Numerical solution of fractional Sturm-Liouville equation in integral form

$: Numerical solution of fractional Sturm-Liouville equation in integral form$
RESEARCH PAPER

NUMERICAL SOLUTION OF FRACTIONAL

STURM–LIOUVILLE EQUATION IN INTEGRAL FORM

Tomasz Blaszczyk 1, Mariusz Ciesielski 2

Abstract

In this paper a fractional differential equation of the Euler–Lagrange/Sturm–Liouville type is considered. The fractional equation with deriva-tives of order α ∈ (0, 1] in the finite time interval is transformed to theintegral form. Next the numerical scheme is presented. In the final partof this paper examples of numerical solutions of this equation are shown.The convergence of the proposed method on the basis of numerical resultsis also discussed.

MSC 2010 : Primary 26A33; Secondary 34A08, 65L10Key Words and Phrases: fractional Euler–Lagrange equation, fractional

Sturm–Liouville equation, fractional integral equation, numerical solution

1. Introduction

The fractional differential equations, both ordinary and partial ones,are very useful tools for modelling many phenomena in physics, mechanics,control theory, biochemistry, bioengineering and economics [10, 12, 21, 22,24, 38]. Therefore, the theory of fractional differential equations is an areathat has developed extensively over the last decades. In the monographs[13, 14, 16, 30, 31, 11] one can find a review of methods of solving fractionaldifferential equations.

c© 2014 Diogenes Co., Sofiapp. 307–320 , DOI: 10.2478/s13540-014-0170-8

308 T. Blaszczyk, M. Ciesielski

In the recent years, a subtopic of the theory of fractional differentialequations gains importance: it concerns the variational principles for func-tionals involving fractional derivatives. These principles lead to equationsknown in the literature as the fractional Euler–Lagrange equations. Theequations of this type were derived when fractional integration by partsrule [13] has been applied.

This approach was initiated by Riewe in [36], where he used non-integerorder derivatives to describe nonconservative systems in mechanics. Next,Klimek [15] and Agrawal [1] noticed that such equations can be investigatedin the sequential approach. A fractional Hamiltonian formalism for thecombined fractional calculus of variations was introduced in [27]. In thework [29] Green theorem for generalized partial fractional derivatives wasproved. Other applications of fractional variational principles are presentedin [2, 19, 25, 26, 28].

Recently, the fractional Sturm–Liouville problems were formulated byKlimek and Agrawal in [18] and Rivero et al. in [37]. The authors in thesepapers considered several types of the fractional Sturm–Liouville equationsand investigated the eigenvalues and eigenfunctions properties of the frac-tional Sturm–Liouville operators.

Unfortunately, the fractional Euler–Lagrange/Sturm–Liouville equationscontain the composition of the left- and right-sided derivatives. It is an ad-ditional drawback for computation of an exact solution (even with simpleLagrangian, see [5, 16, 17]). Consequently, numerous studies have been de-voted to numerical schemes for the fractional equations (see [4, 6, 8, 23, 39]).For numerical methods in the fractional calculus of variations we refer thereader to [33, 34, 35].

In our previous works [7, 8, 9] we proposed numerical scheme on thebasis of a finite difference method of solution for a special case of the prob-lem, namely the fractional oscillator equation. In this paper we proposea numerical solution of the fractional Sturm–Liouville equation. We in-vestigate a new integral form of this equation and a numerical method ofsolution of considered equation in conjunction with analysis of a rate ofconvergence. Another integral form of the fractional Euler-Lagrange equa-tions (containing the Caputo derivatives) has been recently considered in[20].

2. Statement of the problem and definitions

We consider the fractional differential equation with derivatives of orderα ∈ (0, 1] in the finite time interval t ∈ [0, b], for parameter λ ∈ R andvariable potential determined by function q (t)

CDαb− Dα

0+ f (t) + (λ + q (t)) f (t) = 0, (2.1)

NUMERICAL SOLUTION OF FRACTIONAL . . . 309

where f(t) ∈ AC[0, b] is an unknown function (absolutely continuous on[0, b]), satisfying boundary conditions

f (0) = 0, f (b) = L. (2.2)

According to [13, 30, 31] we recall the definitions of the left- and right-sided Riemann-Liouville fractional derivative operators for α ∈ (0, 1)

Dα0+ f (t) := D I1−α

0+ f (t) , (2.3)

Dαb− f (t) := −D I1−α

b− f (t) , (2.4)

where D is the conventional first order derivative and the operators Iα0+

and Iαb− are respectively the left- and right-sided fractional integrals of

order α > 0 defined by

Iα0+f (t) :=

1Γ (α)

∫ t

0

f (τ)(t − τ)1−α dτ (t > 0) , (2.5)

Iαb−f (t) :=

1Γ (α)

∫ b

t

f (τ)(τ − t)1−α dτ (t < b) . (2.6)

The denotations CDα0+ and CDα

b− are used to represent the left- and right-sided Caputo fractional derivatives, respectively, as regularized alternativesof 2.3 and 2.4. Between both definitions the following relationships occur,[13] (valid only for α ∈ (0, 1)):

CDα0+ f (t) := Dα

0+ f (t) − t−α

Γ (1 − α)f (0) , (2.7)

CDαb− f (t) := Dα

b− f (t) − (b − t)−α

Γ (1 − α)f (b) . (2.8)

Further in this paper we will use the following composition rules of theoperators of fractional calculus (for α ∈ (0, 1]), see e.g. [13]

Iα0+

CDα0+ f (t) = f (t) − f (0) (2.9)

Iαb−

CDαb− f (t) = f (t) − f (b) (2.10)

and the fractional integral of a constant C:

Iα0+ C = C

tα

Γ (1 + α), Iα

b−C = C(b − t)α

Γ (1 + α). (2.11)

In particular, when α = 1, then CD1b− D1

0+ = −D2 and for q (t) = 0,Eq. (2.1) becomes

− D2f (t) + λ f (t) = 0 (2.12)


and for λ < 0 (oscillatory character of solutions of the equation) its ana-lytical solution satisfying boundary conditions (2.2) is of the form

f (t) = Lsin

(√−λ t)

sin(√−λ b

) , λ �= −(

kπ

b

)2

, k ∈ Z. (2.13)

The formula for the analytical solution of Eq. (2.1) for α ∈ (0, 1) israther involved. For example, the analytical solution for q (t) = 0 whichcontains the series of fractional integrals is presented in [8, 16, 17]. Thismakes it difficult to carry out any operations on them. There is a problemin calculations of the values of f . The analytical solution of Eq. (2.1) canbe expressed by elementary functions only in a special case (see details infurther description).

3. Transformation of fractional equation to the integral form

Proposition 3.1. The equivalent integral form of Eq. (2.1) withboundary conditions (2.2) is given as

f (t) + Iα0+ Iα

b− (λ + q (t)) f (t) −(

t

b

)α

Iα0+ Iα

b− (λ + q (t)) f (t)|t=b =L

bαtα.

(3.1)

P r o o f. By using the fractional integral operators Iα0+ Iα

b− acting onEq. (2.1), we obtain

Iα0+ Iα

b−CDα

b− Dα0+ f (t) + Iα

0+ Iαb− (λ + q (t)) f (t) = 0. (3.2)

Next we use the composition rule of operators Iαb−

CDαb− (see Eq. (2.10))

in Eq. (3.2), thus we get

Iα0+

(Dα

0+ f (t) − Dα0+ f (t)|t=b

)+ Iα

0+ Iαb− (λ + q (t)) f (t) = 0. (3.3)

The above equation contains an unknown value Dα0+ f (t)

∣∣t=b

and for theunknown function f(t) this value is treated here as a constant.

Using again the composition rule of operators Iα0+ Dα

0+ (see Eq. (2.9)and the fact that if f(0) = 0, then Dα

0+ f (t) = CDα0+ f (t) in Eq. (2.7),

hence Iα0+ Dα

0+ f (t) = f (t)) and the fractional integral of a constant (2.11),we obtain the following form of equation

f (t) − Dα0+ f (t)|t=b

tα

Γ (α + 1)+ Iα

0+ Iαb− (λ + q (t)) f (t) = 0. (3.4)

In order to determine the value Dα0+ f (t)

∣∣t=b

we substitute the valuet = b into Eq. (3.4)

f (b) − Dα0+ f (t)|t=b

bα

Γ (α + 1)+ Iα

0+ Iαb− (λ + q (t)) f (t)|t=b = 0, (3.5)

and obtain


Dα0+ f (t)|t=b =

Γ (α + 1)bα

(f (b) + Iα

0+ Iαb− (λ + q (t)) f (t)|t=b

). (3.6)

Next we substitute the right-hand side of (3.6) into Eq. (3.4) and get theintegral form of Eq. (2.1)

f (t)+Iα0+ Iα

b− (λ+q (t)) f (t)−(

t

b

)α

Iα0+ Iα

b− (λ+q (t)) f (t)|t=b =(

t

b

)α

f (b).

(3.7)Taking into account the boundary conditions (2.2), we obtain Eq. (3.1). �

Remark 3.1. One can easily check that Iα0+ Iα

b− (λ + q (t)) f (t)∣∣t=0

=0 (on the basis of definitions (2.5) and (2.6)). If we put t = 0 and t = b intoEq. (3.1) we can confirm that this equation fulfils the boundary conditions(2.2). In particular, for λ + q (t) = 0, Eq. (3.1) simplifies to the form

f (t) =L

bαtα. (3.8)

4. Numerical algorithm

In this section we present a numerical scheme for solving Eq. (3.1). Weintroduce homogeneous grid of nodes (with the constant time step Δt =b/n, where n+1 is a number of nodes): 0 = t0 < t1 < ... < ti < ti+1 < ... <tn = b, and ti = iΔt. In order to simplify the notation, we introduce a newfunction φ (t) ≡ (λ + q (t)) f (t) and denote the values of functions f(t), q(t)and φ(t) at the node ti by fi = f(ti), qi = q(ti) and φi = φ(ti) = (λ+ qi)fi.

Now we determine the numerical schemes of integration [11, 30, 32] forboth fractional integral operators occurring in Eq. (3.1).

At the node t0 we have Iα0+φ (t)

∣∣t=t0

= 0. The discrete form of theintegral operator (2.5) at nodes ti for i = 1, 2, ..., n is approximated by theformula

Iα0+φ (t)|t=ti

=1

Γ (α)

∫ ti

0

φ (τ)(ti − τ)1−α dτ =

1Γ (α)

i−1∑j=0

∫ tj+1

tj

φ (τ)(ti − τ)1−α dτ

≈ 1Γ (α)

i−1∑j=0

φj + φj+1

2

∫ (j+1)Δt

j Δt

1(iΔt − τ)1−α dτ

=(Δt)α

2Γ (α + 1)

i−1∑j=0

(φj + φj+1) ((i − j)α − (i − j − 1)α)

=i∑

j=0

φj wi,j, (4.1)


where the coefficients wi,j (also including the case for i = 0) are as follows

wi,j =(Δt)α

2Γ (α + 1)

⎧⎪⎪⎨⎪⎪⎩

0 for i = 0 and j = 0iα − (i − 1)α for i > 0 and j = 0(i − j + 1)α − (i − j − 1)α for i > 0 and 0 < j < i1 for i > 0 and j = i

.

(4.2)We determine discrete form of the fractional integral operator (2.6) in a

similar way. This operator at node tn is equal to Iαb−φ (t)

∣∣t=tn

= 0, whereasat nodes ti, i = 0, 1, ..., n − 1, the discrete values are determined by theformula

Iαb−φ (t)|t=ti

=1

Γ (α)

∫ tn

ti

φ (τ)(τ − ti)1−α dτ =

1Γ (α)

n−1∑j=i

∫ tj+1

tj

φ (τ)(τ − ti)1−α dτ

≈ 1Γ (α)

n−1∑j=i

φj + φj+1

2

∫ (j+1)Δt

j Δt

1(τ − iΔt)1−α dτ

=(Δt)α

2Γ (α + 1)

n−1∑j=i

(φj + φj+1) ((j − i + 1)α − (j − i)α)

=n∑

j=i

φj vi,j, (4.3)

where the coefficients vi,j (together with the case for i = n) have the form

vi,j =(Δt)α

2Γ (α + 1)

⎧⎪⎪⎨⎪⎪⎩

0 for i = n and j = n(n − i)α − (n − i − 1)α for i < n and j = n(j − i + 1)α − (j − i − 1)α for i < n and i < j < n1 for i < n and j = i

.

(4.4)The discrete form of the composition of both operators Iα

0+ Iαb−φ (t) at nodes

t = ti for i = 0, 1, ..., n, has the following form

Iα0+ Iα

b−φ (t)|t=ti≈

i∑j=0

wi,j

n∑k=j

φk vj,k, (4.5)

or

Iα0+ Iα

b− (λ + q (t)) f (t)|t=ti≈

i∑j=0

wi,j

n∑k=j

(λ + qi) fi vj,k. (4.6)

One can note that at the node t0 we have Iα0+ Iα

b−φ (t)∣∣t=t0

= 0.


Now we present the discrete form of the integral equation (3.1). Onecan write the solution in the form of the system of n + 1 linear equations.For every grid node ti, i = 0, 1, ..., n, we write the following equation

fi +i∑

j=0

wi,j

n∑k=j

(λ+qi) fivj,k −(

i

n

)α n∑j=0

wn,j

n∑k=j

(λ+qi) fivj,k =(

i

n

)α

L.

(4.7)Analyzing the above system of equations, created equations for node in-dexes i = 0 and i = n, one can reduce to the forms f0 = 0 and fn = L,respectively. In this way the obtained system of linear equations can besolved numerically.

5. Simulation results and numerical error analysis

In this section we present the results of calculation obtained by ournumerical approach to the fractional Sturm–Liouville equation (2.1). Inorder to numerically solve the system of equations (4.7), we used the LUPdecomposition method [32]. We present several examples of calculations fordifferent values of parameters α, λ and forms of function q(t). In all theseexamples we assumed: b = 1 and L = 1 in the right boundary condition.For all presented graphs of functions (Figures 1-3), in the calculations weassume that the time domain t ∈ [0, 1] has been divided into n = 2048subintervals. 5.1. Example of results

Figures 1 and 2 show the example graphs of solutions of Eq. (2.1) forq(t) = 0 (the case of the fractional oscillator equation). Figure 1 presentssolutions for λ ∈ {−3,−10,−20,−25} and variable values of parameter α.We can see how changes of parameter α influence on the frequency of oscil-lations in comparison to the classical oscillator equation (α = 1). Whereasin Figure 3 we show the influence of parameter λ ∈ {−5,−7.5,−10,−12.5}at the constant values of α = 0.6 and α = 0.8 on the solution.

In the last Figure 3 we present solutions of Eq. (2.1) for different formof function q(t) �= 0 (the case of the fractional Sturm–Liouville equation).We show the influence of parameters α, λ and forms of function q(t) on thesolution. 5.2. Error analysis

Next we analyze errors and convergence of the numerical scheme (4.7)for q(t) = 0, any λ and α ∈ (0, 1]. When analytical solution is not available,the rate of convergence p = pi(Δt, α, λ) at nodes ti, for fixed parameters α,λ and variable values of Δt, can be determined from the following formula(see a proposition in [3])

R(Δt,α,λ)i =

f(Δt,α,λ)i − f

(2Δt,α,λ)i

f(Δt/2,α,λ)i − f

(Δt,α,λ)i

= 2pi(Δt,α,λ). (5.1)


Figure 1. Numerical solution of Eq. (2.1) for different pa-rameters α, q(t) = 0, b = 1, L = 1 and λ = −3 (left/top),λ = −10 (right/top), λ = −20 (left/bottom) and λ = −25(right/bottom)

Figure 2. Numerical solution of Eq. (2.1) for λ ∈ {−5,−7.5,−10,−12.5}, b = 1, L = 1, q(t) = 0 and α = 0.6(left-side), α = 0.8 (right-side)

We thus have

pi (Δt, α, λ) = log2

f(Δt,α,λ)i − f

(2Δt,α,λ)i

f(Δt/2,α,λ)i − f

(Δt,α,λ)i

. (5.2)


Figure 3. Numerical solution of Eq. (2.1) for different pa-rameters λ, forms of function q(t) and α = 1 (left-side),α = 0.5 (right-side)

We present numerical values at three selected nodes and rate of con-vergence for α ∈ {0.3, 0.5, 0.7} and λ = −3, q(t) = 0 in Table 1. In Table2 the numerical values at three selected nodes and rates of convergence forα = 0.6, λ ∈ {−5,−7.5,−10} and q(t) = 0 are shown. The values fromTables 1 and 2 are also shown in plots - Figures 1 and 2, respectively.

Analyzing the values in Table 1 and Table 2, one can observe that therate of convergence p is dependent on the fractional order α and does notdepend on parameter λ. The rate of convergence p is close to 1 + α.

6. Conclusions

In this paper the fractional Sturm–Liouville equation with derivativesof order α ∈ (0, 1] in the finite time interval t ∈ [0, b] is considered. Thisequation was transformed using the composition rules for fractional inte-grals and derivatives to the integral form. Next the discrete form of theintegral equation was presented as the system of linear algebraic equations.The obtained system of equations was solved numerically. The equation


was solved for derivatives of different orders α, different values of parame-ter λ and different forms of function q(t). The presented results showed theinfluence these values on the character (i.e. the occurrence of oscillations)of the solution. One can note that for q(t) = 0 the oscillations occur onlyfor λ < 0. The number of oscillations increases when the value of order αdecreases for fixed value of parameter λ. Similarly, for fixed order α, thenumber of oscillations increases when the value of parameter λ decreases.In order to ensure stability of the computation, the convergence study of thenumerical scheme was conducted. The rate of convergence p was estimatedto be close to 1 + α.

Acknowledgements

This work was supported by the Czestochowa University of TechnologyGrant Number BS/MN 1-105-302/13/P.

t = 0.25 t = 0.5 t = 0.75α Δt = 1/n fn/4 p fn/2 p f3n/4 p0.3 1/256 1.53966755 - -6.00003222 - -5.30712828 -

1/512 1.58297521 1.08 -6.17593877 1.12 -5.50843511 1.121/1024 1.60340765 1.18 -6.25702726 1.19 -5.60111461 1.201/2048 1.61245514 1.22 -6.29245343 1.24 -5.64157016 1.241/4096 1.61632922 1.25 -6.30749294 1.26 -5.65873385 1.261/8192 1.61795763 - -6.31377732 - -5.66590217 -

0.5 1/256 3.73516937 - 4.73052344 - 3.57465003 -1/512 3.72842313 1.38 4.72211467 1.39 3.56919099 1.391/1024 3.72583618 1.42 4.71890721 1.42 3.56710826 1.421/2048 3.72486755 1.44 4.71771054 1.44 3.56633103 1.441/4096 3.72451075 1.46 4.71727084 1.46 3.56604538 1.461/8192 3.72438081 - 4.71711100 - 3.56594153 -

0.7 1/256 0.94678077 - 1.40241981 - 1.42764144 -1/512 0.94671534 1.52 1.40231616 1.55 1.42754955 1.571/1024 0.94669252 1.57 1.40228081 1.59 1.42751850 1.601/2048 0.94668482 1.60 1.40226907 1.61 1.42750826 1.621/4096 0.94668228 1.63 1.40226523 1.64 1.42750493 1.641/8192 0.94668146 - 1.40226400 - 1.42750386 -

Table 1. Numerical values of f at nodes ti, i ∈ {n/4, n/2,3n/4} and rates of convergence p for parameters λ = −3,q(t) = 0, b = 1, L = 1


t = 0.25 t = 0.5 t = 0.75λ Δt = 1/n fn/4 p fn/2 p f3n/4 p-5 1/256 -2.16736188 - -2.58746851 - -0.79882549 -

1/512 -2.16934247 1.46 -2.59034981 1.44 -0.80084057 1.451/1024 -2.17006290 1.50 -2.59140830 1.49 -0.80157662 1.491/2048 -2.17031750 1.53 -2.59178487 1.52 -0.80183747 1.521/4096 -2.17040578 1.55 -2.59191604 1.54 -0.80192808 1.541/8192 -2.17043598 - -2.59196106 - -0.80195912 -

-7.5 1/256 -1.51542247 - 0.06057150 - 1.83620282 -1/512 -1.51337627 1.46 0.05880539 1.40 1.83222891 1.451/1024 -1.51263480 1.50 0.05815391 1.46 1.83078071 1.501/2048 -1.51237316 1.53 0.05792141 1.50 1.83026791 1.531/4096 -1.51228251 1.54 0.05784024 1.53 1.83008983 1.551/8192 -1.51225151 - 0.05781234 - 1.83002882 -

-10 1/256 1.58290914 - -1.66780189 - -1.87108209 -1/512 1.58736775 1.42 -1.67135428 1.46 -1.87951726 1.421/1024 1.58904541 1.48 -1.67267356 1.51 -1.88265567 1.481/2048 1.58965039 1.51 -1.67314544 1.53 -1.88377958 1.521/4096 1.58986289 1.54 -1.67331027 1.56 -1.88417251 1.541/8192 1.58993623 - -1.67336695 - -1.88430769 -

Table 2. Numerical values of f at nodes ti, i ∈ {n/4, n/2,3n/4} and rates of convergence p for parameters α = 0.6,q(t) = 0, b = 1, L = 1

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1 Institute of MathematicsCzestochowa University of Technologyal. Armii Krajowej 21, 42–201 Czestochowa, POLAND

e-mail: [email protected] Received: June 18, 2013

2 Institute of Computer and Information SciencesCzestochowa University of Technologyul. Dabrowskiego 73, 42-201 Czestochowa, POLAND

e-mail: [email protected]

Please cite to this paper as published in:Fract. Calc. Appl. Anal., Vol. 17, No 2 (2014), pp. 307–320;DOI: 10.2478/s13540-014-0170-8

Numerical solution of fractional Sturm-Liouville equation in integral form

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