Although liquefaction technology has been extensively applied, plenty of biomass remains tainted with heavy metals (HMs). A meta-analysis of literature published from 2010 to 2023 was conducted to investigate the effects of liquefaction conditions and biomass characteristics on the remaining ratio and chemical speciation of HMs in biochar, aiming to achieve harmless treatment of biomass contaminated with HMs. The results showed that a liquefaction time of 1–3 h led to the largest HMs remaining ratio in biochar, with the mean ranging from 84.09% to 92.76%, compared with liquefaction times of less than 1 h and more than 3 h. Organic and acidic solvents liquefied biochar exhibited the greatest and lowest HMs remaining ratio. The effect of liquefaction temperature on HMs remaining ratio was not significant. The C, H, O, volatile matter, and fixed carbon contents of biomass were negatively correlated with the HMs remaining ratio, and N, S, and ash were positively correlated. In addition, liquefaction significantly transformed the HMs in biochar from bioavailable fractions (F1 and F2) to stable fractions (F3) (P < 0.05) when the temperature was increased to 280–330 ℃, with a liquefaction time of 1–3 h, and organic solvent as the liquefaction solvent. N and ash in biomass were positively correlated with the residue state (F4) of HMs in biochar and negatively correlated with F1 or F2, while H, O, fixed carbon, and volatile matter were negatively correlated with F4 but positively correlated with F3. Machine learning results showed that the contribution of biomass characteristics to HMs remaining ratio was higher than that of liquefaction factor. The most prominent contribution to the chemical speciation changes of HMs was the characteristics of HMs themselves, followed by ash content in biomass, liquefaction time, and C content. The findings of this meta-analysis contribute to factor selection, modification, and application of liquefied biomass to reducing risks.
Citation: Li Ma, Likun Zhan, Qingdan Wu, Longcheng Li, Xiaochen Zheng, Zhihua Xiao, Jingchen Zou. Optimization of liquefaction process based on global meta-analysis and machine learning approach: Effect of process conditions and raw material selection on remaining ratio and bioavailability of heavy metals in biochar[J]. AIMS Environmental Science, 2024, 11(3): 342-359. doi: 10.3934/environsci.2024016
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Although liquefaction technology has been extensively applied, plenty of biomass remains tainted with heavy metals (HMs). A meta-analysis of literature published from 2010 to 2023 was conducted to investigate the effects of liquefaction conditions and biomass characteristics on the remaining ratio and chemical speciation of HMs in biochar, aiming to achieve harmless treatment of biomass contaminated with HMs. The results showed that a liquefaction time of 1–3 h led to the largest HMs remaining ratio in biochar, with the mean ranging from 84.09% to 92.76%, compared with liquefaction times of less than 1 h and more than 3 h. Organic and acidic solvents liquefied biochar exhibited the greatest and lowest HMs remaining ratio. The effect of liquefaction temperature on HMs remaining ratio was not significant. The C, H, O, volatile matter, and fixed carbon contents of biomass were negatively correlated with the HMs remaining ratio, and N, S, and ash were positively correlated. In addition, liquefaction significantly transformed the HMs in biochar from bioavailable fractions (F1 and F2) to stable fractions (F3) (P < 0.05) when the temperature was increased to 280–330 ℃, with a liquefaction time of 1–3 h, and organic solvent as the liquefaction solvent. N and ash in biomass were positively correlated with the residue state (F4) of HMs in biochar and negatively correlated with F1 or F2, while H, O, fixed carbon, and volatile matter were negatively correlated with F4 but positively correlated with F3. Machine learning results showed that the contribution of biomass characteristics to HMs remaining ratio was higher than that of liquefaction factor. The most prominent contribution to the chemical speciation changes of HMs was the characteristics of HMs themselves, followed by ash content in biomass, liquefaction time, and C content. The findings of this meta-analysis contribute to factor selection, modification, and application of liquefied biomass to reducing risks.
The Bessel function has immense applications in the field of engineering, physics, and applied mathematics. Baricz [1], Generalized Bessel functions of the first kind in (2010), which discussed the geometric properties, functional inequalities of generalized Bessel function and also the inequalities involving circular and hyperbolic functions to Bessel function and modified Bessel functions. Tumakov [2] investigated the numerical algorithms for fast computations of the Bessel functions of an integer order with the required accuracy. Choi and Agarwal[3], Abramowitz and Stegun [4], Heymans and Podlubny [5], Watson [6] and Purohit et al. [7] studied the following Bessel function (Bf) defined by
Jα(y)=∞∑n=0(−1)n(y/2)α+2nn!Γ(α+n+1). | (1.1) |
Edward Maitland Wright [8] introduced the generalized form of Bessel function with the name of Bessel-Maitland function (B-M1)
Jαβ(y)=∞∑n=0(−y)nn!Γ(αn+β+1). | (1.2) |
The properties of generalised Bessel function can be found in the work of Srivastava and Singh [9]. Suthar et al. [10,11] discussed the various properties of Bessel-Maitland function. Ali et al. [12] established some fractional operators with the generalized Bessel-Maitland function.
Waseem et al. [13] established the generalized Bessel-Maitland function (B-M11) and discuss the numerous integral formulas for y∈C/(−∞,0]; α,β,γ∈C, ℜ(α)≥0, ℜ(β)≥−1, ℜ(γ)≥0, k∈(0,1)∪N defined by
Jα,γβ,k(y)=∞∑n=0(γ)kn(−y)nn!Γ(αn+β+1). | (1.3) |
Suthar et al. [14] studied the following generalized multi-index Bessel function (Gm-Bf) defined by
J(αj,βj)mγ,k(y)=∞∑n=0(γ)kn(−y)nn!∏mj=1Γ(αjn+βj+1). | (1.4) |
Recently, fractional integrals are widely applied in different branches of mathematics, physics, engineering due to their wide applications (see e.g., [5,15,16,17,18]).
Riemann-Liouville fractional integral operators for ℜ(ρ)>0 are defined by
Iρa+h(u)=1Γ(ρ)∫ya(y−u)ρ−1h(u)du,a<y | (1.5) |
Iρbh(u)=1Γ(ρ)∫by(u−y)ρ−1h(u)du,y<b. | (1.6) |
Riemann-Liouville fractional differentials operators (RLDO) [12,19] for ℜ(ρ)>0; n=[ℜ(n)−1]
Dρa+h(u)=(d/dy)nIn−ρa+h(y) | (1.7) |
Dρbh(u)=(−d/dy)nIn−ρbh(y). | (1.8) |
Srivastava and Singh [9] defined the following fractional integral operator for α1,β1,r∈C, ℜ(α1)>0, ℜ(β1)≥−1 by
h(y)=def∫y0(y−t)β1Jα1β1(r(y−t)α1)h(u)du. | (1.9) |
Srivastava and Tomovski [20] established the fractional integral operator (FIO) having Mittag-Leffler function as a kernel, discuss its boundedness and convergence of integral and also derive the product of FIO with Riemann-Liouville fractional integral operator defined for r,γ∈C, ℜ(α1)>max{0,ℜ(k)−1}; min{ℜ(β1),ℜ(k)}>0
(Er;γ,ka+;α1,β1h)(y)=∫ya(y−t)β1−1Eγ,kα1,β1(r(y−t)α1)h(t)dt. | (1.10) |
Prabhakar fractional integral operators for γ,β1∈C, ℜ(α1)>0 are defined in [21] by
E∗(α1,β1;γ;r)h(y)=∘h(y)=∫ya(y−t)β1−1Eγα1,β1(r(y−t)α1)h(t)dt,a<y, | (1.11) |
E∗(α1,β1;γ;r)h(y)=∫by(t−y)β1−1Eγα1,β1(r(t−y)α1)h(t)dt,y<b. | (1.12) |
Tilahun et al. [22] derived the generalized FIO for ℜ(β1)>0, ℜ(α1)>0 and r,γ∈C as
(Ir;α1,γa+,β1,kh)(y)=∫ya(y−t)β1Jγ,kα1,β1(r(y−t)α1;p)h(t)dt,a<y | (1.13) |
and
(Ir;α1,γa+,β1,kh)(y)=∫by(y−t)β1Jγ,kα1,β1(r(y−t)α1;p)h(t)dt, | (1.14) |
where y<b.
Definition 1.1. (FIO)Fractional integral operator with generalized multi-index Bessel function (Gm-Bf) kernel for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0.
Ir;γ,ka+;(αj,βj)mh(y)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)h(u)du,a<y | (1.15) |
Ir;γ,kb;(αj,βj)mh(y)=m∏j=1∫by(u−y)βjJ(αj,βj)mγ,k(r(u−y)αj)h(u)du,y<b. | (1.16) |
Dirichlet formula (Fubini's theorem) Samko et al. in [23] and Kelelaw et al. in [22] is defined as
∫dbdy∫ybh(y,z)dz=∫dbdz∫dzh(y,z)dy. | (1.17) |
Kilbas [24] analyzed the generalized Wright function for ξi,ζj∈R, (i=1,2⋯r),(j=1,2⋯s) and bi,cj∈C as
rψs(y)=∞∑n=0∏ri=1Γ(bi+ξin)∏sj=1Γ(cj+ζjn)ynn!=rψs[(bi,ξi)1,r(cj,ζj)1,s|y]. | (1.18) |
The integral representation of beta function [25,26] for ℜ(y)>0, ℜ(z)>0 and also in gamma form appearance of beta function is defined as follows
B(y,z)=∫10uy−1(1−u)z−1du=Γ(y)Γ(z)Γ(y+z). | (1.19) |
Pochhammer symbol and its properties can be found [25,26,27] as
(γ)n={γ(γ+1)(γ+2)⋯(γ+n−1),for n≥11,for n=0, γ≠0 | (1.20) |
=Γ(γ+n)Γ(γ) and (γ)kn=Γ(γ+kn)Γ(γ) (k>0). | (1.21) |
The space of Lebesgue measurable for complex and real valued functions defined by Kelelaw et al. [22] as follows
L(a,y)={h:||h||1:=∫ya|h(u)|du<∞}. | (1.22) |
The following some conditions of fractional integral operators can be obtained by setting the integrals according to requirements:
1). Setting r=0, j=1=m and β1=β1−1 in (FIO) defined in Eqs (1.15) and (1.16), we get the Riemann-Liouville fractional integral operator defined in [28] as
I0;γ,ka+;(α1,β1−1)mh(y)=Iβ1a+h(y) | (1.23) |
I0;γ,kb;(α1,β1−1)mh(y)=Iβ1bh(y). | (1.24) |
2). Setting j=m=1, β1=β1−1 in Eq (1.15), we have a fractional integral defined in Eq (1.10) as
Ir;γ,ka+;(α1,β1−1)mh(y)=(Er;γ,ka+;α1,β1h)(y). | (1.25) |
3). Setting j=m=1, k=1, β1=β1−1, in Eqs (1.15) and (1.16), we get the FIO defined in Eqs (1.11) and (1.12) respectively
Ir;γ,1a+;(α1,β1−1)mh(y)=E∗(α1,β1;γ;r)h(y)=∘h(y) | (1.26) |
Ir;γ,1b;(α1,β1−1)mh(y)=E∗(α1,β1;γ;r)h(y). | (1.27) |
4). Setting j=m=1, k=0 and limits from [0,y] in Eq (1.15), we get a fractional integral defined in Eq (1.9) as
Ir;γ,0a+;(α1,β1)mh(y)=∫y0(y−t)β1Jα1β1(r(y−t)α1)h(u)du=h(y). | (1.28) |
5). Setting j=m=1 in Eq (1.15) then, we get the generalized fractional integral operator defined in Eq (1.13) as
Ir;γ,ka+;(α1,β1)h(y)=(Ir;α1,γa+,β1,kh)(y). | (1.29) |
Lemma 1.1. Consider the Riemann-Liouville fractional integral operator with multi-index power function for αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and ℜ(ρ)>0 as
m∏j=1Iρa+[(u−a)βj+αjn](y)=m∏j=1[Γ(βj+αjn+1)Γ(ρ+βj+αjn+1)(y−a)ρ+βj+αjn]. | (1.30) |
Remark 1.1. Setting j=m=1 in lemma 1.1 then we obtain the result that defined the Mathai Haubold [29] and Kelelaw et al. [22] as
Iρa+[(u−a)β1+α1n](y)=(y−a)ρ+β1+α1nΓ(β1+α1n+1)Γ(ρ+β1+α1b+1). | (1.31) |
The preliminary results for generalized multi-index Bessel function which used to proceed the new results is given in this section. We calculate the nth-differential and also develop some results with the coordination of Riemann-Liouville fractional operator and (Gm-Bf).
Theorem 2.1. Consider the nth-differential of generalized multi-index Bessel function with power function for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=(y−u)βj−nJ(αj,βj−n)mγ,k(r(y−u)αj). | (2.1) |
Proof. Let the nth-differential of generalized multi-index Bessel function with power function as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)], | (2.2) |
using the behavior of (1.4), we take as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=(d/dy)n[(y−u)βj∞∑n=0(γ)kn(−r(y−u)αj)nn!∏mj=1Γ(αjn+βj+1)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)βj+αjn]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)(β1+β2+⋯+βm)+(α1n+α2n+⋯+αmn)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)(β1+α1n)+(β2+α2n)+⋯+(βm+αmn)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)×(d/dy)n[(y−u)(β1+α1n)(y−u)(β2+α2n)⋯(y−u)(βm+αmn)]. | (2.3) |
Using the identity result for simplification of (2.3), we get
(d/dy)nyθ=Γ(θ+1)Γ(θ−n+1)yθ−n,θ≥n | (2.4) |
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)×Γ(β1+α1n+1)Γ(β2+α2n+1)⋯Γ(βm+αmn+1)(y−u)αjn+βj−nΓ(β1+α1n−n+1)Γ(β2+α2n−n+1)⋯Γ(βm+αmn−n+1)=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+βj−n+1)(y−u)αjn+βj−n=(y−u)βj−n∞∑n=0(γ)kn(−r(y−u)αj)nn!∏mj=1Γ(αjn+βj−n+1)=(y−u)βj−nJ(αj,βj−n)mγ,k(r(y−u)αj). | (2.5) |
Corollary 2.1. Suppose that αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N then theorem 2.1 can be expressed as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=1Γ(γ)(y−u)βj−n2ψm+1{(γ,k)(βj+1,αj)(βj−n+1,αj)(βj+1,αj)|mj=1|r(y−u)αj}. | (2.6) |
Corollary 2.2. Suppose that r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N and setting that βj=βj−1, (r(y−u)αj)=(−r(y−u)αj) in theorem 2.1 we see that
(d/dy)n[(y−u)βj−1J(αj,βj−1)mγ,k(r(y−u)αj)]=(y−u)βj−1−nE(αj,βj−n)mγ,k(r(y−u)αj), | (2.7) |
where E(αj,βj−n)mγ,k(.) is generalized multi-index Mittag-Leffler function.
Theorem 2.2. Consider the Riemann-Liouville fractional integral operator defined in Eq (1.5) with generalized multi-index Bessel function for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1(y−a)βj+ρJ(αj,βj+ρ)mγ,k(r(y−a)αj). | (2.8) |
Proof. Let (RLIO) with (Gm-Bf) is defined in Eq (1.4), we have
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1Iρa+[(u−a)βj∞∑n=0(γ)kn(−r(u−a)αj)nn!∏mj=1Γ(αjn+βj+1)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1Iρa+[(u−a)βj+αjn]. | (2.9) |
By using Lemma 1.1 in Eq (2.9) then we attain the equation as
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1Γ(αjn+βj+1)(y−a)αjn+ρ+βjΓ(αjn+βj+ρ+1)=m∏j=1(y−a)ρ+βj∞∑n=0(γ)kn(−r(y−a)αj)nn!∏mj=1Γ(αjn+βj+ρ+1)=m∏j=1(y−a)βj+ρJ(αj,βj+ρ)mγ,k(r(y−a)αj). | (2.10) |
Corollary 2.3. Consider the right-sided Riemann-Liouville fractional integral operator with generalized multi-index Bessel function for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Iρb[(b−u)βjJ(αj,βj)mγ,k(r(b−u)αj)](y)=m∏j=1(b−y)βj+ρJ(αj,βj+ρ)mγ,k(r(b−y)αj). | (2.11) |
Corollary 2.4. Consider the Riemann-Liouville fractional differential operator defined in Eq (1.7) with generalized multi-index Bessel function for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Dρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1(y−a)βj−ρJ(αj,βj−ρ)mγ,k(r(y−a)αj). | (2.12) |
Corollary 2.5. Consider the right-sided Riemann-Liouville fractional differential operator with generalized multi-index Bessel function for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Dρb[(b−u)βjJ(αj,βj)mγ,k(r(b−u)αj)](y)=m∏j=1(b−y)βj−ρJ(αj,βj−ρ)mγ,k(r(b−y)αj). | (2.13) |
In this section, we discuss some properties of the generalized fractional integrals with non singular function as a kernel.
Theorem 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for ϑ,ρ,σ,r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, when h(u)=(u−a)ϑ+ρσ−1, then
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ−1](y)=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(y−a)αj). | (3.1) |
Proof. (FIO) defined in Eq (1.15) with Eq (1.4), we obtain as
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ−1](y)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)(u−a)ϑ+ρσ−1du=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1∫ya(u−a)ϑ+ρσ−1(y−u)βj+αjndu=∞∑n=0(γ)kn(−r)nn!1∏mj=1Γ(αjn+βj+1)m∏j=1∫ya(u−a)ϑ+ρσ−1(y−u)βj+αjndu | (3.2) |
Substituting u=y−z(u−a) and using the definition of beta function and the following relation in Eq (3.2), we get
Iλa+[(y−a)u−1](y)=Γ(u)Γ(λ+u)(y−a)λ+u−1. |
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ](y)=m∏j=1∞∑n=0(γ)kn(−r)nn!Γ(ϑ+ρσ)∏mj=1Γ(ϑ+ρσ+αjn+βj+1)(y−a)ϑ+ρσ+βj+αjn=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)∞∑n=0(γ)kn(−r(y−a)αj)nn!∏mj=1Γ(ϑ+ρσ+αjn+βj+1)=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(y−a)αj). | (3.3) |
Corollary 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for ϑ,ρ,σ,r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, when h(u)=(a−u)ϑ+ρσ, then
[Ir;γ,kb;(αj,βj)m(b−u)ϑ+ρσ](y)=m∏j=1(b−y)ϑ+ρσ+βj+1Γ(ϑ+ρσ+1)J(αj,βj+ϑ+ρσ+1)mγ,k(r(b−y)αj). | (3.4) |
Theorem 3.2. Consider the composition of Riemann-Liouville fractional integral operator with (FIO) for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(ρ)>0 then
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)={Ir;γ,ka+;(αj,βj+ρ)m}(y)={Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y). | (3.5) |
Proof. Let the left side of Eq (3.5), we seen as
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=1Γ(ρ)∫ya(y−u)ρ−1[Ir;γ,ka+;(αj,βj)mh](u)du=1Γ(ρ)∫ya(y−u)ρ−1[m∏j=1∫ua(u−t)βjJ(αj,βj)mγ,k(r(u−t)αj)h(t)dt]du. | (3.6) |
By interchanging the order of integrations and using the Eq (1.17) in Eq (3.6), we attain as
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=∫ya[1Γ(ρ)m∏j=1∫yt(y−u)ρ−1(u−t)βjJ(αj,βj)mγ,k(r(u−t)αj)du]×h(t)dt. | (3.7) |
Setting u−t=η, we have
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=∫ya[m∏j=11Γ(ρ)∫y−t0(y−t−η)ρ−1ηβjJ(αj,βj)mγ,k(r(η)αj)dη]×h(t)dt=∫yam∏j=1Iρ0+[ηβjJ(αj,βj)mγ,k(r(η)αj)](y−t)×h(t)dt. | (3.8) |
applying theorem 2.2, we see
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=m∏j=1∫ya[(y−t)βj+ρJ(αj,βj+ρ)mγ,k(r(y−t)αj)]h(t)dt={Ir;γ,ka+;(αj,βj+ρ)m}(y). | (3.9) |
We start the right side to determine the second part of (3.5) with (FIO) defined in Eq (1.15) as
{Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)[Iρa+h](u)du=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)[1Γ(ρ)∫ua(u−t)ρ−1h(t)dt]du. | (3.10) |
Interchanging the order of integration and using Eq (1.17), we get
{Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=m∏j=1∫ya[1Γ(ρ)∫yt(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)(u−t)ρ−1du]h(t)dt. | (3.11) |
Setting y−u=x and using theorem (2.2) then
{Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=m∏j=1∫ya[1Γ(ρ)∫0y−t(x)βjJ(αj,βj)mγ,k(r(x)αj)(y−x−t)ρ−1(−dx)]h(t)dt=m∏j=1∫ya[1Γ(ρ)∫y−t0(x)βjJ(αj,βj)mγ,k(r(x)αj)(y−t−x)ρ−1dx]h(t)dt=m∏j=1∫yaIρ0+[(x)βjJ(αj,βj)mγ,k(r(x)αj)](y−t)×h(t)dt=m∏j=1∫ya(y−t)βj+ρJ(αj,βj+ρ)mγ,k(r(y−t)αj)h(t)dt={Ir;γ,ka+;(αj,βj+ρ)m}(y). | (3.12) |
Thus, we obtain the desired results by combining Eqs (3.9) and (3.12).
CCorollary 3.2. Composition of right-sided (FIO) with right-sided Riemann-Liouville fractional integral for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(ρ)>0 then
{Iρb[Ir;γ,kb;(αj,βj)mh]}(y)={Ir;γ,kb;(αj,βj+ρ)m}(y)={Ir;γ,kb;(αj,βj)m[Iρbh]}(y). | (3.13) |
Corollary 3.3. Consider the composition of Riemann-Liouville fractional differential operator with (FIO) for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(ρ)>0 then
{Dρa+[Ir;γ,ka+;(αj,βj)mh]}(y)={Ir;γ,ka+;(αj,βj−ρ)m}(y)={Ir;γ,ka+;(αj,βj)m[Dρa+h]}(y). | (3.14) |
Theorem 3.3. If r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 then
||Ir;γ,ka+;(αj,βj)mh||1≤A||h||1, | (3.15) |
where
A=m∏j=1∞∑n=0|(y−a)ℜ(βj)+1||(γ)kn||(−r(y−a)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(βj)+1. | (3.16) |
Proof. Let Vn be the nth-terms of (3.16); we have
|Vn+1Vn|=m∏j=1|(γ)kn+k|(γ)kn|n!(n+1)!|∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+αj+βj+1)|×ℜ(αj)n+ℜ(βj)+1ℜ(αj)n+ℜ(αj)+ℜ(βj)+1|(−1)n+1(−1)n||r(y−a)ℜ(αj)|≈m∏j=1(kn)k|−r(y−a)ℜ(αj)|(n+1)|∏mj=1|(αj)|n(αj)|,asn→∞ | (3.17) |
hence, |Vn+1Vn|→0 as n→∞, and k<ℜ(αj) which means that right hand side of (3.16) is convergent and finite under the given condition. The condition of boundedness of the integral operator (Ir;γ,ka+;(αj,βj)mh)(y) is discussed in the space of Lebesgue measure L(a,y) of a continuous function, where y>a. Consider the Lebesgue measurable space (1.22) and FIO (1.15), we have
||Ir;γ,ka+;(αj,βj)mh||1=∫ya|Ir;γ,ka+;(αj,βj)mh|du=∫ya|m∏j=1∫ua(u−τ)βjJ(αj,βj)mγ,k(r(u−τ)αj)h(τ)dτ|du≤∫ya[m∏j=1∫yτ(u−τ)βj|J(αj,βj)mγ,k(r(u−τ)αj)|du]|h(τ)|dτ. | (3.18) |
Putting u−τ=λ,u=y⇒λ=y−τ;u=τ⇒λ=0,du=dλ in Eq (3.18), we get
||Ir;γ,ka+;(αj,βj)mh||1≤∫ya[m∏j=1∫y−τ0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτ≤∫ya[m∏j=1∫y−a0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτ, |
where
m∏j=1∫y−a0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ≤m∏j=1∫y−a0λℜ(βj)|∞∑n=0(γ)kn|(−r)n(λ)αjb|n!∏mj=1Γ(αjn+βj+1)|dλ≤∞∑n=0|(γ)kn||(−r)n|n!|∏mj=1Γ(αjn+βj+1)|m∏j=1∫y−a0λℜ(αj)n+ℜ(βj)dλ≤m∏j=1∞∑n=0|(y−a)ℜ(βj)+1||(γ)kn||(−r(y−a)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(βj)+1=A. | (3.19) |
Therefore,
||Ir;γ,ka+;(αj,βj)mh||1≤∫yaA|h(τ)|dτ≤A||h||1⇒||Ir;γ,ka+;(αj,βj)mh||1≤A||h||1 |
Corollary 3.4. If r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 then
||Ir;γ,kb;(αj,βj)mh||1≤A||h||1, | (3.20) |
where
A=m∏j=1(b−y)ℜ(βj)+1∞∑n=0|(γ)kn||(−r(b−y)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(βj)+1. | (3.21) |
The results we discussed in this paper is creating a chain of fractional operators with kernels, having convergence and boundedness, continuity, symmetric properties and composition with Riemann-Liouville operators [9,12,20,21,22,25,30,31] in fractional calculus. We constructed the fractional operator with generalized multi-index Bessel function as a kernel, and discussed its properties, continuity, and check the behaviour with Riemann-Liouville fractional operators. We analyzed the generalized multi-index Bessel function nth-derivative and integral in the field of fractional calculus.
Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.
The authors declare that they have no competing interests.
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