An efficient numerical approach for solving two-point fractional order nonlinear

R E S E A R C H Open Access

An efficient numerical approach for solving two-point fractional order nonlinear

boundary value problems with Robin boundary conditions

Hyunju Kim¹, Junseo Lee²and Bongsoo Jang^2*

*Correspondence:

bsjang@unist.ac.kr

2Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), 44919, Ulsan, Republic of Korea Full list of author information is available at the end of the article

Abstract

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To

approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point.

To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476,2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs,O(h²) andO(h³) for shooting techniques with Newton’s method and Halley’s method, respectively.

A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

Keywords: Caputo fractional derivative; Nonlinear shooting method;

Predictor–corrector scheme; Robin boundary condition

1 Introduction

Fractional calculus has proven to describe many phenomena in science and engineer- ing more accurately than integer-order calculus because of the nonlocal property of the fractional derivative [2–7]. Many authors have introduced numerical methods for solving fractional differential equations arising in science and engineering. The authors in Refs.

[8,9] proposed a computational algorithm based on reproducing kernel Hilbert space for solving time-fractional partial differential equations in porous media and nonlinear homogeneous and nonhomogeneous time-fractional equations. In Ref. [10], a numerical method based on multiple fractional power series solution was introduced to deal with the

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Schrödinger equation. The authors in Refs. [11–14] proposed several numerical methods based on collocation method, finite difference method, and L1 approximation for solving time-fractional diffusion equations. Regarding Fractional order Boundary Value Problems (FBVPs), Ref. [15] (and the references therein) investigated a Caputo fractional hybrid two- point boundary value problem describing the thermostat models. In Ref. [16], the authors studied a fractional-order nonlocal continuum model of a Euler–Bernoulli beam whose governing equation is described as a FBVP, using the fractional finite element model.

Recently, the authors in Ref. [17] developed a spectral collocation method to deal with two-point linear multi-term FBVPs with Caputo fractional operator. In Ref. [18], the au- thors reformulated two-point FBVPs with a Riemann–Liouville fractional operator to a Volterra integral equation of the second kind and then developed an integral discrete scheme based on finite difference method.

However, numerical methods for solving FNBVPs with Robin Boundary Conditions (RBCs) have been paid less attention to and, in this paper, we consider the two-point Frac- tional order Nonlinear Boundary Value Problem (FNBVP) with RBCs:

⎧⎨

⎩

D^αa²y(t) =f(t,y,D^αa¹y(t)), t∈[a,b],

a1y(a) +b1y(a) =γ₁, a2y(b) +b2y(b) =γ₂,

(1)

where 0 <α₁≤1, 1 <α₂< 2,α₁,α₂,γ₁,γ₂∈R.D^αa¹andD^αa² are Caputo fractional differen- tiations defined as follows.

Definition 1.1 Letα∈R⁺. The operatorJ_a,t^α, defined onL1[a,b] by

J_a^αy(t) := 1 (α)

t a

(t–τ)^α–1y(τ)dτ,

fora≤t≤b, is called the Riemann–Liouville fractional integral operator of orderα.

We setα= 0,J_a⁰=I, the identity operator.

Definition 1.2 letα∈R⁺. The operatorD^α_ais defined by

D^α_ay(t) :=J_a^α–αD^αy(t) = 1 (α–α)

_t

a

(t–τ)^α–αy^(α)(τ)dτ,

where is the ceiling function and is the floor function.

The multi-term Caputo sense FIVP can be transformed into the system of FIVPs by Theorem1.1[19].

Theorem 1.1 Let us consider the following multi-term Caputo sense fractional differential equation with initial conditions:

⎧⎨

⎩

D^α_aⁿy(t) =f(t,y(t),D^αa¹y(t),D^αa²y(t), . . . ,D^αa^n–1y(t)),

y^(j)(a) =y^(j)a, j= 0, 1, . . . ,α_n, (2)

where y^(j)(a)is the jth derivative at t=a,α_n>α_n–1>· · ·α₁> 0,α_i–α_i–1≤1 for all i= 2, 3, . . . ,n and0 <α1< 1.Then we can defineβi,

⎧⎨

⎩ β₁:=α₁,

β_i:=α_i–α_i–1, i= 2, 3, . . . ,n.

Then the multi-term fractional differential equation with initial conditions(2)is equivalent to the following system of fractional differential equations:

D^β_a¹y1(t) =y2(t), D^β_a²y2(t) =y3(t), ...

D^β_a^n–1yn–1=yn(t),

D^β_aⁿy_n=f(t,y₁,y₂, . . . ,y_n–2,y_n–1),

(3a)

together with the initial conditions:

yi(t0) =

⎧⎪

⎪⎨

⎪⎪

⎩

y⁽⁰⁾ if i= 1, y^(l) ifαi–1=l∈N, 0 else,

(3b)

in the following sense:

1.Whenever the function y∈C^αn[a,b]is a solution of the multi-term equation with initial conditions(2),the vector-valued function Y:= (y₁, . . . ,y_n)^Twith

yi(t) :=

⎧⎨

⎩

y(t) if i= 1, D^αa^i–1y(t) if i= 1,

is a solution of the system of fractional differential equations(3a)with initial conditions (3b)

2.Whenever the vector-valued function Y := (y1, . . . ,yn)^T is a solution of the system of multi-order fractional differential equations(3a)with initial conditions(3b),the function y:=y1is a solution of the multi-term equation with initial conditions(2).

In this paper, we propose new schemes to deal with FNBVPs and the algorithms are summarized as follows:

1 In the case that0 <α₁< 1, we transform the FNBVP (1) witha= 0into a system of FIVPs using Theorem1.1.

In the case ofα₁= 1, i.e. the FNBVP (1) has a single term of fractional orderα₂, we substitute the integer orderα₁= 1with the fractional orderα₁= 1 –ε,ε→0+so that the FNBVP satisfies the assumption,0 <α1< 1in Theorem1.1. First, the Gronwall inequality for two-term equations in [19] guarantees that the difference between the solution of FNBVP withα₁= 1and withα₁= 1 –εapproaches0asε→0+. The

FNBVP withα₁= 1 –εis transformed into a system of FIVPs and then we reduce the number of equations in the system. We prove the reduced system is equivalent to the original system asε→0+in Sect.2.

2 To deal with FIVPs, we adopt high-order predictor–corrector methods (HPCMs) with linear interpolation and quadratic interpolation [1] into Volterra integral equations which are equivalent to FIVPs.

3 ICs of the FIVPs in the system equivalent to (1) are obtained by RBC att= 0. But ICs includes:=y(0)and sincesis unknown, we approximatesby means of nonlinear shooting techniques based on Newton’s method and Halley’s method. The error function|a2y(b,s) +b2y(b,s) –γ2|is used to construct the root-finding problem in order to make the approximate solution toy(t)satisfy the RBC att=b.

4 The algorithm of the proposed shooting technique is as follows: The system of FIVPs is solved with an initial approximation tos,s_kat thekth iteration. Using the

approximate solution to the system obtained by HPCMs withs0, we finds1by solving Newton’s (Halley’s) formula. We update the approximate solution to the system with s₁and measure the norm of the error function. We repeat this process until the norm of the error function is within the tolerance.

Similar to our proposed schemes, the authors in Refs. [20,21] introduced numerical meth- ods for solving FBVPs with RBCs. In Refs. [20,21], the FBVP with RBCs is turned into the FIVP by using a shooting method with a guess for the unknown ICy(0) and then the FIVP is transformed into the Volterra integral equation. The integral–differential term in the Volterra integral equation is approximated by an integral discretization scheme with constant and first-order interpolating polynomials in paper [20] and [21], respectively.

However, the integral discretization schemes can only handle linear FBVPs and the rate of convergence depends on the fractional orderα. This is elaborately addressed in Sect.4.

The main advantages of our proposed schemes are as follows:

1 The proposed schemes can handle both linear and nonlinear FBVPs with general RBCs.

2 Our proposed schemes can deal with multi-term FBVPs where0 <α₁≤1and 1 <α₂< 2.

3 Our proposed methods with HPCMs have uniform convergence ratesO(h²)and O(h³)for shooting techniques based on Newton’s method and Halley’s method, respectively, with enough iterations, regardless of fractional orders thanks to the global error estimates of HPCMs in [1].

4 It is not required to solve a matrix system as Newton’s method and Halley’s method are applied into a system of FIVPs.

This article is organized as follows. In Sect.2, we describe an idea about the transfor- mation of FNBVP with RBCs (1) into a system of FIVPs. In Sect.3, we describe nonlin- ear shooting methods based on Newton’s method and Halley’s method, to approximate unknown ICs:=y(0) of FIVPs in the system. Also, we briefly mention how to apply the HPCMs into a system of FIVPs in Sect.3. In Sect.4, we demonstrate numerical examples verifying that the proposed shooting techniques combined with HPCMs guarantee the global convergence rates of HPCMs. We also confirm the performance and effectiveness of the proposed methods by comparing with the modified integral discretization scheme in Ref. [21]. A conclusion will be given in Sect.5. Finally tables of numerical results and

the linear explicit method which is an alternative method for solving FIVPs are described in theAppendix.

2 Problem formulation

In this section, we describe how to transform the FNBVP with RBCs (1) into a system of FIVPs according to the value ofα₁. Basically, we apply Theorem1.1withβ₁:=α₁,β₂:=

1 –α₁,β₃:=α₂– 1 to the FNBVP in the case of 0 <α₁< 1. Ifα₁is equal to 1, then we replace α₁with 1 –,→0+ and setβ₁:= 1 –,β₂:=,β₃:=α₂– 1. We reduce the size of system using the factβ₂→0+.

Case 1:0 <α₁< 1

First, we consider a FNBVP with Dirichlet boundary conditions as follows:

⎧⎨

⎩

D^α₀²y(t) =f(t,y(t),D^α₀¹y(t)), t∈[0,b]

y(0) =y0, y(b) =yb. (4)

Applying Theorem1.1withβ₁:=α₁,β₂:= 1 –α₁,β₃:=α₂– 1, the FNBVP with Dirichlet boundary conditions (4) can be transformed as follows:

⎧⎪

⎪⎨

⎪⎪

⎩

D^α₀¹y(t) =w(t), y(0) =y0,y(b) =yb, D^1–α₀ ¹w(t) =z(t), w(0) = 0,

D^α₀^2–1z(t) =f(t,y(t),w(t)), z(0) =y⁽¹⁾(0).

(5)

From the system of fractional differential equations (5), we obtain the following system of FIVPs:

⎧⎪

⎪⎨

⎪⎪

⎩

D^α₀¹y(t) =w(t), y(0) =y0, D^1–α₀ ¹w(t) =z(t), w(0) = 0, D^α₀^2–1z(t) =f(t,y(t),w(t)), z(0) =s,

(6)

where the ICsis unknown and so needs to be approximated.

Similar to the case of Dirichlet boundary conditions, the FNBVP with RBCs (1) can be written as follows:

⎧⎪

⎪⎨

⎪⎪

⎩

D^α₀¹y(t) =w(t), y(0) =y₀=^γ1–b_a ^1s

1

D^1–α₀ ¹w(t) =z(t), w(0) = 0 D^α₀^2–1z(t) =f(t,y(t),w(t)), z(0) =s.

(7)

Case 2:α₁= 1

We consider the following FBVP with Dirichlet boundary conditions:

⎧⎨

⎩

D^α₀²y(t) =f(t,y(t),y(t)), t∈[0,b], y(0) =y0, y(b) =yb,

(8)

where 1 <α₂< 2,α₂∈R. Since the fractional differential equation in (8) does not satisfy the assumption, 0 <α₁< 1 in Theorem1.1, we cannot apply the strategy used in (4) to

(8). So we modify the equation in (8) to meet the assumption, with the same boundary conditions as in (8) as follows:

⎧⎨

⎩

D^α₀²y(t) =f(t,y(t),D^1–ε₀ y(t)), t∈[0,b], y(0) =y0, y(b) =yb,

(9)

whereα2∈(1, 2),→0+. By Lemma2.1, solutions of the two FBVPs (8) and (9) are ap- proximately equal and the absolute error depends on.

Lemma 2.1 (First Gronwall inequality for two-term equations in [19]) Let α₂> 0and α₁,α˜₁∈(0,α₂)be chosen so that the equation

D^α₀²y(t) =f

t,y(t),D^α₀¹y(t) , subject to the initial conditions

y(0) =y0, y⁽¹⁾(0) =y¹₀, . . . , y^(α2–1)(0) =y^α₀^2–1 and

D^α₀²z(t) =f

t,z(t),D^α₀^˜1z(t)˜ subject to the same initial conditions

z(0) =y₀, z⁽¹⁾(0) =y¹₀, . . . , z^(α2–1)(0) =y₀^α2–1

(where f satisfies a Lipschitz condition in its second and third arguments on a suitable domain)has unique continuous solutions y,z: [0,T]→R.We assume further thatα₁= ˜α1.Then there exist constants K1and K2such that

y(t) –z(t) ≤K1|α₁–α˜₁|Eαn

K2T^α2

, ∀t∈[0,T], where Eα_ndenotes the Mittag-Leffler function of orderα_n.

Applying Theorem1.1to (9), we obtain the following system of FIVPs:

⎧⎪

⎪⎨

⎪⎪

⎩

D^1–₀ y(t) =w(t), y(0) =y0, D₀w(t) =z(t), w(0) = 0, D^α₀^2–1z(t) =f(t,y(t),w(t)), z(0) =s.

(10)

Now, we show that the system of FIVPs (10) is equivalent to the following system as→0 using Lemmas2.2through2.4and Theorem2.1:

⎧⎨

⎩

D^1–₀ ˜y(t) =˜z(t), y(0) =˜ y0,

D^α₀^2–1˜z(t) =f(t,˜y(t),z(t)),˜ z(0) =˜ s. (11)

Lemma 2.2(Theorem 2.10 in [19]) Let f ∈C[a,b]andα≥0.Moreover,assume thatα_k is a sequence of positive numbers such thatαk→αas k→ ∞.Then,for everyδ> 0,

k→∞lim sup

t∈[a+δ,b]

J_a^αkf(t) –J_a^αf(t) = 0.

Lemma 2.3(Lemma 6.19 in [19]) Letα,T,φ₁,φ₂∈R⁺.Moreover,assume thatδ: [0,T]→ Ris a continuous function satisfying the inequality

δ(t) ≤φ1+ φ₂ (α)

t 0

(t–τ)^α–1 δ(τ) dτ, ∀t∈[0,T].

Then

δ(t) ≤φ₁Eα

φ₂t^α

, ∀t∈[0,T].

Lemma 2.4 Let0 <γ ≤α≤β.Then,for any t∈[a,b], J_a^αy(t) –J_a^βy(t) ≤

(γ)

(α)(b–a)^α–γ+(γ)

(β)(b–a)^β–γ

J_a^γ y(t) . Proof By Definition1.1,

J_a^αy(t) +J_a^βy(t) = 1 (α)

t a

(t–τ)^α–1y(τ)dτ+ 1 (β)

t a

(t–τ)^β–1y(τ)dτ

= 1

(α) _t

a

(t–τ)^{γ–1+(α–γ)}y(τ)dτ

+ 1

(β) _t

a

(t–τ)^{γ–1+(β–γ)}y(τ)ds.

Then

J_a^αy(t) –J_a^βy(t) ≤(b–a)^α–γ (α)

_t

a

(t–τ)^γ–1 y(τ) dτ +(b–a)^β–γ

(β) _t

a

(t–τ)^γ–1 y(τ) dτ

≤ (γ)

(α)(b–a)^α–γ+(γ)

(β)(b–a)^β–γ 1

(γ) _t

a

(t–τ)^γ–1 y(τ) dτ

≤ (γ)

(α)(b–a)^α–γ+(γ)

(β)(b–a)^β–γ

J_a^γ y(t) . Theorem 2.1 Let1 <α₂< 2,^∀T∈R⁺and f : [0,T]×R×R→Rsatisfies the Lipschiz con- dition in its second and third arguments on a suitable domain.Then we have the following inequality:

f(t,x1,y1) –f(t,x2,y2) ≤L

|x2–x1|+|y2–y1| , where^∀t∈[0,T],x1,x2,y1,y2: [0,T]→Rand0 <L.

If,for any0 <1,y andˆ y are solutions of the following systems,˜ respectively:

⎧⎪

⎪⎨

⎪⎪

⎩

D^1–₀ ˆy(t) =w(t), y(0) =y0, D₀w(t) =z(t), w(0) = 0, D^α₀^2–1z(t) =f(t,ˆy(t),w(t)), z(0) =s, and

⎧⎨

⎩

D^1–₀ ˜y(t) =˜z(t), y(0) =˜ y₀, D^α₀^2–1˜z(t) =f(t,˜y(t),z(t)),˜ z(0) =˜ s, then

y(t) –ˆ y(t)˜ →0, as→0.

Proof By Lemma 6.2 in [19], FIVPsD^1–₀ y(t) =ˆ w(t),y(0) =y0, andD^1–₀ y(t) =˜ z(t),˜ y(0) =˜ y0

are equivalent to Volterra integral equations of the second kind, respectively, as follows:

ˆ

y(t) =y0+J₀^1–w(t), y(t) =˜ y0+J₀^1–z(t).˜

Thenˆy(t) –˜y(t) can be expressed as a Riemann–Liouville fractional integral ofw(t) –˜z(t) as follows:

ˆ

y(t) –y(t) =˜ J₀^1–

w(t) –z(t)˜

. (12)

Sincew(t) =Jz(t) by Lemma 6.2 in [19] and rewritingw(t) –z(t) as˜ w(t) –J₀z(t) +˜ J₀z(t) –˜

˜

z(t), we obtain the following inequality:

w(t) –z(t)˜ ≤J₀ z(t) –˜z(t) + J₀z(t) –˜ z(t)˜ . (13) SinceD^α₀^2–1z(t) =f(t,y(t),ˆ w(t)) andD^α₀^2–1z(t) =˜ f(t,y(t),˜ z(t)) are equivalent to Volterra in-˜ tegral equationsz(t) =s+J₀^α2–1f(t,y(t),ˆ w(t)) andz(t) =˜ s+J₀^α2–1f(t,y(t),˜ w(t)) by Lemma 6.2 in [19] respectively, using the Lipschitz condition, (12), and (13), we obtain the following inequalities:

z(t) –z(t)˜ = J₀^α2–1 f

t,y(t),ˆ w(t) –f

t,y(t),˜ ˜z(t)

≤J₀^α2–1 f

t,y(t),ˆ w(t) –f

t,y(t),˜ z(t)˜

≤LJ₀^α2–1 ˆy(t) –˜y(t) + w(t) –z(t)˜

≤LJ₀^α2–1

J₀^1– w(t) –z(t)˜ +J₀ z(t) –z(t)˜ (14)

+ J₀˜z(t) –z(t)˜ . (15)

Since 1 <α₂ < 2 and (α₂) = (α₂– 1)(α₂– 1), we have the following inequalities for J₀^α2–1|Jz(t) –˜ z(t)˜ |:

J₀^α2–1 Jz(t) –˜ z(t)˜ = 1 (α₂– 1)

_t

0

(t–τ)^α2–2 J₀˜z(t) –z(t)˜ dτ

≤ T^α2–1

(α₂– 1)J₀z(t) –˜ z(t)˜ _∞

≤T^α2–1

(α2)J₀z(t) –˜ z(t)˜ _∞. (16)

Similarly, we can obtain the following inequality:

J₀^α2– J₀z(t) –˜ z(t)˜ ≤ T^α2–

(α2–+ 1)J₀z(t) –˜ z(t)˜

∞. (17) Applying the inequality (16) into (14), we have the following inequality:

z(t) –z(t)˜ ≤L

J₀^α2– w(t) –z(t)˜ +J₀^α2–1+ z(t) –z(t)˜ +T^α2–1

(α2)J₀z(t) –˜ z(t)˜ _∞

. (18)

Using the inequalities (13) and (17), we have the following inequalities:

J₀^α2– w(t) –z(t)˜ ≤J₀^α2–

J₀ z(t) –z(t)˜ + J₀z(t) –˜ ˜z(t)

≤J₀^α2 z(t) –z(t)˜ + T^α2–

(α2–+ 1)J₀z(t) –˜ ˜z(t)_∞. (19) Applying the inequality (19) into (18) and using Lemma2.4withγ=α₂– 1,α=α₂– 1 +, β=α₂, we have the following inequalities:

z(t) –z(t)˜ ≤L

J₀^α2 z(t) –z(t)˜ + T^α2–

(α2–+ 1)J₀˜z(t) –z(t)˜ _∞ +J₀^α2–1+ z(t) –z(t)˜ +T^α2–1

(α2)J₀z(t) –˜ z(t)˜ _∞

=L

J₀^α2– z(t) –z(t)˜ +J₀^α2–1+ z(t) –z(t)˜ +

T^α2–

(α2–+ 1)+T^α2–1 (α2)

J₀z(t) –˜ ˜z(t)_∞

≤L

(α2– 1)

(α2– 1 +)T+(α2– 1) (α2) T

J₀^α2–1 z(t) –z(t)˜ +

T^α2–

(α₂–+ 1)+T^α2–1 (α₂)

J₀z(t) –˜ ˜z(t)_∞ , so that

z(t) –z(t)˜ ≤LC¹J₀^α2–1 z(t) –z(t)˜ +LC²J₀z(t) –˜ z(t)˜

∞, (20) where

C¹≡ (α2– 1)

(α2– 1 +)T+(α2– 1)

(α2) T, C²≡ T^α2–

(α2–+ 1)+T^α2–1 (α2).

Applying Lemma2.3to inequality (20), we obtain the following inequality:

z(t) –z(t)˜ ≤LC²J₀z(t) –˜ z(t)˜ _∞Eα₂–1

LC¹T^α2–1 . Therefore,

y(t) –˜ y(t)ˆ ≤J₀^1– w(t) –z(t)˜

≤J₀¹ z(t) –z(t)˜ +J₀^1– J₀z(t) –˜ z(t)˜

≤T z(t) –˜z(t) + T^1–

(2 –)J₀z(t) –˜ z(t)˜ _∞

≤TLC²J₀z(t) –˜ z(t)˜ _∞Eα2–1

LC¹T^α2–1 + T^1–

(2 –)J₀˜z(t) –˜z(t)_∞

=

TLC²Eα₂–1

LC¹T^α2–1

+ T^1–

(2 –)

J₀z(t) –˜ z(t)˜ _∞.

We haveC≡TLC²Eα2–1[LC¹T^α2–1] +_(2–)^T1– . Then y(t) –˜ y(t)ˆ ≤CJ₀z(t) –˜ z(t)˜ _∞.

Thus, by Lemma2.2,

y(t) –˜ y(t)ˆ ≤CJ₀z(t) –˜ z(t)˜ _∞→0, as→0.

For the FNBVP with RBCs (1), similarly, by Lemma2.1and using Theorem1.1, we obtain the following FIVPs:

⎧⎪

⎪⎨

⎪⎪

⎩

D^1–₀ y(t) =w(t), y(0) =^γ1–b_a ^1s

1 , D₀w(t) =z(t), w(0) = 0, D^α₀^2–1z(t) =f(t,y(t),w(t)), z(0) =s.

(21)

The system (21) can be reduced as follows:

⎧⎨

⎩

D^1–₀ ˜y(t) =˜z(t), y(0) =˜ ^γ1_a^–b1s

1 ,

D^α₀^2–1˜z(t) =f(t,˜y(t),z(t)),˜ z(0) =˜ s. (22) 3 Nonlinear shooting methods and high-order predictor–corrector methods FBVPs have been transformed to systems of FIVPs in Sect.2. Before we address how to deal with systems of FIVPs (6), (7), (11), and (22) using High-order Predictor–Corrector Methods (HPCMs), the unknown ICz(0) =sshould be handled first. In this section, we de- scribe two nonlinear shooting techniques based on Newton’s method and Halley’s method to approximates. Both Newton’s formula and Halley’s formula are designed to determine the solution of a system of FIVPs satisfying the RBC at the right end point of an interval.

Without loss of generality, we consider the system of FIVPs (7) that is equivalent to the FNBVP with RBCs (1).

In order that the RBC at the right end pointa2y(b) +b2y(b) =γ₂is involved in approxi- matings, we definey(s) :=y(s,t)|t=band let the error function beF(s) :=a₂y(s) +b_2∂t^∂y(s) – γ₂. We approximate the solution of the root-finding problemF(s) = 0 by using Newton’s method and Halley’s method, respectively. For convenience, we denote

gs(t) =∂g(s,t)

∂s , gss=∂²g(s,t)

∂s²

throughout this section.

3.1 Shooting with Newton’s method

The conventional Newton formula forF(s) = 0 can be expressed as follows:

s_k+1=s_k– F(sk)

Fs(sk), k= 0, 1, 2, . . . ,m, (23)

wheremis the maximum number of iterations and Fs(sk) = ∂F

∂s(s) s=sk,t=b

=a2

∂y

∂s(s,t) s=s_k,t=b

+b2

∂

∂s ∂y

∂t(s,t) s=s_k,t=b

. (24)

Observing y_s(s_k) and y_ts(s_k), it turns out that they are equal to _∂s^∂y(t)|s=sk,t=b and

∂

∂sz(t)|s=sk,t=b, respectively, in the system of FIVPs (7). Thus we solve the following sys- tem obtained from the system of FIVPs (7) by applying the operator_∂s^∂ using HPCMs for eachk:

⎧⎪

⎪⎨

⎪⎪

⎩

D^α₀¹ys(t) =ws(t), ys(0) = –b₁/a₁, D^1–α₀ ¹ws(t) =zs(t), ws(0) = 0, D^α₀^2–1z_s(t) =f_s(t,y(t),w(t)), z_s(0) = 1.

(25)

Since bothtandsare independent variables,f_s(t,y(t),w(t)) can be written as fs

t,y(t),w(t)

=fy·ys(t) +fw·ws(t). (26)

The detailed description of HPCMs dealing with a system of FIVPs is in Sect.3.3. By solv- ing the system (25),sk+1 in Newton’s formula (23) is computed. Using the updated ap- proximate value of ICs,sk+1, we update approximate solutions of systems of FIVPs (6), (7), (11), and (22). Repeating this process, we obtain anskhaving an acceptable error of the root-finding problemF(s) = 0 at an appropriate number of iterationsk.

3.2 Shooting with Halley’s method

The conventional Halley formula forF(s) = 0 is as follows:

sk+1=sk– 2F(s_k)F_s(s_k)

2F_s²(sk) –F(sk)Fss(sk), k= 0, 1, 2, . . . ,m, (27)

whereFs(sk) is described in (24) and Fss(sk) =∂²F

∂s²(s) s=s_k,t=b

=a2

∂²y

∂s²(s,t) s=sk,t=b

+b2

∂²

∂s² ∂y

∂t(s,t) s=sk,t=b

.

Similar to the way we foundys(sk) andyts(sk) in the shooting with Newton’s method, we findyss(sk) andzss(sk) by solving the following system of FIVPs obtained by applying the operator_∂s^∂22 using HPCMs for eachk:

⎧⎪

⎪⎨

⎪⎪

⎩

D^αa¹y_ss(t) =w_ss(t), y_ss(0) = 0, D^1–αa ¹wss(t) =zss(t), wss(0) = 0, D^αa^2–1zss(t) =fss(t,y(t),w(t)), zss(0) = 0.

(28)

Sincetandsare independent variables,f_ss(t,y(t),w(t)) can be written as

fy·yss(t) +fw·wss(t) +fyy·ys(t)²+fww·ws(t)²+fwy·ws(t)ys(t). (29) 3.3 High-order predictor–corrector methods for system of FIVPs

In order to find askwith an acceptable accuracy, we iteratively solve systems of FIVPs (25) or (28). Once we find thesk, we solve systems of FIVPs (6), (7), (11), or (22). In this subsec- tion, we describe how to deal with those systems of FIVPs using High-order Predictor–

Corrector Methods (HPCMs) introduced in Ref. [1]. Without loss of generality, we con- sider the following FIVP:

⎧⎨

⎩

D^α₀y(t) =f(t,y(t)), t∈[0,b],

D⁽ⁱ⁾y(0) =ci, i= 0, . . . ,α. (30)

For convenience, let us denoteyjas approximated value ofy(tj) except fory0=c0and let fj≡f(tj,yj),y^c_jbe a corrector ofyj,y^p_j be a predictor ofyj, andf_j^p≡f(tj,y^p_j),j= 1, . . . ,N. If j= 0 thenf0=f(0,c0). We divide the domainas follows:

_N:={t_j|a=t₀<· · ·<t_j<· · ·<t_n<t_n+1<· · ·<t_N =b}.

For simplicity, let the step size be uniform, which meanstj+1–tj=h,j= 0, 1, . . . ,N– 1.

Then (30) can be rewritten at timetn+1as follows:

y(tn+1) =g(tn+1) + 1 (α)

n j=0

_tj+1

tj

(tn+1–τ)^α–1f τ,y(τ)

dτ, (31)

whereg(tn+1) =_α

i=0 (tn+1)ⁱ

i! ci. We interpolatef(τ,y(τ)) using linear or quadratic Lagrange polynomials over each intervalI_j= [t_j,t_j+1],j= 0, 1, . . . ,N– 1. Then we obtain the following predictor–corrector schemes.

1 HPCM with linear Lagrange polynomial:

y^c_n+1=g(t_n+1) + 1 (α)

_n–1

j=0

B^1,j_n+1f_j+B^2,j_n+1f_j+1

+B^1,n_n+1fn+B^2,n_n+1f_n+1^P

,

(32)

where B^1,j_n+1=1

h t_j+1

tj

(t_n+1–τ)^α–1(t_j+1–τ)dτ,

B^2,j_n+1= –1 h

_tj+1

tj

(tn+1–τ)^α–1(tj–τ)dτ,

y^P_n+1=g(tn+1) +Gα,f(tn+1) +b¹_n+1fn–1+b²_n+1fn, (33) b¹_n+1=1

h tn+1

tn

(t_n+1–τ)^α–1(t_n–τ)dτ, b²_n+1= –1

h _tn+1

tn

(t_n+1–τ)^α–1(t_n–1–τ)dτ,

G_α,f(t_n+1) = 1 (α)

n–1 j=0

B^1,j_n+1f_j+B^2,j_n+1f_j+1 .

2 HPCM with quadratic Lagrange polynomial:

y^c_n+1=g(tn+1) + 1 (α)

A^1,0_n+1f0+A^2,0_n+1f1/2+A^3,0_n+1f1

+ n–1

j=1

A^1,j_n+1f_j–1+A^2,j_n+1f_j+A^3,j_n+1f_j+1

+A^1,n_n+1fn–1+A^2,n_n+1fn+A^3,n_n+1f_n+1^P

,

(34)

where A^1,0_n+1= 2

h² t_j+1

tj

(t_n+1–τ)^α–1(t_1/2–τ)(t₁–τ)dτ,

A^2,0_n+1= –4 h²

_tj+1

tj

(tn+1–τ)^α–1(t0–τ)(t1–τ)dτ,

A^3,0_n+1= 2 h²

_tj+1

tj

(tn+1–τ)^α–1(t0–τ)(t1/2–τ)dτ, A^1,j_n+1= 1

2h² tj+1

tj

(t_n+1–τ)^α–1(t_j–τ)(t_j+1–τ)dτ,

A^2,j_n+1= –1 h²

_tj+1

tj

(tn+1–τ)^α–1(tj–1–τ)(tj+1–τ)dτ,

A^3,j_n+1= 1 2h²

_tj+1

tj

(tn+1–τ)^α–1(tj–1–τ)(tj–τ)dτ,

and the predictorf_n+1^p is found as follows:

y^P_n+1=g(tn+1) +Gα,f(tn+1) +a¹_n+1fn–2+a²_n+1fn–1+a³_n+1fn, (35) where

Gα,f(t_n+1) = 1 (α)

A^1,0_n+1f0+A^2,0_n+1f1/2+A^3,0_n+1y(t₁)

+ n–1

j=0

A^1,j_n+1fj–1+A^2,j_n+1fj+A^3,j_n+1fj+1

,

2a¹_n+1= 1 2h²

_tn+1

tn

(tn+1–τ)^α–1(tn–1–τ)(tn–τ)dτ, a²_n+1= –1

h² _tn+1

tn

(tn+1–τ)^α–1(tn–2–τ)(tn–τ)dτ, a³_n+1= 1

2h² tn+1

tn

(tn+1–τ)^α–1(tn–2–τ)(tn–1–τ)dτ.

Remark3.1 Using the HPCM with linear and quadratic interpolations, we find predictors y^p_n+1,w^p_n+1,z^p_n+1 individually and then find correctorsy^c_n+1,w^c_n+1,z^c_n+1. The entire steps of the proposed schemes are summarized in AppendixB.

The following theorems [1] bound the global errorEn+1of the HPCM with linear and quadratic interpolations, respectively.

Theorem 3.1(Theorem 2.4 in [1]: Global error of HPCM with linear interpolation) De- fine En+1to be global error.Suppose f(·,y(·))∈C²[a,b]and furthermore is Lipschitz contin- uous in the second argument,then we have

En+1= y(tn+1) –˜yn+1 ≤O h²

, given E1≤Ch².

Theorem 3.2(Theorem 3.4 in [1]: Global error of HPCM with quadratic interpolation) Suppose f(·,y(·))∈C³[a,b]and is Lipschitz continuous in the second argument,then we have

En+1≤O h³

,

given E1,E2≤O(h³)and E1/2≤O(h^3–α), 0 <α< 1.

4 Numerical examples

In this section, we experimentally illustrate the performance of the proposed schemes. Nu- merically, we verify that our proposed schemes can deal with more complex FBVPs than