(4.1.1)
u(x, 0) =g(x), u(−1, t) =0,
v ∈ (0, 1) ,
u(−1, t) =u(1, t) = 0,
v ∈ (1, 2) .
Here τ ∈ (0, 1) and Lv denotes a fractional differential operator of order v. The solution to (4.1.1) is represented as non-polynomial basis functions called Jacobi polyfractonomials given as α−µ+1,−β+µ−1 (x), x ∈ [−1, 1] , (4.1.2) Pnα,β,µ = (1 + x)−β+µ−1 Pn−1 α−µ+1,−β+µ−1 where Pn−1 (x) are the standard Jacobi polynomials in which µ ∈ (0, 1), −1 ≤ α < 2 − µ and −1 ≤ β < µ − 1.
We now consider the polyfractonomial that corresponds to α = β = −1 which is −µ,µ Pnµ (x) = (1 + x)µ Pn−1 (x),
x ∈ [−1, 1] .
(4.1.3)
From the properties of the eigensolution in (Zayernouri and Karniadakis, 2013), we can take the Riemann–Liouville and Caputo left-sided fractional derivative of (4.1.3), which by (2.1.9) gives the same result, namely −µ,µ µ µ µ µ −1 Dx (Pn (x)) = −1 Dx (1 + x) Pn−1 (x) =
Γ(n + µ) Pn−1 (x) , Γ(n)
(4.1.4)
where Pn−1 is Legendre polynomial of order n − 1. We seek solutions uN ∈ VNµ = span{Pnµ (x),
1 ≤ n ≤ N },
of the form uN (x) =
N X j=1
16
µ ∈ (0, 1),
u bj Pjµ (x).
x ∈ [−1, 1],
(4.1.5)
(4.1.6)
Section 4.2. Fractional Differentiation Matrix Dµ .
Page 17
The polyfractonomials can be expressed as uN (x) =
N X
u bj hµj (x) ,
(4.1.7)
j=0
where hµj (x) represents the fractional Lagrange interpolants, defined on the interpolation points, −1 = x0 < x2 < . . . < xN = 1. Define the interpolants hµj (x), then we have hµj (x)
=
x − x1 xj − x1
µ Y N x − xk , xj − xk
2≤j≤N,
(4.1.8)
k=1 k6=j
satisfying the Kronecker delta property ( 1 hj (xk ) = 0
if j = k if j 6= k
j, k = 0, 1, . . . , N ,
at the interpolation points. 4.1.1 Remark. We only construct hµj (x) for j = 2, 3, . . . , N when the maximum fractional order v ∈ (0, 1) because of the homogeneous Dirichlet boundary condition(s) in (4.1.1). So, we set uN (−1) = 0 (Zayernouri and Karniadakis, 2014).
4.2
Fractional Differentiation Matrix Dµ .
We now obtain the differentiation matrices , as illustrated in Zayernouri and Karniadakis (2014), for Dµ of a general fractional order µ ∈ (0, 1). Substituting (4.1.8) into (4.1.7) and taking the µth order fractional derivative, we have:
µ −1 Dx uN (x)
= −1 Dxµ
N X
uN (xj ) hµj (x)
j=2
=
N X
uN (xj ) −1 Dxµ (hµj (x))
j=2
=
N X j=2
=
N X j=2
uN (xj ) −1 Dxµ
x − x1 xj − x1
µ Y N k=1 k6=j
uN (xj ) −1 Dxµ ((1 + x)µ `j ) cj ,
x − xk xj − xk
(4.2.1)
Section 4.2. Fractional Differentiation Matrix Dµ . where cj =
Page 18
1 . (xj − x1 )µ
The Lagrange polynomials `j =
N Y x − xk , xj − xk
j = 2, 3, . . . , N ,
k=1 k6=j
−µ,µ are polynomials of order (N − 1) and can be represented in terms of the Jacobi polynomials Pn−1 (x) as N X −µ,µ `j = γn,j Pn−1 (x). (4.2.2) n=1
The coefficients γn,j can be computed by collocation and inverting the Jacobi–Vandermonde matrix. Substituting (4.2.2) into (4.2.1), we have: ! N N X X −µ,µ µ uN (xj ) −1 Dxµ (1 + x)µ γn,j Pn−1 (x) cj −1 Dx uN (x) = n=1
j=2
=
N X
uN (xj )cj
N X
γn,j
µ −1 Dx
−µ,µ . (1 + x)µ Pn−1
(4.2.3)
n=1
j=2
We recall the polyfractonomial (4.1.3), and substituting it into (4.2.3), we obtain: µ −1 Dx uN (x) =
N X
N X
uN (xj )cj
γn,j
µ µ −1 Dx (Pn (x))
.
(4.2.4)
n=1
j=2
Now, substituting (4.1.4) into (4.2.4) we obtain: µ −1 Dx uN (x)
=
N X
uN (xj )cj
N X
γn,j
n=1
j=2
Γ(n + µ) Pn−1 (xi ) . Γ(n)
(4.2.5)
Evaluating (4.2.5) at the collocation points {xi }N i=2 which are same as the interpolation points, we obtain: N N X X Γ(n + µ) µ uN (xj )cj γn,j Pn−1 (xi ) −1 Dx uN (x)|xi = Γ(n) n=1
j=2
=
N X
µ Dij uN (xj ),
(4.2.6)
j=2
where µ Dij
= cj
N X
γn,j
n=1
Γ(n + µ) Pn−1 (xi ) Γ(n)
.
Equation (4.2) implies that µ Dij
N X 1 Γ(n + µ) = γn,j Pn−1 (xi ) . (xj + 1)µ Γ(n) n=1
(4.2.7)
Section 4.3. Fractional Differentiation Matrix D1+µ .
Page 19
µ Dij are the entries of the (N − 1) × (N − 1) fractional differentiation matrix Dµ .
We can express this in matrix form as D = P ΛΓ V −1 Λx , where P = Legendre–Vandermonde matrix, Γ(n + µ) ΛΓ = diagonal matrix of , Γ(n) V −1 = Inverse of Jacobi–Vandermonde matrix , 1 . Λx = diagonal matrix of (xj + 1)µ Python code for computing Dµ is in Appendix A.
4.3
Fractional Differentiation Matrix D1+µ .
We now consider the fractional differentiation matrix of the form D1+µ , for µ ∈ (0, 1) as illustrated in Zayernouri and Karniadakis (2014). We obtain the fractional differentiation matrix, following the property of the Riemann–Liouville derivative (2.1.5) and take the first derivative of (4.2.5). We obtain: 1+µ −1 Dx uN (x)
d (−1 Dxµ uN (x)) dx N N X d X Γ(n + µ) = Pn−1 (x) uN (xj )cj γn,j dx Γ(n) =
n=1
j=2
=
=
N X
uN (xj )cj
N X
j=2
n=1
N X
N X
uN (xj )cj
Γ(n + µ) d (Pn−1 (x)) Γ(n) dx
Γ(n + µ) n 1,1 P (x) . Γ(n) 2 n−2
γn,j
γn,j
n=1
j=2
(4.3.1)
Similarly, evaluating (4.3.1) at the collocation points {xi }N i=1 we obain: 1+µ −1 Dx uN (x)|xi
=
N X
uN (xj )cj
n=1
j=2
=
N X
N X
γn,j
Γ(n + µ) n 1,1 P (x) Γ(n) 2 n−2
1+µ Dij uN (xj ) ,
(4.3.2)
j=1
where 1+µ Dij
N X 1 Γ(n + µ) n 1,1 = γn,j P (xi ) (1 + xj )µ Γ(n) 2 n−2 n=2
(4.3.3)
Section 4.3. Fractional Differentiation Matrix D1+µ .
Page 20
are the entries of the fractional differentiation matrix D1+µ . Another way to implement fractional derivatives of the form D1+µ is to multiply the ordinary differentiation matrix D by fractional differentiation matrix Dµ . For example, D3/2 can be computed by multiplying D and D1/2 . Having derived the fractional differentiation matrix, we now implement this as we would see in the example below. 4.3.1 Example. Steady-state fractional advection equation. We reproduce this problem from Zayernouri and Karniadakis (2014) to demonstrate exponential convergence. Now, consider the space-fractional advection equation of order v ∈ (0, 1) given as ν −1 Dx u(x)
= f (x),
x ∈ [−1, 1]
(4.3.4)
u(−1) = 0 . We solve for a force term f (x) =
9 9 Γ(7 + 17 ) (1 + x)6+ 17 −ν , 9 Γ(7 + 17 − ν)
(4.3.5)
which leads to the analytic solution 9
uexact (x) = (1 + x)6+ 17 .
(4.3.6)
Solution To solve this problem, we seek a solution of the form uN (x) =
N X
uN (xj )hµj (x) ,
(4.3.7)
j=2
with the initial condition uN (x0 ) = uN (−1) = 0 .
(4.3.8)
RN (x) =−1 Dxν u(x) − f (x) ,
(4.3.9)
We require the residual to vanish at the collocation points. Setting ν = µ, we obtain N X
µ Dij uN (xj ) − f (xi ) = 0,
i = 2, 3, . . . , N .
(4.3.10)
j=2
Therefore, the collocation scheme leads to the linear system below: Dµ uN (x) = f (x) ,
(4.3.11)
where Dµ is the (N − 1) × (N − 1) fractional differentiation matrix. Using a fractional order µ = 0.5 and n = 30, we obtain the numerical solution of (4.3.4), shown in Figure 4.1.
Section 4.3. Fractional Differentiation Matrix D1+µ .
100 80
Page 21
Max Error = 1.5632e-13 Exact Approximate
u(x)
60 40 20 0 −20 −1.0
−0.5
0.0
0.5
x
1.0
Figure 4.1: Solution of (4.3.4). Plotting the log-linear L∞ norm error of the numerical solution against n, we obtain Figure 4.2 which shows subgeometric convergence of the solutions. (The subgeometric convergence is consistent with result for (ν = 1/2) in Zayernouri and Karniadakis (2014)). Log-Linear plot
102
Error e−6.5
100
n
10-2
L∞ error
10-4 10-6 10-8 10-10 10-12 10-14
0
5
10
n
15
20
25
Figure 4.2: Subgeometric convergence of approximation to solution of (4.3.4) as the number of points is increased.
5. Applications. We now see the application of our spectral collocation method for ODEs and FDEs in the model of gas in a fluid and the Bagley–Torvik equation. The implementation of the solutions is done using Python software.
5.1
Model of gas in a fluid.
We give a mathematical model given by (Y. I. Babenko, 1986) as seen in (Podlubny, 1998), describing the dynamics of a solution of a gas in a fluid. Here we do not always get exponential convergence. The model is given as: ∂ M ∂C V0 f (τ /θ) · P (τ, 0) = FD , (5.1.1) ∂τ RT ∂x x=0 √ ∂C − D = 0 Dτ1/2 (Cs (τ ) − C0 ), (0 < τ < θ) , (5.1.2) ∂x x=0
C(0, x) = C0 ,
P (0, x) = P0 = C0 /κ ,
(0 < x < ∞) ,
(5.1.3)
where the parameters are: V0 = initial gas volume, θ = the time of the gas compression to zero volume, f (f /θ) = the function describing a change of the gas volume, M = gas molar weight, R = universal gas constant, D = coefficient of the diffusion of the gas in the fluid, F = the contact surface between the gas and the fluid, C(t, x) = gas concentration, P (t, x) = gas pressure, T = gas temperature, κ = equilibrium constant. We are interested in finding the pressure near the contact surface P (t, 0). The description of the mass of the gas volume due to diffusion through the contact surface is given by (5.1.1). Equation (5.1.2) illustrates the mass change that depends on the change of the gas concentration near the contact surface (Podlubny, 1998). In dimensionless form, the model is written as ∂ ∂c (c(t, 0)f (t)) = λ , ∂t ∂ξ ξ=0 ∂c = 0 Dτ1/2 (c(t, 0) − 1) , − ∂ξ
(5.1.4) (5.1.5)
ξ=0
c(0, ξ) =1 , 22
(5.1.6)
Section 5.1. Model of gas in a fluid.
Page 23
where τ t= , θ
P C = , p=c= C0 P0
√
√
ξ = x Dθ ,
λ = κRT
Dθ . M V0
Now, substituting (5.1.5) into (5.1.4) we obtain, d 1/2 f (t)p(t) + λ0 Dt p(t) − 1 = 0 , dt p(0) = 1 .
(0 < t < 1)
(5.1.7) (5.1.8)
Let y(t) = p(t) − 1 =⇒ p(t) = y(t) + 1 , then, (5.1.7) and (5.1.8) become: d 1/2 (f (t)(y(t) + 1)) + λ0 Dt y(t) = 0 , dt y(0) = 0 .
(0 < t < 1) ,
(5.1.9) (5.1.10)
which is the non-homogeneous linear fractional differential equation with a zero initial condition (Podlubny, 1998). We now solve (5.1.9)-(5.1.10) numerically. To do this, we first rescale the interval (0 < t < 1) to Chebyshev grid points by setting x = 2t − 1 so that the new interval becomes (−1 ≤ x ≤ 1). This implies that y(t) = y( x+1 2 ) = Y (x). Equation (5.1.9) and (5.1.10) become 2
√ d (F (x)(Y (x) + 1) + 2λ0 Dx1/2 Y (x) = 0, dx Y (−1) = 0 .
Simplifying and discretizing (5.1.11), we obtain √ 2ΛF D + 2λ 0 Dx1/2 + 2ΛF 0 Y (x) = −2ΛF 0 (x) } | {z } | {z } | {z y c
(5.1.11) (5.1.12)
(5.1.13)
L
L y = c,
(5.1.14)
where L is an N × N matrix, y are the solutions to be determined, c is an N × 1 matrix, D and 0 Dx1/2 are the ordinary and fractional differentiation matrices respectively, ΛF is the diagonal matrix of the function F (x) , ΛF 0 is the diagonal matrix of the derivative of the function F (x) , λ0 is a given constant. We implement the initial condition Y (−1) = 0 as follows: here, Y (−1) represents the first row of matrix L, 0 represents the first row of matrix c. Since the first row of matrix c is zero, we delete the
Section 5.1. Model of gas in a fluid.
Page 24
first row and the first column of matrix L to have an (N − 1) × (N − 1) matrix. In the same, we delete the first row of matrix c to have an (N − 1) × 1 matrix. Then the solutions y are obtained by solving the linear system (5.1.14). Suppose we solve numerically for a case where f (t) = 1 −
√
2 λ0 = √ , π
t,
and the exact solution is y(t) =
√
t.
We obtain in Figure 5.1 the graphical solutions for n = 40 and n = 80.
1.0 0.8
n=40
1.0
Approximate Exact
0.8 0.6
0.4
0.4
0.2
0.2
Approximate Exact
Y(x)
0.6
n=80
0.0 −1.0
−0.5
0.0
x
0.5
1.0
0.0 −1.0
−0.5
0.0
x
Figure 5.1: Numerical solutions of gas in a fluid. Plotting the convergence of the approximated solutions we obtain Figure 5.2.
0.5
1.0
Section 5.2. Bagley–Torvik Equation.
Page 25
Log-Linear
L∞ error
10-1
Error 1/n
10-2
10-3
10
20
30
40
n
50
60
70
80
Figure 5.2: Algebraic convergence of the approximated solutions. Figure 5.2 shows algebraic convergence, contrary to the examples shown in Zayernouri and Karniadakis (2014).
5.2
Bagley–Torvik Equation.
The Bagley–Torvik Equation as given in Podlubny (1998), is an initial value problem for FDEs. It arises in the mathematical modelling of the motion of a large thin plate immersed in a Newtonian fluid. It is given by 3/2
Ay 00 (t) + B 0 Dt y(t) + Cy(t) = f (t) ,
(t > 0),
(5.2.1)
with initial conditions given as y(0) = 0,
y 0 (0) = 0 ,
3/2
(5.2.2)
where 0 Dt is a fractional derivative of order 1.5; y(t) is the solution of the equation; and A, B and C are constants given as: • A = M , the mass of the thin plate; √ • B = 2S µρ is the damping force, where – S is the area of the plate immersed in the Newtonian fluid, – µ is the viscosity of the Newtonian fluid, – ρ is the density of the Newtonian fluid; • C = k, is the stiffness of the spring.
Section 5.2. Bagley–Torvik Equation.
Page 26
5.2.1 Example. First let us consider a nonfractional version of (5.2.1), given as Ay 00 (t) + BDy(t) + Cy(t) = f (t),
0 < t < 1,
(5.2.3)
0 < t < 1,
(5.2.4)
which gives rise to damped oscillations. Let us consider the specific case of y 00 (t) + 2y 0 (t) + y(t) = sin(4πt) , with the initial conditions y 0 (0) = 0 .
y(0) = 0 ,
(5.2.5)
The exact solution is e−t−1 4π(t + 16π 2 (t + 1) + 3) − 8πet+1 cos(4πt) − (16π 2 − 1)et+1 sin(4πt) . y(t) = (1 + 16π 2 )2
(5.2.6)
To solve this problem, we rescale the interval 0 < t < 1 to −1 ≤ x ≤ 1 using x = 2t − 1. Therefore the boundary conditions become y(−1) = 0, y 0 (−1) = 0. Now approximating (5.2.4) we have : x+1 2 D y(x) + 2Dy(x) + y(x) = sin 4π 2 (D2 + 2D + I) y(x) = sin(2π(x + 1)) {z } |{z} | {z } | y
L
c
L y = c, where L is an N × N matrix, c is an N × 1 matrix, D is the ordinary differentiation matrix, I is an identity matrix and y are the solutions to be determined. The boundary conditions y(−1) = 0 and y 0 (−1) = 0 are implemented as follows: the condition y(−1) which is represented as [1, 0, 0, · · · , 0] is placed into L as the first row while y 0 (−1) represents the first row of the D and is placed into L as the second row. The first and second row of c are replaced with zeros. Then, the solutions are obtained by solving the linear system. Implementing this in Python, we obtain the graphical solution, as shown in Figure 5.3 (left). The solution converges geometrically as shown in Figure 5.3 (right).
O[e−1.25n ]
10-1 10-2
0.030
10-3
L∞ error
0.025
y(x)
Error
100
0.035
0.020 0.015 0.010
10-4 10-5 10-6 10-7 10-8 10-9
0.005
10-10
Exact Approximate
0.000 −0.005 −1.0
Log-linear
101
0.040
−0.5
0.0
x
0.5
10-11 10-12 1.0
10-13
10
15
20
25
n
30
35
40
Figure 5.3: Left: Solution of (5.2.4)-(5.2.5) using Chebyshev points. Right: Geometric rate of convergence of the approximated solutions as n increases.
Section 5.2. Bagley–Torvik Equation.
Page 27
5.2.2 Example. Suppose we compare the solutions of (5.2.1)-(5.2.2) for an ordinary derivative D, the fractional derivative D0.5 and the fractional derivative D1.5 , for the values of A = 1, B = 2, C = 1 and f (t) = sin(4πt) for 0 < t < 1, we obtain the differential equations: y 00 (t) + 2Dy(t) + y(t) = sin(4πt) ;
(5.2.7)
00
0.5
y(t) + y(t) = sin(4πt) ;
(5.2.8)
00
1.5
y(t) + y(t) = sin(4πt) .
(5.2.9)
y (t) + 2D y (t) + 2D
Solving these equations the same way we solved example (5.2.1), using the fractional differentiation matrix from Chapter 4, we obtain the graphical solutions of the approximated values in Figure 5.4 (left). We know the exact solution of (5.2.7) on the Chebyshev grid [−1, 1] and we have constructed the error of convergence as seen in Figure 5.4. We do not know the exact solutions for (5.2.8) and (5.2.9), but we can use their computed values at y(1) to estimate the error. That is, choosing a range of values of n, say from 10 to 100, the error at each n for the solutions to (5.2.8) and (5.2.9) is given by computed error(n) = |y100 (1) − yncomputed (1)| .
From here, we can now plot the errors of convergence for the solutions of (5.2.8) and (5.2.9) as seen in Figure 5.4 (right).
0.05
0.04
Log-linear
101
D1 D0.5 D1.5
D1 D0.5 D1.5
100 10
-1
10-2 10-3
L∞ error
y(x)
0.03
0.02
0.01
10-4 10-5 10-6 10-7 10-8 10-9
10-10 0.00
10-11 10-12
−0.01 −1.0
−0.5
x
0.0
0.5
1.0
10-13
10
20
30
40
50
n
60
70
80
90
100
Figure 5.4: Left: Solution of (5.2.7)-(5.2.9) using Chebyshev points. Right: Convergence of Errors. The legend notaton D, D0.5 and D1.5 represents equations (5.2.7), (5.2.8) and (5.2.9) respectively. The solutions to (5.2.7) converge exponentially while the solutions to (5.2.8) and (5.2.9) converge algebraically.
6. Conclusion and Further work. 6.1
Conclusion.
Our work so far has been on the spectral collocation method for FDEs. We have considered the collocation method using the Chebyshev points. We have used Chebyshev differentiation matrices in the numerical solutions of our FDEs and ODEs. We implemented an example of spectral collocation for ODEs using a second order linear boundary value problem, the numerical result yielded exponential convergence. Spectral collocation for FDEs was implemented using a steady-state fractional advection equation, in which the numerical solutions converged exponentially. We applied our spectral collocation method to the model of gas in a fluid. The result gave an algebraic convergence of the numerical solutions of the gas pressure near the contact surface. We also applied our spectral collocation method to the Bagley−Torvik equation, which is a model of the motion of a large thin plate in a Newtonian fluid. Here, we considered an ordinary case of the Bagley−Torvik equation (damped oscillations), the numerical solutions converged exponentially. Thereafter we compared the ordinary case to the fractional order cases of the Bagley−Torvik equation, the numerical solutions of the fractional order cases converged algebraically. Contrary to the implication in the paper by Zayernouri and Karniadakis (2014), we find that geometric convergence is not achieved in general. In fact, in the approximate solution of the model of gas in a fluid, we found algebraic convergence for the solutions (Figure 5.2).
6.2
Further work.
We have solved FDEs in one dimension, we recommend that two-dimensional FDEs be considered. Fractional partial differential equations (FPDEs) can be looked into. It could be interesting to consider time-dependent problems and extensions can be made to nonlinear FDE problems.
28
Appendix A. Differentiation matrices. The following Python codes are the codes for ordinary D and fractional Dµ differentiation matrices. The type of data returned by the code is an array. Comments are preceded by the hash symbol (#) or enclosed in three single quotes. Additional code for the implemented examples and applications can be found in Nnakenyi (2015). ------------------------------------------------------------------------------------‘‘‘ Libraries to be imported. ’’’ from __future__ import division import numpy as np from math import gamma from numpy.linalg import inv ‘‘‘Functions to be used for the differentiation matrix code.’’’ def V_leg(n): ‘‘‘ A function that takes ‘n’ and returns the (n+1)by(n+1) Vandermonde-Legendre matrix. ’’’ x = np.array([-np.cos(np.pi*j/n) for j in range(n+1)]) # For Chebyshev points A = np.outer(x,x) A[0] = np.ones((n+1)) # First Legendre polynomial A[1] = x # Second Legendre polynomial for k in range(1,n): # For other recursive terms for i in range(n+1): A[k+1,i] = ((2*k+1)*A[1,i]*A[k,i] - k*A[k-1,i])/(k+1) return A.transpose() # The matrix.
def V_jac(n,mu): ‘‘‘ A function that takes ‘n’ and ‘mu’ (order), computes the Vandermonde’s matrix of Jacobi polynomial and returns an (n+1)by(n+1) matrix. ’’’ x = np.array([-np.cos(np.pi*j/n) for j in range(n+1)]) # Chebyshev points A = np.outer(x,x) a = -mu b = mu A0 = 0.5*(a+b)+1 B0 = 0.5*(a-b) A[0] = np.ones((n+1)) # First Jacobi polynomial A[1] = A0*x + B0 # Second Jacobi polynomial for k in range(1,n): # For other recursive terms for i in range(n+1): An =((2*k+a+b+1)*(2*k+a+b+2))/(2*(k+1)*(k+a+b+1)) Bn = ((a**2-b**2)*(2*k+a+b+1))/(2*(k+1)*(k+a+b+1)*(2*k+a+b)) 29
Page 30 Cn = ((k+a)*(k+b)*(2*k+a+b+2))/((k+1)*(k+a+b+1)*(2*k+a+b)) A[k+1,i] = (An*x[i] + Bn)*A[k,i] - Cn*A[k-1,i] return A.transpose() # The matrix ‘‘‘ The differentiation matrix code ’’’ def cheb(n,mu): ‘‘‘ A function that takes ‘n’ and ‘mu’ (the differentiation order), computes the Chebyshev differntiation matrix and returns: i. (n+1)by(n+1) matrix for ordinary differentiation and the Chebyshev points ii. (n-1)by(n-1) matrix for fractional differentiation and the Chebyshev points. Here, the Chebyshev points (x) goes from -1 to 1. ’’’ if mu == 1: ‘‘‘ computes for the ordinary differentiation matrix ’’’ if n == 0: print ‘‘n should be greater than zero" else: k= np.linspace(0/n,n/n,n+1) D=np.outer(k,k) D[0][0]= -(2* n**2 +1)/6 D[n][n]=(2* n**2 +1)/6 D[n][0] = (-1)**n /2 D[0][n] = -(-1)**n /2 for j in range(n+1): for l in range(n+1): if j!= l: if k[j]==0 or k[j]==n: c = 2 else: c =1 D[j][l] = -c *(-1)**(j+l)/float((c)*(((np.cos(k[j]*np.pi)) -(np.cos(k[l]*np.pi))))) for i in range(1,n): D[i][i] = -(-1)* np.cos(k[i]*np.pi)/(2*(1-(np.cos(k[i]*np.pi))**2)) D[0][i] = -2*(-1)**(i)/(1 - (np.cos(k[i]*np.pi))) D[n][i] = 2*(-1)**(i+n)/(1 + (np.cos(k[i]*np.pi))) D[i][0] = 1*(-1)**(i)/(2*(1 - (np.cos(k[i]*np.pi)))) D[i][n] = -(-1)**(i+n)/(2*(1 + (np.cos(k[i]*np.pi)))) x = np.array([-np.cos(np.pi*j/(n)) for j in range(n+1)]) return D,x # The matrix D and the grid points. else: ‘‘‘ Computes for fractional differentiation matrix ’’’ k = np.zeros(n) A = np.outer(k,k) # Matrix to store values into A1 = np.outer(k,k) # Matrix to store values into B = [] x = np.array([-np.cos(np.pi*j/(n-1)) for j in range(n)])
Page 31 B1 = 1/((x+1)**mu) for k in range(1,n+1): B.append(gamma(k+mu)/gamma(k)) for i in range(n): for j in range(n): if i == j: A[i,j] = B[i] A1[i,j] = B1[i] A = np.mat(A) A1 = np.mat(A1) P = np.mat(V_leg(n-1)) j1 = inv(np.matrix(V_jac(n-1,mu))) D = P*A*j1*A1
# # # #
Matrix for 1/(x+1)^mu Matrix for gamma(n+mu)/gamma(n) The legendre polynomial Inverse of the Jacobi polynomial
D = D[1:n,1:n] # Boundary conditions x = x[1:] return np.array(D),x # The matrix D and the grid points. -----------------------------------------------------------------------------------
Acknowledgements I thank God, the all-knowing, who has given me the wisdom, knowledge and understanding to have completed this project. May His name be praised forever. My sincere gratitude goes to my supervisors Prof. J.A.C Weideman and Dr. N. Hale for their tireless effort in guiding me all through the phase of this project. May God bless and reward you. With gratitude I thank the founder of AIMS, Prof. Neil Turok, the director of AIMS, Prof. Barry Green, the academic director Prof. Jeff Sanders, the IT manager, Jan Groenewald, the facilities and logistics manager, Igsaan Kamalie and other Pioneers, Staffs and Tutors of AIMS, for this wonderful opportunity to have a taste of research in utmost splendour. Thanks for providing my need all through this project phase and my study at AIMS. I thank my father Mr. Bernard Nnakenyi and my siblings for their moral support and prayers for my success. May God continue to protect, provide and keep you. My excellent appreciation goes to my friends; Steve, Nneka, Emmanuel, Zainab, Adebayo, Reine and other AIMS students for being there when I needed them. Thank you so much. May we continue to rise to greater heights through Christ our Lord (Amen).
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References W. Bao. Orthogonal Polynomials and Polynomial Approximations. http://www.math.nus.edu.sg/∼bao/ teach/ma5251/chap3and4.pdf, December 2011. [Online Lecture notes; accessed 12-May-2015]. A. Bhrawy, M. Tharwat, and A. Yildirim. A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations. Applied Mathematical Modelling, 37(6):4245–4252, 2013. J. P. Boyd. Chebyshev and Fourier spectral methods. Dover Publications, 2000. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods in fluid dynamics. Technical report, Springer, 1988. A. Chatterjee. Statistical origins of fractional derivatives in viscoelasticity. Journal of Sound and Vibration, 284(3):1239–1245, 2005. B. Costa. Spectral methods for Partial Differential Equations. Cubo-Revista de Matem´atica, 6:1–32, 2004. K. Diethelm, N. J. Ford, and A. D. Freed. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1-4):3–22, 2002. E. Doha, A. Bhrawy, and S. Ezz-Eldien. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Applied Mathematical Modelling, 35(12):5662–5672, 2011. E. Doha, A. Bhrawy, and S. Ezz-Eldien. A new Jacobi operational matrix: an application for solving fractional differential equations. Applied Mathematical Modelling, 36(10):4931–4943, 2012. T. A. Driscoll and N. Hale. Rectangular spectral collocation. IMA Journal of Numerical Analysis, 2015. S. Esmaeili and M. Shamsi. A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16 (9):3646–3654, 2011. B. Fornberg. A practical guide to pseudospectral methods, volume 1. Cambridge University Press, 1998. F. Ghoreishi and S. Yazdani. An extension of the spectral Tau method for numerical solution of multiorder fractional differential equations with convergence analysis. Computers & Mathematics with Applications, 61(1):30–43, 2011. J.-H. He. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 167(1):57–68, 1998. M. Khader. On the numerical solutions for the fractional diffusion equation. Communications in Nonlinear Science and Numerical Simulation, 16(6):2535–2542, 2011. A. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and applications of fractional differential equations, volume 204. Elsevier Science Limited, 2006. T. Kisela. Fractional Differential Equations and Their Applications. PhD thesis, Diploma Thesis, Faculty of Mechanical Engineering-Institute of Mathematics, Supervisor: doc. RNDr. J. Cerm´ak, Brno University of Technology, Brno, 2008. 33
REFERENCES
Page 34
R. Koekoek. Orthogonal polynomials. http://homepage.tudelft.nl/11r49/documents/wi4006/orthopoly. pdf, April 2015. [Online Lecture notes; accessed 12-May-2015]. D. Kumar, J. Singh, and S. Kumar. A fractional model of Navier–Stokes equation arising in unsteady flow of a viscous fluid. Journal of the Association of Arab Universities for Basic and Applied Sciences, 2014. C. Lanczos. Trigonometric interpolation of empirical and analytic functions. Journal of Mathematical Physics, 17:123–199, 1938. G. Leibniz. Letter from Hanover, Germany, September 30, 1695 to GA L’Hospital. JLeibnizen Mathematische Schriften, 2:301–302, 1849. X. Li and C. Xu. A space-time spectral method for the time fractional diffusion equation. SIAM Journal on Numerical Analysis, 47(3):2108–2131, 2009. X. Li and C. Xu. Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Communications in Computational Physics, 8(5): 1016, 2010. C. Lubich. On the stability of linear multistep methods for Volterra convolution equations. IMA Journal of Numerical Analysis, 3(4):439–465, 1983. C. Lubich. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3):704–719, 1986. R. L. Magin. Fractional calculus in bioengineering, part 1. Critical ReviewsTM in Biomedical Engineering, 32(1), 2004. N. Mai-Duy. An effective spectral collocation method for the direct solution of high-order ODEs. Communications in numerical methods in engineering, 22(6):627–642, 2006. NIST Digital Library of Mathematical Functions. Orthogonal Polynomials. http://dlmf.nist.gov/18.9, August 2014. [Online Lecture notes; accessed 12-May-2015]. C. Nnakenyi. Python codes. https://bitbucket.org/Assumpta chinenye/my python codes/src/ beb255101d63ec6f895dbb125f6230356c661bfa?at=master, May 2015. I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198. Academic press, 1998. E. Sifakis. Introduction to numerical methods. http://pages.cs.wisc.edu/∼sifakis/courses/cs412-s13/ lecture notes/CS412 12 Feb 2013.pdf, October 2013. [Online Lecture notes; accessed 12-May-2015]. L. N. Trefethen. Finite difference and spectral methods for ordinary and partial differential equations. Cornell University, Department of Computer Science and Center for Applied Mathematics, 1996. L. N. Trefethen. Spectral methods in MATLAB, volume 10. SIAM, 2000. Y. I. Babenko. Heat and Mass Transfer. Khimiya, Leningrad, 1986. (In Russian). M. Zayernouri and G. E. Karniadakis. Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. Journal of Computational Physics, 252:495–517, 2013.
REFERENCES
Page 35
M. Zayernouri and G. E. Karniadakis. Fractional spectral collocation method. SIAM Journal on Scientific Computing, 36(1):A40–A62, 2014. X. Zhang, X. Guo, and A. Xu. Computations of fractional differentiation by Lagrange interpolation polynomial and Chebyshev polynomial. Inform. Technol. J, 11(4):557–559, 2012. Y. Zhang, D. A. Benson, and D. M. Reeves. Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources, 32(4): 561–581, 2009.
