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Research article Special Issues

The parameter-Newton iteration for the second-order cone linear complementarity problem

  • In this paper, we propose the parameter-Newton (PN) method to solve the second-order linear complementarity problem (SOCLCP). The key idea of PN method is that we transfer the SOCLCP into a system of nonlinear equations by bringing in a parameter. Then we solve the system of nonlinear equations by Newton method. At last, we prove that the PN method has quadratic convergence. Compared with the bisection-Newton (BN) method, the PN method has less CPU time and higher accuracy in numerical tests.

    Citation: Peng Zhou, Teng Wang. The parameter-Newton iteration for the second-order cone linear complementarity problem[J]. Electronic Research Archive, 2022, 30(4): 1454-1462. doi: 10.3934/era.2022076

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  • In this paper, we propose the parameter-Newton (PN) method to solve the second-order linear complementarity problem (SOCLCP). The key idea of PN method is that we transfer the SOCLCP into a system of nonlinear equations by bringing in a parameter. Then we solve the system of nonlinear equations by Newton method. At last, we prove that the PN method has quadratic convergence. Compared with the bisection-Newton (BN) method, the PN method has less CPU time and higher accuracy in numerical tests.



    Studying the theory of fractional differential equations (FDEs) and fractional integral equations (FIEs) is crucial because they are used in many modeling applications. Fractional equations are essential for many areas of fundamental analysis and their applications in economics, physics, and other disciplines. Quadratic integral equations (QIE), in particular, tend to be helpful in describing a wide range of everyday issues, such as theory of radiative transfer, the theory of neutron transport, the kinetic theory of gases, the queuing theory, and the traffic theory (see, for example, [1,2,3,4]).

    One of the studied QIEs is called the hybrid integral equation (HIE); see [5]. This issue has received great attention in the last few years; see [6,7,8,9]. As we see, the hybrid fixed point theory is used to develop the existing solution of the hybrid equations; see [10,11,12,13]. Other researchers focused on the analysis of QIEs in Orlicz spaces [14], equations of QIEs with fractional order arising in the queuing theory and biology [15], and the analysis of QIEs depending on both Schauder and Schauder–Tychonoff fixed point principles [16].

    The Chandrasekhar quadratic integral equation (CQIE) occurs in the theory of radiative transfer in a plane-parallel atmosphere [2]. The radiative transfer process and the integral equation for the scattering function and transmitted functions were developed by Chandrasekhar's work in the 1950s; see [14]. This work quickly turned into a major scientific topic in both mathematics and astrophysics, see [15,16]. The radiative transfer process and the simultaneous integral equation for the transmitted and scattering functions were developed in Chandrasekhar's seminal work from 1960. In [17], the simultaneous integral equation of Chandrasekhar was presented, along with the iteration scheme for the transmitted and scattering functions.

    CQIE takes the form

    {Φ(ϱ)=φ(ϱ)+ϱΦ(ϱ)ϱ0ϱϱ+sˆg1(s,Φ(s))ds+f1(ϱ,Φ(ϱ))RLJμˆg2(ϱ,Φ(ϱ)),ϱˆI,ˆI=[0,b],μ(0,1), (1.1)

    where RLJμ denotes the Riemann-Liouville fractional integral (RLFI) of order μ, f1(ϱ,Φ(ϱ))C(ˆI×R,R{0}), and ˆgκ(ϱ,Φ(ϱ))C(ˆI×R,R), for κ=1,2.

    As a result of the applications of these equations, researchers were interested in studying them, and as a result of the difficulty of finding exact solutions, CQIEs are solved using the ADM and the PM. The ADM provides many advantages, including the ability to solve a variety of linear and nonlinear equations in deterministic or stochastic fields effectively and present an analytical solution for all of these equation types without requiring linearization or discretization. Additionally, it is reliable and provides faster convergence than other classical methods. Moreover, we use measures of weak noncompactness to study existence and uniqueness results. These results demonstrate that the two solutions provide nearly equal accuracy; however, when comparing the time required in each case, the ADM is found to take less time than the PM.

    It can be summarized as follows: first, the second section introduces the basic concepts of the measure of noncompactness and the hypothesis. In the third section, we show that if the solution exists, then it will be unique and convergent. After that, we solve some nonlinear Chandraseker QIEs with fractional orders with a comparison between ADM and PM techniques. Finally, graphs are also constructed to illustrate the effectiveness of these two approaches and to compare them.

    In the complement of this work, the classical Banach space C(ˆI)=C[0,b] is used, which contains all real continuous functions defined on ˆI having the norm

    Φ=max{|Φ(ϱ)|:ϱˆI}. (2.1)

    Let us recall the MNC definition in C(ˆI) which is used in this investigation and fix a bounded nonempty subset Ω of C(ˆI). For ηΩ and ϵ>0, the modulus of continuity of the function η on the interval ˆI is defined by

    ω(η,ϵ)=sup{|η(ϱ)η(s)|:ϱ,sˆI,|ϱs|ϵ} (2.2)

    and

    ω(Ω,ϵ)=sup{ω(η,ϵ):ηΩ},ω0(Ω)=limϵ0ω(Ω,ϵ). (2.3)

    It is well-known that ω0(Ω) is a measure of noncompactness inC(ˆI)such that the Hausdorff measure χ may be expressed by the formula

    χ(Ω)=12ω0(Ω), (2.4)

    see [18]. We use the following theorem to prove our investigation.

    Theorem 1. [18] Let Q be a bounded, nonempty, and closed convex subset in the space E. Also, let ˆH:QQ be a continuous operator such that χ(ˆHΩ)l χ(Ω) to any nonempty subset Ω of Q, as l[0,1) is constant, so ˆH has a fixed point in the set Q.

    Considering the hypothesis:

    i. φ :ˆIR is continuous.

    ii. f1:ˆI×RR is continuous and the function f1(ϱ,0) C(R),R is the space of all bounded continuous functions, and there exists a positive constant M=supϱˆI|f1(ϱ,0)|.

    iii. |f1(ϱ,Φ)f1(ϱ,ζ)|L|Φζ| for any ϱˆI for all Φ,ζR.

    iv. ˆgκ:ˆI×RR,κ =1,2 satisfies the Carathéodory condition (CC) so, it is measurable in ϱ for all ΦR and continuous in Φ for all ϱˆI, and there exist functions mκ, κ =1,2L1 such that:

    |ˆgκ(ϱ,Φ)|mκ(ϱ) for all (ϱ,Φ)ˆI×R. (2.5)

    v. RLJγm2(ϱ)<M2, γμ, C0, and b01ϱ+sm1(s)dsM1.

    vi. There exists a number R0>0, such that

    R0=[φ+M2LbμγΓ(μγ+1)+M2MbμγΓ(μγ+1)][1b2M1]1. (2.6)

    Define the ball BR0 as

    BR0={ΦC(ˆI):ΦR0}. (3.1)

    Theorem 2. Using the hypotheses (i)(vi), if [M2LbμγΓ(μγ+1)+M1b2]<1, then there exists at least a solution ΦC(ˆI) for HIE (1.1).

    Proof. Let the operator ˆH defined on C(ˆI) be

    (ˆHΦ)(ϱ)=φ(ϱ)+ϱΦ(ϱ)ϱ0ϱϱ+sˆg1(s,Φ(s))ds+f1(ϱ,Φ(ϱ))ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ(s))ds,ϱˆI.

    From the hypotheses (i)(vi), the function ˆHΦ is continuous on ˆI for any ΦBR0. Further, applying the given hypothesis, we derive the following estimate:

    |ˆHΦ(ϱ)||φ(ϱ)|+|ϱΦ(ϱ)|ϱ0ϱϱ+sm1(s)ds+|f1(ϱ,Φ)|ϱ0(ϱs)μ1Γ(μ)m2(s)dsφ+|ϱΦ(ϱ)|ϱ0ϱϱ+sm1(s)ds+[|f1(ϱ,Φ)f1(ϱ,0)|+|f1(ϱ,0)|]ϱ0(ϱs)μ1Γ(μ)m2(s)dsφ+|b2Φ(ϱ)|b01ϱ+sm1(s)ds +[|f1(ϱ,Φ)f1(ϱ,0)|+|f1(ϱ,0)|]RLJμγRLJγm2(ϱ)φ+b2R0M1+M2LR0bμγΓ(μγ+1)+M2MbμγΓ(μγ+1). (3.2)

    So ˆHΦ is bounded on the interval ˆI. Also, we get

    ˆHΦφ+b2R0M1+M2LR0bμγΓ(μγ+1)+M2MbμγΓ(μγ+1),

    for the operator ˆH, which transforms the ball BR0 into itself, and R0 is:

    R0=[φ+M2MbμγΓ(μγ+1)][1b2M1+M2LbμγΓ(μγ+1)]1>0.

    Now, we are going to show that the operator ˆH is continuous on the ball BR0. So, we need to prove that the operator G2 defined by

    G2Φ(ϱ)=ϱ0(ϱs)μ1Γ(μ)g2(s,Φ(s))ds,ϱˆI,

    is continuous on BR0. To do this, fix ϵ>0, let Φ0BR0, and from hypothesis (ii), we find δ>0 such that ||ΦΦ0||δ, and then we have |ˆg2(s,Φ)ˆg2(s,Φ0)|ϵ for s ˆI, where Φ is any arbitrary element in BR0. For arbitrary fixed ϱˆI, we get

    |G2Φ(ϱ)G2Φ0(ϱ)|=1Γ(μ)ϱ0(ϱs)μ1|ˆg2(s,Φ(s))ˆg2(s,Φ0(s))|dsϵΓ(μ)ϱ0(ϱs)μ1dsϵΓ(μ+1).

    The operator f1Φ(ϱ)=f1(ϱ,Φ(ϱ)) is continuous, and then the operator f1.G2 is continuous on BR0, and similarly, we can prove that the operator

    G1Φ(ϱ)=ϱΦ(ϱ)ϱ0ϱϱ+sˆg1(s,Φ(s))ds,ϱˆI,

    is continuous on BR0. This shows that ˆH is continuous on BR0. Let χ be a nonempty subset of BR0. Fix ϵ>0, and choose Φχ and ϱ1,ϱ2ˆI such that |ϱ2ϱ1|ϵ. Let ϱ1ϱ2, and then

    |(ˆHΦ)(ϱ2)(ˆHΦ)(ϱ1)|=|φ(ϱ2)φ(ϱ1)
    +ϱ2Φ(ϱ2)ϱ20ϱ2ϱ2+sˆg1(s,Φ(s))dsϱ1Φ(ϱ1)ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))ds+ϱ2Φ(ϱ2)ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))dsϱ2Φ(ϱ2)ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))ds+f1(ϱ2,Φ(ϱ2))RLJμˆg2(ϱ2,Φ(ϱ2))f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ1,Φ(ϱ1))+f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ2,Φ(ϱ2))f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ2,Φ(ϱ2))|.

    Hence,

    |(ˆHΦ)(ϱ2)(ˆHΦ)(ϱ1)||φ(ϱ2)φ(ϱ1)|+|ϱ2Φ(ϱ2)|ϱ20|ϱ2ϱ2+sϱ1ϱ1+s|m1(s)ds+|ϱ2Φ(ϱ2)ϱ1Φ(ϱ1)|ϱ10ϱ1ϱ1+sm1(s)ds+|ϱ2Φ(ϱ2)|ϱ2ϱ1|ϱ2ϱ2+s|m1(s)ds+[f1(ϱ2,Φ(ϱ2))f1(ϱ2,Φ(ϱ1))]RLJμˆg2(ϱ2,Φ(ϱ2))+[f1(ϱ2,Φ(ϱ1))f1(ϱ1,Φ(ϱ1))]RLJμˆg2(ϱ2,Φ(ϱ2))+f1(ϱ1,Φ(ϱ1))[RLJμˆg2(ϱ2,Φ(ϱ2))RLJμˆg2(ϱ1,Φ(ϱ1))],

    and so

    |RLJμˆg2(ϱ2,Φ(ϱ2))RLJμˆg2(ϱ1,Φ(ϱ1))|=|ϱ10(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))ds+ϱ2ϱ1(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))dsϱ10(ϱ1s)μ1Γ(μ)ˆg2(s,Φ(s))ds||ϱ10(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))ds+ϱ2ϱ1(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))dsϱ10(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))ds||ϱ2ϱ1(ϱ2s)μ1Γ(μ)ˆg2(s,Φ(s))ds|.

    Then

    |RLJμˆg2(ϱ2,Φ(ϱ2))RLJμˆg2(ϱ1,Φ(ϱ1))|RLJμϱ1|ˆg2(ϱ2,Φ(ϱ2))|RLJμϱ1m2(ϱ2)RLJμγϱ1 RLJγϱ1m2(ϱ2)M2(ϱ2ϱ1)μγΓ(μγ+1).

    Then

    |(ˆHΦ)(ϱ2)(ˆHΦ)(ϱ1)||φ(ϱ2)φ(ϱ1)|+b2R0ϱ10|ϱ2ϱ1|ϱ1+s(ϱ2+s)m1(s)ds+b2R0ϱ2ϱ11ϱ2+sm1(s)ds+b[|ϱ2Φ(ϱ2)ϱ2Φ(ϱ1)|+|ϱ2Φ(ϱ1)ϱ1Φ(ϱ1)|]b01ϱ1+sm1(s)ds+|f1(ϱ2,Φ(ϱ2))f1(ϱ2,Φ(ϱ1))|RLJμγ RLJγm2(ϱ2)+|f1(ϱ2,Φ(ϱ1))f1(ϱ1,Φ(ϱ1))|RLJμγRLJγm2(ϱ2)+|f1(ϱ1,Φ(ϱ1))f1(ϱ1,0)|M2(ϱ2ϱ1)μγΓ(μγ+1)+|f1(ϱ1,0)|M2(ϱ2ϱ1)μγΓ(μγ+1).

    We get

    |(ˆHΦ)(ϱ2)(ˆHΦ)(ϱ1)||φ(ϱ2)φ(ϱ1)|+bR0|ϱ2ϱ1|b01ϱ1+sm1(s)ds+b2R0ϱ2ϱ1m1(s)ds+M1b2|Φ(ϱ2)Φ(ϱ1)|+bR0|ϱ2ϱ1|+L|Φ(ϱ2)Φ(ϱ1)|RLJμm2(ϱ2)+η(f1,ϵ)RLJμm2(ϱ2)+L|Φ(ϱ1)|M|ϱ2ϱ1|μηΓ(μγ+1)+M2M|ϱ2ϱ1|μγΓ(μγ+1),

    so,

    |(ˆHΦ)(ϱ2)(ˆHΦ)(ϱ1)||φ(ϱ2)φ(ϱ1)|+bR0M1|ϱ2ϱ1|+b2R0ϱ2ϱ1m1(s)ds+M1b2|Φ(ϱ2)Φ(ϱ1)|+bR0M1|ϱ2ϱ1|+|Φ(ϱ2)Φ(ϱ1)|M2LbμγΓ(μγ+1)+M2Ψ(f1,ϵ)bμγΓ(μγ+1)+L|Φ(ϱ1)|M|ϱ2ϱ1|μγΓ(μγ+1)+M2M|ϱ2ϱ1|μηΓ(μγ+1),

    where Ψ(f1,ϵ)=sup{|f1(ϱ2,Φ(ϱ1)f1(ϱ1,Φ(ϱ1))|:ϱ1,ϱ2ˆI,|ϱ2ϱ1|ϵ,ΦBR0}.

    Knowing that f1 is uniformly continuous on the set ˆI ×BR0, we derive the inequality

    ω0(ˆHΦ)[M2LbμγΓ(μγ+1)+M1b2]ω0(Φ).

    Then

    ω0(ˆHΦ)[M2LbμγΓ(μγ+1)+M1b2]ω0(Φ).

    From (2.4), this inequality leads to

    χ(ˆHΦ)[M2LbμγΓ(μγ+1)+M1b2]χ(Φ).

    If [M2LbμγΓ(μγ+1)+M1b2]<1 and using Theorem 1, there exists at least a solution ΦC(ˆI) for HIE (1.1).

    The following hypotheses are satisfied if there exist ˆgκ, κ =1,2, such that (iv) ˆgκ: ˆI×RR, κ =1,2, satisfies CC, so it can be measurable in ϱ for all ΦR and continuous in Φ for all ϱˆI. Then

    |ˆgκ(ϱ,Φ)ˆgκ(ϱ,ζ)|Lκ|Φζ|,  κ=1,2, (3.3)

    for all ϱˆI and Φ,ζ R. Let Φ1 and Φ2 be two solutions for the HIE (1.1), and hence

    |Φ1(ϱ)Φ2(ϱ)|=|ϱΦ1(ϱ)ϱ0ϱϱ+sˆg1(s,Φ1(s))dsϱΦ2(ϱ)ϱ0ϱϱ+sˆg1(s,Φ2(s))ds+ϱΦ2(ϱ)ϱ0ϱϱ+sˆg1(s,Φ1(s))dsϱΦ2(ϱ)ϱ0ϱϱ+sˆg1(s,Φ1(s))ds+f1(ϱ,Φ1(ϱ))ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ1(s))dsf1(ϱ,Φ2(ϱ))ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ2(s))ds+f1(ϱ,Φ1(ϱ))ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ2(s))dsf1(ϱ,Φ1(ϱ))ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ2(s))ds| (3.4)
    ϱ|Φ1(ϱ)Φ2(ϱ)|ϱ0ϱϱ+sm1(s)ds+ϱ|Φ2(ϱ)|ϱ0ϱϱ+sL1|Φ1(s)Φ2|ds+|f1(ϱ,Φ1(ϱ))f1(ϱ,Φ2(ϱ))|ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ2(s))ds+|f1(ϱ,Φ1(ϱ))|ϱ0(ϱs)μ1Γ(μ)ˆg2(s,Φ2(s))ˆg2(s,Φ1(s)))ds(b2M1+L1b2R0+LM2bμγΓ(μγ+1)+L2bμ(LR0+M)Γ(μ+1))|Φ1(ϱ)Φ2(ϱ)|Υ|Φ1(ϱ)Φ2(ϱ)|, (3.5)

    where Υ=(b2M1+L1b2R0+LM2bμγΓ(μγ+1)+L2bμ(LR0+M)Γ(μ+1)). Then we get the theorem:

    Theorem 3. Assume that the hypotheses (i)–(vi) are satisfied, and Υ<1. Then, the solution ΦC(ˆI) of (1.1) is unique.

    In the 1980s, Adomian presented the ADM [19,20,21], which is an analytical method used to solve a lot of different equations such as DEs, IEs, integro-differential equations, and partial DEs [22,23,24,25]. The obtained solution is an infinite series that converges to the exact solution. An important benefit of the ADM is that there is no linearization or perturbation that can change the main problem that has been solved, which is serious. A lot of researchers are interested in using the ADM, as it is successfully applied to many applications that appear in applied sciences [26,27,28]. In this research, the ADM is used as the first method to solve the HIE (1.1).

    Applying the ADM to (1.1), the ADM solution algorithm is

    Φ0(ϱ)=φ(ϱ), (4.1)
    Φκ(ϱ)=ϱˆAκ1(ϱ)+ˇDκ1(ϱ), (4.2)

    where ˆAκ, and ˇDκ are Adomian polynomials of the nonlinear terms ˆg1(ϱ,Φ),f1(ϱ,Φ), and ˆg2(ϱ,Φ) which take the forms

    ˆAκ=1κ![dκdλκ(κ=0λκΦκϱ0ϱϱ+sˆg1(s,κ=0λκΦκ(s))ds)]λ=0, (4.3)
    ˇDκ=1κ![dκdλκ(f1(ϱ,κ=0λκΦκ)RLJμˆg2(ϱ,κ=0λκΦκ(ϱ)))]λ=0. (4.4)

    Finally, the ADM solution is

    Φ(ϱ)=κ=0Φκ(ϱ). (4.5)

    Theorem 4. If hypotheses (i)(vi) are satisfied, Υ2<1, and |Φ1(ϱ)|<k, where k is a positive constant, then the series solution (4.5) of (1.1) using the ADM is convergent.

    Proof. Define the sequence {ˆSρ} such that ˆSρ=ρκ=0Φκ(ϱ) is a sequence of partial sums taken from the series (4.5), and

    Φ(ϱ)ϱ0ϱϱ+sˆg1(s,Φ(s))ds=κ=0ˆAκ,f1(ϱ,Φ(ϱ))RLJμˆg2(ϱ,Φ(ϱ))=κ=0ˇDκ.

    Let ˆSρ and ˆSθ be two partial sums of the ADM series solution such that ρ>θ. We want to prove that {ˆSρ} is a Cauchy sequence (CS) in this Banach space (Bs).

    ˆSρˆSθ=ρκ=0Φκθκ=0Φκ=ϱρκ=0ˆAκ1(ϱ)+ρκ=0ˇDκ1(ϱ)ϱθκ=0ˆAκ1(ϱ)θκ=0ˇDκ1(ϱ),

    hence,

    ˆSρˆSθ=[ϱ(ρκ=0ˆAκ1(ϱ)θκ=0ˆAκ1(ϱ))]+[ρκ=0ˇDκ1(ϱ)θκ=0ˇDκ1(ϱ)]=[ϱ(ρκ=θ+1ˆAκ1(ϱ))]+[ρκ=θ+1ˇDκ1(ϱ)].

    Thus, by applying || to both sides, we find

    |ˆSρˆSθ|=|[ϱ(ρκ=θ+1ˆAκ1(ϱ))]+[ρκ=θ+1ˇDκ1(ϱ)]|
    |ϱρ1κ=θˆAκ(ϱ)|+|ρ1κ=θˇDκ(ϱ)|
    |ϱ(ˆSρ1ˆSθ1)|ϱ0|ϱϱ+s[ˆg1(s,(ρ1κ=θΦκ))]|ds+|f1(ϱ,(ρ1κ=θΦκ))RLJμˆg2(ϱ,(ρ1κ=θΦκ))|b2|(ˆSρ1ˆSθ1)|b01ϱ+s|ˆg1(s,Φ)|ds+|f1(ϱ,ˆSρ1)f1(ϱ,ˆSθ1)|RLJμ|ˆg2(ϱ,Φ)|,
    |ˆSρˆSθ|b2|(ˆSρ1ˆSθ1)|b01ϱ+s|ˆg1(s,Φ)|ds+L|ˆSρ1ˆSθ1|RLJμγ RLJγm2(ϱ)b2|(ˆSρ1ˆSθ1)|M1+LM2|ˆSρ1ˆSθ1|RLJμγ(1)
    ˆSρˆSθ[b2M1+LM2(b)μγΓ(μγ+1)]ˆSρ1ˆSθ1Υ2ˆSρ1ˆSθ1,

    where Υ2=[b2M1+LM2(b)μγΓ(μγ+1)]. Let ρ=θ+1, and we get

    ˆSθ+1ˆSθΥ2ˆSθˆSθ1Υ22ˆSθ1ˆSθ2Υθ2ˆS1ˆS0.

    Using the triangle inequality, we arrive at

    ˆSρˆSθˆSθ+1ˆSθ+ˆSθ+2ˆSθ+1++ˆSρˆSρ1 [Υθ2+Υθ+12++Υρ12]ˆS1ˆS0Υθ2[1+Υ2++Υρθ12]ˆS1ˆS0Υ2[1Υρθ21Υ2]Φ1.

    If 0<Υ2<1 and ρ>θ, this leads to (1Υρθ2)1. Then

    ˆSρˆSθΥθ21Υ2Φ1Υθ21Υ2maxϱˆI|Φ1(ϱ)|.

    If |Φ1(ϱ)|<k, θ, then ˆSρˆSθ0, which leads to {ˆSρ} being a CS in this BS and the series (4.5) is convergent.

    Theorem 5. The maximum absolute error of the ADM series solution (4.5) is

    maxϱˆI|Φ(ϱ)θκ=0Φκ(ϱ)|Υθ21Υ2maxϱˆI|Φ1(ϱ)|. (4.6)

    Proof. In Theorem 2, we see that

     ˆSρˆSθΥθ21Υ2maxϱˆI|Φ1(ϱ)|,

    and ˆSρ=ρκ=0Φκ(ϱ), ρ. Then, ˆSρΦ(ϱ), and hence

    Φ(ϱ)ˆSθΥθ21Υ2maxϱˆI|Φ1(ϱ)|,

    and the maximum absolute error is written as

    maxϱˆI|Φ(ϱ)θκ=0Φκ(ϱ)|Υθ21Υ2maxϱˆI|Φ1(ϱ)|.

    The method of successive approximations (PM) was presented by Emile Picard in 1891. PM and ADM methods were first compared by Rach and Bellomo in 1987 [26,29]. In 1999, Golberg deduced that these two methods were equivalent for linear differential equations [30]. But this equivalence is not achieved in the nonlinear case. In 2010, El-Sayed et al. used them to solve QIE [31]. In 2012, El-Sayed et al. used them to solve a coupled system of fractional QIEs [32]. In 2014, El-Sayed et al. used them to solve FQIE [33]. In 2024, Ziada used them to solve a nonlinear FDE system containing the Atangana–Baleanu derivative [34]. In this research, we use them to get the solution for a nonlinear HDE and compare their results.

    Applying the PM to the QIE (1.1), the solution is a sequence constructed by

    Φ0(ϱ)=φ(ϱ),Φκ(ϱ)=Φ0(ϱ)+ϱΦκ1(ϱ)ϱ0ϱϱ+sˆg1(s,Φκ1(s))ds+f1(ϱ,Φκ1(ϱ))RLJμˆg2(ϱ,Φκ1(ϱ)). (4.7)

    All the functions Φκ(ϱ) are continuous functions and Φκ are the sum of successive differences

    Φκ(ϱ)=Φ0(ϱ)+κ=1(ΦκΦκ1). (4.8)

    Therefore, the sequence Φκ convergence is the same as the infinite series (ΦκΦκ1) convergence. The final PM solution takes the form

    Φ(ϱ)=limκΦκ(ϱ). (4.9)

    From the above relations, we can deduce that if the series (ΦκΦκ1) is convergent, then the sequence Φκ(ϱ) is convergent to Φ(ϱ). To prove that the sequence {Φκ(ϱ)} is informally convergent, consider the associated series

    κ=1[Φκ(ϱ)Φκ1(ϱ)]. (4.10)

    From (4.7) for κ =1, we get

    Φ1(ϱ)Φ0(ϱ)=ϱΦ0(ϱ)ϱ0ϱϱ+sˆg1(s,Φ0(s))ds+f1(ϱ,Φ0(ϱ))RLJμˆg2(ϱ,Φ0(ϱ)). (4.11)

    So, we have

    |Φ1(ϱ)Φ0(ϱ)|=|ϱΦ0(ϱ)ϱ0ϱϱ+sˆg1(s,Φ0(s))ds+f1(ϱ,Φ0(ϱ))RLJμˆg2(ϱ,Φ0(ϱ))||ϱ||Φ0(ϱ)|ϱ0|ϱϱ+s||ˆg1(s,Φ0(s))|ds+|f1(ϱ,Φ0(ϱ))|RLJμˆg2(ϱ,Φ0(ϱ)). (4.12)

    Thus,

    |Φ1(ϱ)Φ0(ϱ)|b2|Φ0(ϱ)|b01ϱ+sm1(s)ds+[|f1(ϱ,Φ0(ϱ)))f1(ϱ,0)+f1(ϱ,0)|] RLJμRLJηm2(ϱ)[b2R0M1+M2bμη(LR0+M)Γ(μη+1)]:=ψ. (4.13)

    Now, we get an estimate for Φκ(ϱ)Φκ1(ϱ), κ :

    (4.14)

    and

    (4.15)

    Thus,

    (4.16)

    From the hypotheses and we have

    (4.17)

    so,

    (4.18)

    where

    In the above relation, if we put and use (4.13), we get

    (4.19)

    Doing the same for gives us

    Then the general form of this relation is

    (4.20)

    Since then the series

    (4.21)

    is uniformly convergent. Hence, the sequence is uniformly convergent. Since , and are continuous in , then

    (4.22)

    Hence, the solution exists.

    Example 1. For the HIE of Chandraseker type:

    (5.1)

    where

    and its exact solution is

    Applying the ADM to Eq (5.1), we get

    (5.2)
    (5.3)

    Using the PM in Eq (5.1), the solution algorithm is

    (5.4)
    (5.5)

    Figure 1 shows ADM solutions at different values of ( ), and Figure 2 shows PM solutions at the same values.

    Figure 1.  ADM solutions at different values of .
    Figure 2.  PM solutions at different values of .

    Remark 1. A comparison between the absolute relative error (ARE) of ADM and PM solutions with the exact solution (where ) is given in Table 1. It is clear from these results that the two solutions nearly give the same accuracy, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 22 sec., PM time = 319.188 sec.). Figure 3(a) shows the ADM and the exact solution, while Figure 3(b) shows the PM with the exact solution.

    Table 1.  ARE of (ADM, PM) for Example 1.
    2.48379 2.19628
    2.47045 7.0281
    2.04146 5.33709
    6.32111 0.0000224935
    5.3704 0.0000686815
    0.0000360447 0.000171175
    0.000215013 0.000371487
    0.000750286 0.000731218
    0.0020904 0.00134688
    0.00511876 0.00240123
    0.0115464 0.00436134
    0.024631 0.00870185
    0.0503602 0.0201514
    0.0990462 0.0519358
    0.186618 0.132278

     | Show Table
    DownLoad: CSV
    Figure 3.  The exact solution versus ADM and Picard solutions to (5.1).

    Example 2. For the HIE of Chandraseker type:

    (5.6)

    where

    and its exact solution is

    Applying the ADM to Eq (5.6), we get

    (5.7)
    (5.8)

    Using the PM in Eq (5.6), we have

    (5.9)
    (5.10)

    Figure 4(a) shows the ADM and exact solution, while Figure 4(b) shows the PM bwith the exact solution.

    Figure 4.  The exact solution versus ADM and Picard solutions to (5.6).

    Remark 2. The absolute difference (AD) between ADM and PM solutions (where ) is for . It is clear from these results that the two solutions are nearly the same, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 42 sec., PM time = 253.2 sec.). Figure 5 shows ADM and PM solutions at (.

    Figure 5.  PM and ADM solutions at .

    Example 3. For the HIE of Chandraseker type:

    (5.11)

    where

    and its exact solution is

    Applying the ADM to Eq (5.11), we have

    (5.12)
    (5.13)

    Using the PM in Eq (5.11), we get

    (5.14)
    (5.15)

    Remark 3. A comparison between the ARE of ADM and PM solutions with the exact solution is given in Table 2. It is clear from these results that the two solutions nearly give the same accuracy, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 69.124 sec., PM time = 70.875 sec.). Figure 6 shows ADM solutions at different values of ( ), and Figure 7 shows PM solutions at the same values.

    Table 2.  ARE of (ADM, PM) for Example 3.
    1.0964*10 0.0000724208
    0.0000268761 0.00060454
    0.000179457 0.00212539
    0.000701695 0.0052377
    0.00204302 0.0106117
    0.00493079 0.0189737
    0.0104472 0.0310892
    0.0201116 0.0477376
    0.0359674 0.0696801
    0.0606745 0.0976178

     | Show Table
    DownLoad: CSV
    Figure 6.  ADM solutions at different values of .
    Figure 7.  PM solutions at different values of .

    In this research, two analytical methods (ADM and PM) are used to solve the fractional CQIE that was found in the nonlinear analysis and its applications. The existence of a unique solution and its convergence to the two methods are proved (see Theorems 2, 4, and 5). This article focused on making a comparison between them with the exact solution (see the results in Tables 1 and 2). It is observed from the obtained results that the difference between their accuracy is too small to consider, but when we compare their used time, it was clear that the ADM takes less time than the PM (it is more clear in Example 1). These results showed that the two methods satisfied certain criteria that were provided by the solutions.

    Table 3.  Abbreviations.
    IVP Initial value problem
    ADM Adomian decomposition method
    PM Picard method
    FDEs Fractional differential equations
    HDE Hybrid differential equation
    CQIE Chandrasekhar quadratic integral equation
    RLFI Riemann--Liouville fractional integral
    MNC Measure of noncompactness
    BS Banach space
    CS Cauchy sequence
    ARE Absolute relative error

     | Show Table
    DownLoad: CSV

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    Conceptualization, Eman A. A. Ziada and Hind Hashem; Formal analysis, Hind Hashem; Funding acquisition, Asma Al-Jaser; Investigation, Asma Al-Jaser and Osama Moaaz; Methodology, Eman A. A. Ziada and Monica Botros; Software, Asma Al-Jaser and Monica Botros; Writing—original draft, Eman A. A. Ziada and Monica Botros; Writing—review and editing, Osama Moaaz. All authors have read and agreed to the published version of the manuscript.

    The authors declare there is no conflict of interest.



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