In this article, we solved pantograph delay differential equations by utilizing an efficient numerical technique known as Chebyshev pseudospectral method. In Caputo manner fractional derivatives are taken. These types of problems are reduced to linear or nonlinear algebraic equations using the suggested approach. The proposed method's convergence is being studied with particular care. The suggested technique is effective, simple, and easy to implement as compared to other numerical approaches. To prove the validity and accuracy of the presented approach, we take two examples. The solutions we obtained show greater accuracy as compared to other methods. Furthermore, the current approach can be implemented for solving other linear and nonlinear fractional delay differential equations, owing to its innovation and scientific significance.
Citation: M. Mossa Al-Sawalha, Azzh Saad Alshehry, Kamsing Nonlaopon, Rasool Shah, Osama Y. Ababneh. Fractional view analysis of delay differential equations via numerical method[J]. AIMS Mathematics, 2022, 7(12): 20510-20523. doi: 10.3934/math.20221123
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In this article, we solved pantograph delay differential equations by utilizing an efficient numerical technique known as Chebyshev pseudospectral method. In Caputo manner fractional derivatives are taken. These types of problems are reduced to linear or nonlinear algebraic equations using the suggested approach. The proposed method's convergence is being studied with particular care. The suggested technique is effective, simple, and easy to implement as compared to other numerical approaches. To prove the validity and accuracy of the presented approach, we take two examples. The solutions we obtained show greater accuracy as compared to other methods. Furthermore, the current approach can be implemented for solving other linear and nonlinear fractional delay differential equations, owing to its innovation and scientific significance.
In 1878, Clifford algebra was defined in [1]. In 1982, Brackx et al. [2] generalized some results of the complex analysis to Clifford analysis. Malonek and Ren [3] studied the Almansi-type decomposition theorems for the k-order monogenic functions and k-order λ-weighted monogenic functions in 2002. In the unweighted case, the star-like condition of the domain is needed. This fact accounts for the greater generality of the decomposition in the weighted case, which indeed holds in any domain. When k=1, the origin of the notion of λ-weighted monogenic functions is given. In 2017, García et al. [4] studied an integral representation for the solution to the sandwich Dirac equation in Clifford analysis. Yang et al. [5] obtained the Cauchy theorem for the solution to the k-order Dirac equation with α-weight in 2018, where k>0 is an integer and α is a nonzero real number. In 2020, Blaya et al. [6] gave the integral representation for the solution of the bilateral higher-order Dirac equation and proved some properties for Cauchy and Teodorescu transforms. In 2022, Peláez et al. [7] took the sum of the left Dirac operator multiplied by α and the right Dirac operator multiplied by β as a new operator, and studied the integral representation of solutions to higher-order new operators, where α,β are real numbers. In 2023, Dinh [8] introduced (α,β)-monogenic functions and isotonic functions, where α,β are real numbers and α≠β; they gave the integral representation formulae of these functions respectively by using the new proof method and proved the series representation of polynomial Dirac equations. In 2024, Gao et al. [9] got an integral representation for the solution of the bilateral higher order Dirac equation with α-weight, where α is a nonzero real number. Liu et al. [10] investigated some Riemann-Hilbert boundary value problems for perturbed Dirac operators in the Clifford algebra Cl(V3,3). D. A. Santiesteban et al.[11] examined well-posed boundary value problems for second-order elliptic systems of partial differential equations in bounded regular domains of Euclidean spaces.
In 2008, Clifford algebras depending on parameters emerged as an extension of the classical Clifford algebra. Its applications in partial differential equations were introduced by Tutschke and Vanegas [12]. In 2012, Di Teodoro et al. [13] studied solutions for the first order homogeneous meta-Dirac equation and then gave a solution of the inhomogeneous equation by using Fubini′s theorem. In 2013, the integral representations for the meta-Dirac operator of n-order and its conjugate operators of n-order are derived by Balderrama et al. [14]. In 2014, some achievements of hypercomplex analysis were expounded and some of its development trends were presented in reference [15]. Ariza et al. [16] gave the integral formulae to solutions for second order elliptic Dirac equation in 2015. In 2017, Ariza García et al. [17] obtained the correlation between first-order differential operators and q-Dirac operators, with the aim of studying initial value problems, where q is a n-dimensional vector. In 2021, Cuong et al. [18] studied the integral expression of monogenic functions in the Clifford algebra depending on three parameters and solved two boundary value problems related to this function.
Based on the above work, we have conducted certain work with the aim of extending the results from the classical Clifford algebra to the framework of parameter dependent Clifford algebra. In Section 2, we investigate some important properties of functions valued in this Clifford algebra. In Section 3, integral representations for p-order λ-weighted monogenic functions and right q-order λ-weighted monogenic functions are derived. Furthermore, in Section 4, we present an integral representation for (p+q)-order λ-weighted monogenic functions. Finally, Section 5 contains the conclusion and discussion of this paper. This paper mainly generalizes some results of references [5,9].
In this section, we present some basic results on the parameter dependent Clifford analysis, meanwhile, we prove some important properties of some functions valued in the parameter dependent Clifford algebra.
Suppose that αj, γij=γji are nonnegative real numbers for i,j=1,2,…,n,i≠j, the set of base element is {e0=1,e1,…,en}, and the base element satisfies the following multiplication rule
{e2j=−αj,eiej+ejei=2γij. | (2.1) |
From this, we obtain a parameter dependent Clifford algebra Bn(2,αj,γij) which is generated by the structural relationship (2.1). Every element of the algebra is of the form c=∑A1cA1eA1, cA1∈R, where A1:={j1,…,jk}⊆{1,…,n}, j1<j2<⋯<jk, eA1=ej1⋯ejk, and e0=e∅=1. As indices we use the elements A1 of the set containing the ordered subsets of {1,2,…,n}, with the empty subset corresponding to the index 0. The set A1 runs over all the possible ordered sets A1={1≤j1<…<jk≤n}, or A1=∅. The dimension of this algebra is 2n.
Let N∗ be the set of positive integers. If 1≤j≤n and j∈N∗, the base element satisfies the involution ¯ej=−ej. If eA=eh1⋯hr=eh1⋯ehr, then ¯eA=¯ehr⋯¯eh1=(−1)rehr⋯eh1. For any ξ=∑A1ξA1eA1∈Bn(2,αj,γij), we define ¯ξ=∑A1ξA1¯eA1, |ξ|2=∑A1ξ2A1, where ξA1∈R.
The Euclidean Clifford algebra Bn(2,1,0) is one of the special cases of Bn(2,αj,γij).
The function f:Ω→Bn(2,αj,γij) is denoted by f(x)=∑AfA1(x)eA1, where fA1(x) is a real-valued function and Ω is an open connected bounded domain in Rn. f is a r-times continuously differentiable function, which means fA1 is a r-times continuously differentiable function, where r∈N∗. The set consisting of the r-times continuously differentiable function is denoted by Fr(Ω,Bn(2,αj,γij)).
When f∈F1(Ω,Bn(2,αj,γij)), Dirac operators and its conjugate operators acting on function f are defined respectively as follows:
Dxf=n∑k=1ek∂f∂xk,fDx=n∑k=1∂f∂xkek,¯Dxf=n∑k=1¯ek∂f∂xk,f¯Dx=n∑k=1∂f∂xk¯ek. |
After a direct calculation, we have
Dx¯Dx=¯DxDx=n∑j=1αj∂2j−2∑1≤i<j≤nγij∂i∂j, |
the corresponding quadratic form is
n∑j=1αjξ2j−2∑1≤i<j≤nγijξiξj, | (2.2) |
which has a coefficient matrix
B=(α1−γ12⋯−γ1n−γ12α2⋯−γ2n⋯⋯⋱⋯−γ1n−γ2n⋯αn). | (2.3) |
Denote
B1=α1,B2=(α1−γ12−γ12α2),B3=(α1−γ12−γ13−γ12α2−γ12−γ13−γ12α3),⋯,Bn=B. |
See references [19,20]. By using the Sylvester′s criterion, (2.2) is a positive definite quadratic form if and only if the determinant of each Bj is a positive number for all j=1,2,…,n, i.e.,
det(Bj)>0. | (2.4) |
In this situation, Dx¯Dx=¯DxDx becomes an elliptic Dirac operator, so we denote Dx¯Dx=¯DxDx by ˜Δn.
Suppose that (2.4) holds in this paper, then the inverse matrix of matrix B exists and can be represented by
A=(a11a12⋯a1na12a22⋯a2n⋯⋯⋱⋯a1na2n⋯ann), | (2.5) |
where aij=aji, i,j=1,2,…,n.
See reference [12]. For two points x=(x1,⋯,xn) and ζ=(ζ1,⋯,ζn) in Rn, x≠ζ, the representation of the non-Euclidean distance ρ as follows:
ρ2:=ρ2(x,ζ)=n∑i,j=1aij(xi−ζi)(xj−ζj), | (2.6) |
the representation of the Euclidean distance is ι=|x−ζ|.
See reference [14]. Suppose that for some Y∈Rn and Y satisfying |Y|=1, we denote x−ξ=ιY, then the infimum of ρ(Y,0) for all Y is positive, i.e., ρ2(Y,0)≥c0>0, where c0 is a constant, so ρ2(x,ξ)≥c0ι2.
For f∈F1(Ω,Bn(2,αj,γij)), the first-order λ-weighted Dirac operators acting on the function f are defined as follows:
Dλxf=ρ−λxH(x)(Dxf),fDλx=(fDx)ρ−λxH(x), |
where ρx=(∑ni,j=1aijxixj)12, H(x)=∑ni,j=1¯eiaijxj, and λ is a fixed nonzero real number.
Definition 2.1. [12] Suppose f∈F1(Ω,Bn(2,αj,γij)), then a solution f of the Dirac equation Dxf(x)=0 (f(x)Dx=0) is called a left (right) monogenic function.
See reference [12]. We know that ρ−nxH(x) is not only a left monogenic function but also a right monogenic function.
Definition 2.2. Suppose f∈Fp+q(Ω,Bn(2,αj,γij)), p,q are positive integers.
(i) A solution f of the p-order λ-weighted Dirac equation (Dλx)pf(x)=0 is called a left p-order λ-weighted monogenic function, where (Dλx)p=Dλx∘⋯∘Dλx (p-times), ∘ is a composite operation of operators.
(ii) A solution f of the right q-order λ-weighted equation f(x)(Dλx)q=0 is called a right q-order λ-weighted monogenic function, where (Dλx)q=Dλx∘⋯∘Dλx (q-times).
Remark 2.1. (i) When p=1(q=1) in Definition 2.2, a solution f of the λ-weighted Dirac equation Dλxf(x)=0(f(x)Dλx=0) is called a left (right) λ-weighted monogenic function.
(ii) A left p-order λ-weighted monogenic function can be called a p-order λ-weighted monogenic function for short. A left λ-weighted monogenic function can be called a λ-weighted monogenic function for short.
Definition 2.3. Suppose f∈Fp+q(Ω,Bn(2,αj,γij)), p and q are positive integers, then the solution f of equation ((Dλx)pf(x))(Dλx)q=0 is called a (p+q)-order λ-weighted monogenic function.
If f is a left p-order λ-weighted monogenic function, then f is a (p+q)-order monogenic λ-weighted function.
Remark 2.2. ρ−nxH(x), ρ−n+(p−1)λxH(x), and ρ−n+(p+q−1)λxH(x) are (p+q)-order λ-weighted monogenic functions, where λ is a fixed nonzero real number.
For x∈∂Ω, its outer unit normal vector is N(x)=(N(x1),…,N(xn))=(N1,…,Nn), dσx=n∑i=1Nieidμ is the Clifford-algebra-valued measure element of ∂Ω, dμ represents the scalar measure element of ∂Ω, and ∂Ω is a sufficiently smooth boundary.
Lemma 2.1. [12] Suppose f,g∈F1(Ω,Bn(2,αj,γij)), then
∫∂Ωf(x)dσxg(x)=∫Ω[(f(x)Dx)g(x)+f(x)(Dxg(x))]dx, |
where dx=dx1∧dx2∧⋯∧dxn.
Similar to the proof of the theorem in reference [21], we can prove that Lemma 2.2 holds.
Lemma 2.2. Suppose f,g∈F1(Ω,Bn(2,αj,γij)), then
Dx(f(x)g(x))=(Dxf(x))g(x)+n∑k=1ekf(x)∂g(x)∂xk,(f(x)g(x))Dx=n∑k=1∂f(x)∂xkg(x)ek+f(x)(g(x)Dx). |
Proof. We suppose that f(x)=∑A1fA1(x)eA1, g(x)=∑A2gA2(x)eA2, where fA1(x) and gA2(x) are real-valued functions, then
Dx(f(x)g(x))=n∑k=1ek∂(f(x)g(x))∂xk=n∑k=1ek∂[(∑A1fA1(x)eA1)(∑A2gA2(x)eA2)]∂xk=n∑k=1ek∑A1∑A2∂(fA1(x)gA2(x))∂xkeA1eA2=n∑k=1ek∑A1∑A2[∂fA1(x)∂xkgA2(x)+fA1(x)∂gA2(x)∂xk]eA1eA2=n∑k=1ek∂∑A1fA1(x)eA1∂xk∑A2gA2(x)eA2+n∑k=1ek∑A1fA1(x)eA1∂∑A2gA2(x)eA2∂xk=(Dxf(x))g(x)+n∑k=1ekf(x)∂g(x)∂xk. |
Similarly, we can prove the other equality.
Suppose that Mλs(x)=Es(x)ρ−λxH(x), where Es(x)=Csρn−sλx, Cs=1ωnλs−1(s−1)!, s∈N∗, ωn represents the Euclidean surface measure of the unit sphere.
Proposition 2.1. When s>1, we have
DxMλs(x)=Mλs(x)Dx=Es−1(x). |
Proof. Since AB=E and E is the identity matrix, we obtain that for m,k=1,2,…,n,
αkakm−n∑i=1,i≠kγikaim={0,m≠k,1,m=k, |
then
n∑i,k=1ek¯eiamkaim=n∑k=1ek¯ekamkakm+n∑i,k=1,k<iek¯eiamkaim+n∑i,k=1,k>iek¯eiamkaim=n∑k=1ek¯ekamkakm+n∑i,k=1,k<iek¯eiamkaim+n∑j,l=1,j<lel¯ejamlajm=n∑k=1ek¯ekamkakm+n∑i,k=1,k<i(ek¯ei+ei¯ek)amkaim=n∑k=1αkamkakm−2n∑i,k=1,k<iγikamkaim=n∑k=1αkamkakm−n∑i,k=1,k≠iγikamkaim=n∑k=1(αkamk−n∑i=1,k≠iγikaim)akm=n∑k=1δkmakm=amm, |
therefore,
n∑i,k,m=1ek¯eiamkaimx2m=n∑m=1ammx2m. |
Also,
n∑m,i,j,k=1,j≠mek¯eiamkaijxmxj=n∑m,i,j,k=1,j<mek¯eiamkaijxmxj+n∑m,i,j,k=1,j<mek¯eiajkaimxjxm=n∑m,j,k=1,j<mek¯ek(amkakj+ajkakm)xmxj+n∑m,i,j,k=1,i<k,j<mek¯ei(amkaij+ajkaim)xmxj+n∑m,i,j,k=1,i<k,j<mei¯ek(amkaij+ajkaim)xmxj=n∑m,j,k=1,j<mαk(amkakj+ajkakm)xmxj−2n∑m,i,j,k=1,i<k,j<mγik(amkaij+ajkaim)xmxj=n∑m,j,k=1,j<mαk(amkakj+ajkakm)xmxj−n∑m,i,j,k=1,i≠k,j<mγik(amkaij+ajkaim)xmxj=n∑m,k,j=1,j<m(αkamk−n∑i=1,i≠kγikaim)akjxmxj+n∑m,k,j=1,j<m(αkajk−n∑i=1,i≠kγikaij)akmxmxj=n∑m,k,j=1,j<mδkmakjxmxj+n∑m,k,j=1,j<mδkjakmxmxj=n∑m,j=1,j≠mamjxmxj. |
Consequently,
¯H(x)H(x)=(n∑i,j=1eiaijxj)(n∑k,m=1¯ekakmxm)=(n∑i,k,m=1ek¯eiamkaimx2m)+(n∑m,i,j,k=1,m≠jek¯eiamkaijxmxj)=n∑m=1ammx2m+n∑j,m=1,m≠jamjxmxj=ρ2x. |
By ¯H(x)=−H(x), we can conclude that H(x)¯H(x)=−H2(x)=ρ2x.
By AB=E, we have
DxH(x)=n∑i,j=1eiaji¯ej=n∑i=1aiiei¯ei+n∑i,j=1,i<jaij(ei¯ej+ej¯ei)=n∑j=1ajjαj−n∑i,j=1,i≠jaijγij=n∑j=1(ajjαj−n∑i=1,i≠jaijγij)=n. |
Similarly, we can prove that H(x)D=n.
By equalities ¯H(x)H(x)=ρ2x and DxH(x)=n, we can conclude that
DxMλs(x)=Cs[(Dxρ−n+(s−1)λx)H(x)+ρ−n+(s−1)λx(DxH(x))]=Cs[n∑k=1ek∂(n∑i,j=1aijxixj)−n+(s−1)λ2∂xkH(x)+nρ−n+(s−1)λx]=Cs[n∑k=1ek−n+(s−1)λ2(n∑i,j=1aijxixj)−n+(s−1)λ2−1(n∑i=1aikxi+n∑j=1akjxj)H(x)+nρ−n+(s−1)λx]=Cs[n∑k=1ek(−n+(s−1)λ)ρ−n+(s−1)λ−2xn∑i=1aikxiH(x)+nρ−n+(s−1)λx]=Cs[(−n+(s−1)λ)ρ−n+(s−1)λ−2xn∑i,k=1aikxiekH(x)+nρ−n+(s−1)λx]=Cs[(−n+(s−1)λ)ρ−n+(s−1)λ−2x¯H(x)H(x)+nρ−n+(s−1)λx]=Cs(s−1)λρ−n+(s−1)λx=Es−1(x). |
Similarly, we have Mλs(x)Dx=Es−1(x).
Proposition 2.2. Let f∈Fk(Ω,Bn(2,αj,γij)), k∈N∗, s=1,2,…,k.
(1) Suppose that f is a solution of the Dirac equation Dxf=0, then
(Dλx)s(ρkλxf(x))=k!(k−s)!λsρ(k−s)λxf(x). |
(2) Suppose that f is a solution of the Dirac equation fDx=0, then
(ρkλxf(x))(Dλx)s=k!(k−s)!λsρ(k−s)λxf(x). |
Proof. (1) When s=1, by using the equality H(x)¯H(x)=ρ2x, it is easy to deduce that
Dx(ρkλxf(x))=(Dρkλx)f(x)+n∑m=1emρkλx∂f(x)∂xm=n∑m=1em∂(n∑i,j=1aijxixj)kλ2∂xmf(x)+ρkλx(Df(x))=n∑m=1emkλ2ρkλ−2x(n∑i=1aimxi+n∑j=1amjxj)f(x)=kλρkλ−2x(n∑m,i=1aimxiem)f(x)=kλρkλ−2x¯H(x)f(x), |
and
Dλx(ρkλxf(x))=ρ−λxH(x)[Dx(ρkλxf(x))]=kλρ(k−1)λxf(x). |
We suppose that (Dλx)s−1(ρkλxf(x))=k!(k−s+1)!λs−1ρ(k−s+1)λxf(x) holds, then
(Dλx)s(ρkλxf(x))=Dλx(k!(k−s+1)!λs−1ρ(k−s+1)λxf(x))=k!(k−s+1)!λs−1ρ−λxH(x)[Dx(ρ(k−s+1)λxf(x))]=k!(k−s+1)!λs−1ρ−λxH(x)[(k−s+1)λρ(k−s+1)λ−2x¯H(x)f(x)]=k!(k−s)!λsρ(k−s)λxf(x). |
According to the mathematical induction, we get the conclusion.
Similarly, we can prove that (2) holds
Proposition 2.3. Suppose 1≤s≤p, 1≤t≤q, and 1≤k≤p+q, where s,t,k,p,q∈N∗.
(1) Let f∈Fp(Ω,Bn(2,αj,γij)) be a solution of the equation Dxf=0. Then, ρ(p−s)λxf(x) is a solution of the p-order λ-weighted Dirac equation (Dλx)pf(x)=0.
(2) Let f∈Fq(Ω,Bn(2,αj,γij)) be a solution of the right equation fDx=0. Then, ρ(q−t)λxf(x) is a solution of the right q-order λ-weighted Dirac equation f(x)(Dλx)q=0.
(3) Let f∈Fp+q(Ω,Bn(2,αj,γij)) be a solution of the equation system Dxf=0 and fDx=0. Then, ρ(p+q−k)λxf(x) is a solution of the (p+q)-order λ-weighted Dirac equation ((Dλx)pf(x))(Dλx)q=0.
Proof. By Proposition 2.2, (1) and (2) hold.
(3) (i) When q≤k≤p+q, i.e., 1≤k−q≤p, by (1) in Proposition 2.3, we conclude that ρ(p−(k−q))λxf(x) satisfies equation (Dλx)p(ρ(p−(k−q))λxf(x))=0. Therefore, (3) is clearly valid.
(ii) When 1<k≤q, as f satisfies condition Dxf=0 and based on (1) in Proposition 2.2, we can deduce that
(Dλx)p(ρ(p+q−k)λxf(x))=(p+q−k)!(q−k)!λpρ(q−k)λxf(x). |
As f satisfies condition fDx=0 and based on (2) in Proposition 2.3, we obtain
[(Dλx)p(ρ(p+q−k)λxf(x))](Dλx)q=(p+q−k)!(q−k)!λp[(ρ(q−k)λxf(x))(Dλx)q]=0, |
therefore, (3) is established.
Theorem 2.1. Let 1≤s≤p, 1≤t≤q, 1≤k≤p+q, where s,t,k,p,q∈N∗.
(1)Ep(x)ρ−sλxH(x) is a solution of the p-order λ-weighted Dirac equation (Dλx)pf(x)=0.
(2)Eq(x)ρ−tλxH(x) is a solution of the right q-order λ-weighted Dirac equation f(x)(Dλx)q=0.
(3)Ep+q(x)ρ−kλxH(x) is a solution of the (p+q)-order λ-weighted Dirac equation ((Dλx)pf(x))(Dλx)q=0.
Proof. By Proposition 2.3, (1) and (2) hold.
(3) It is obvious that
Ep+q(x)ρ−kλxH(x)=Cp+qρn−(p+q)λxρ−kλxH(x)=ρ(p+q−k)λxCp+qH(x)ρnx. |
By Proposition 2.3 and the equality Dx(ρ−nxH(x))=(ρ−nxH(x))Dx=0, we conclude that (3) in Theorem 2.1 is established.
In this section, we prove two Cauchy-Pompeiu integral formulae for functions valued in Bn(2,αj,γij), and obtain the Cauchy integral formulae for the null solution to higher order λ-weighted Dirac operators as their corollary, respectively.
In this paper, we denote {x|y0=x+x0∈Ω} as Ω∗x0, for any x0∈Ω.
Theorem 3.1. Let p,q∈N∗, s=0,1,…,p; r=0,1,…,q.
(1) If f∈Fp(¯Ω,Bn(2,αj,γij)), then for any x0∈¯Ω, when 0<λ<1p, (Dλx)sf(y0) is a bounded function in ¯Ω∗x0.
(2) If f∈Fq(¯Ω,Bn(2,αj,γij)), then for any x0∈¯Ω, when 0<λ<1q, f(y0)(Dλx)r is a bounded function in ¯Ω∗x0.
Proof. (1) When s=0, as f∈Fp(¯Ω,Bn(2,αj,γij)), (Dλx)0f(y0) is a bounded function in ¯Ω∗x0.
When s=1,2,…,p, we denote H(x)fs(x) by gs(x), and let
f1(x)=Dxf(y0),f2(x)=−λf1(x)+Dxg1(x),f3(x)=−2λf2(x)+Dxg2(x),⋯fp−1(x)=−(p−2)λfp−2(x)+Dxgp−2(x),fp(x)=−(p−1)λfp−1(x)+Dxgp−1(x). |
As f∈Fp(¯Ω,Bn(2,αj,γij)), f1,f2,...,fp are bounded functions in ¯Ω∗x0.
When s=1, we have Dλxf(y0)=ρ−λxH(x)(Dxf(y0))=ρ−λxg1(x).
We suppose that t<p, t∈N∗, and (Dλx)tf(y0)=ρ−tλxgt(x), then
(Dλx)t+1f(y0)=ρ−λxH(x)[Dx(ρ−tλxgt(x))]=ρ−λxH(x)[((−tλ)ρ−tλ−2x¯H(x))gt(x)+ρ−tλx(Dxgt(x))]=ρ−(t+1)λxH(x)[−tλft(x)+Dx(gt(x))]=ρ−(t+1)λxgt+1(x). |
According to the mathematical induction, we get
(Dλx)sf(y0)=ρ−sλxgs(x). |
So for any x0∈¯Ω, if 0<λ<1s, then we conclude that (Dλx)sf(y0) is bounded in ¯Ω∗x0.
Hence, for any x0∈¯Ω, when λ∈⋂ps=1(0,1s)=(0,1p), we conclude that (Dλx)sf(y0) is bounded in ¯Ω∗x0, where s=1,2,…,p.
Similarly, we can prove that (2) holds.
Theorem 3.2. Suppose that f∈Fp(¯Ω,Bn(2,αj,γij)), p≤n,0<λ<1p,p∈N∗, for arbitrary x0∈Ω, then we have
c1(αj,γij)f(x0)=p∑s=1(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))−(−1)p∫Ω∗x0Ep(x)((Dλx)pf(y0))dx, | (3.1) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function f∈Fp(¯Ω,Bn(2,αj,γij)).
Proof. For any x0∈Ω, we have 0+x0∈Ω, so 0∈Ω∗x0.
We can choose an arbitrarily small positive number δ, and make a small ball Bδ={x:|x|<δ} such that ¯Bδ is a subset of Ω∗x0.
For any s=2,3,…,p, by Lemma 2.1, Proposition 2.1, and Theorem 3.1, we have
∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))−limδ→0∫∂BδMλs(x)dσx((Dλx)s−1f(y0))=limδ→0∫Ω∗x0∖Bδ(Mλs(x)Dx)((Dλx)s−1f(y0))dx+limδ→0∫Ω∗x0∖BδMλs(x)[Dx((Dλx)s−1f(y0))]dx=limδ→0∫Ω∗x0∖BδEs−1(x)((Dλx)s−1f(y0))dx+limδ→0∫Ω∗x0∖BδEs(x)((Dλx)sf(y0))dx=∫Ω∗x0Es−1(x)((Dλx)s−1f(y0))dx+∫Ω∗x0Es(x)((Dλx)sf(y0))dx. |
From Theorem 3.1, it can be derived that (Dλx)s−1f(y0) is a bounded function in ¯Ω∗x0, then |(Dλx)s−1f(y0)|≤M1.
For x∈∂Bδ, suppose that x=δX, where X∈∂B1={X:|X|=1}, dμ=δn−1dμ1, dμ1 is the surface element of the unit sphere ∂B1, and since ρ2x≥c0δ2, we can obtain
|∫∂BδMλs(x)dσx((Dλx)s−1f(y0))|≤M1∫∂Bδ|Es(x)|ρ1−λx|dσx|=M1∫∂Bδ|Cs|ωnρn−sλxρ1−λxdμ=M1∫∂Bδ|Cs|ωnρn−1−(s−1)λxδn−1dμ1≤M2∫∂Bδ1δn−1−(s−1)λδn−1dμ1=M2∫∂Bδ1δ−(s−1)λdμ1≤M3δ(s−1)λ+1≤M3δ2, |
where Mi>0 are constants, i=1,2,3, then we can conclude that
limδ→0∫∂BδMλs(x)dσx((Dλx)s−1f(y0))=0. |
Hence,
∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))=∫Ω∗x0Es−1(x)(f(y0)(Dλx)s−1)dx+∫Ω∗x0Es(x)((Dλx)sf(y0))dx, | (3.2) |
where s=2,3,…,p.
For s=1, by Lemma 2.1 and the equality (ρ−nxH(x))D=0, we have
∫∂Ω∗x0Mλ1(x)dσxf(y0)−limδ→0∫∂BδMλ1(x)dσxf(y0)=limδ→0∫Ω∗x0∖Bδ(Mλ1(x)Dx)f(y0)dx+limδ→0∫Ω∗x0∖BδMλ1(x)(Dxf(y0))dx=limδ→0∫Ω∗x0∖Bδ[(H(x)ωnρnx)Dx]f(y0)dx+limδ→0∫Ω∗x0∖BδE1(x)(Dλxf(y0))dx=∫Ω∗x0E1(x)(Dλxf(y0))dx. |
We can calculate that
limδ→0∫∂Bδ1ωnρn−1xdμ=limδ→0∫∂B11ωnδn−1(n∑i,j=1aijXiXj)n−12δn−1dμ1=∫∂B11ωn(n∑i,j=1aijXiXj)n−12dμ1=c1(αj,γij), | (3.3) |
we can conclude that c1(αj,γij) is a Clifford constant, and c1(αj,γij) does not depend on δ but only on the values of the parameters αj and γij; see Remark 2.6 in reference [14].
Hence,
limδ→0∫∂BδMλ1(x)dσxf(y0)=limδ→0∫∂BδH(x)ωnρnxH(x)ρxf(y0)dμ=limδ→0∫∂Bδ−1ωnρn−1x(f(y0)−f(x0))dμ+limδ→0∫∂Bδ−1ωnρn−1xf(x0)dμ=−[limδ→0∫∂Bδ1ωnρn−1xdμ1]f(x0)=−c1(αj,γij)f(x0), |
therefore,
∫∂Ω∗x0Mλ1(x)dσxf(y0)+c1(αj,γij)f(x0)=∫Ω∗x0E1(x)(Dλxf(y0))dx. | (3.4) |
By Equalities (3.2) and (3.4), we have
p∑s=2(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))−∫∂Ω∗x0Mλ1(x)dσxf(y0)−c1(αj,γij)f(x0)=(−1)p∫Ω∗x0Ep(x)((Dλx)pf(y0))dx+(−1)p∫Ω∗x0Ep−1(x)((Dλx)p−1f(y0))dx+(−1)p−1∫Ω∗x0Ep−1(x)((Dλx)p−1f(y0))dx+(−1)p−1∫Ω∗x0Ep−2(x)((Dλx)p−2f(y0))dx+⋯+∫Ω∗x0E2(x)((Dλx)2f(y0))dx+∫Ω∗x0E1(x)(Dλxf(y0))dx−∫Ω∗x0E1(x)(Dλxf(y0))dx=(−1)p∫Ω∗x0Ep(x)((Dλx)pf(y0))dx. |
Consequently, we prove that the conclusion holds.
Remark 3.1. When c1(αj,γij) is not required to be invertible, the value of f(x0) is not uniquely determined by the integral transform.
Corollary 3.1. Suppose that f∈Fp(¯Ω,Bn(2,αj,γij)) is a solution of the equation (Dλx)pf(y0)=0 in ¯Ω∗x0, p≤n,0<λ<1p, p∈N∗, for arbitrary x0∈Ω, then we have
c1(αj,γij)f(x0)=p∑s=1(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0)), | (3.5) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy integral formula of the p-order λ-weighted monogenic function.
Theorem 3.3. Suppose that f∈Fq(¯Ω,Bn(2,αj,γij)), q≤n,0<λ<1q,q∈N∗, for arbitrary x0∈Ω, then we have
f(x0)c1(αj,γij)=q∑r=1(−1)r∫∂Ω∗x0(f(y0)(Dλx)r−1)dσxMλr(x)−(−1)q∫Ω∗x0Eq(x)(f(y0)(Dλx)q)dx, | (3.6) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function f∈Fq(¯Ω,Bn(2,αj,γij)).
Proof. Similar to the proof of Theorem 3.2, we can prove Theorem 3.3.
Corollary 3.2. Suppose that f∈Fq(¯Ω,Bn(2,αj,γij)) is a solution of right q-order λ-weighted Dirac equation f(y0)(Dλx)q=0 in ¯Ω∗x0, q≤n,0<λ<1q, q∈N∗, for arbitrary x0∈Ω, then we have
f(x0)c1(αj,γij)=q∑r=1(−1)r∫∂Ω∗x0(f(y0)(Dλx)r−1)dσxMλr(x), | (3.7) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy integral formula of the right q-order λ-weighted monogenic function.
In this section, we obtain the integral representation for the (p+q)-order λ-weighted monogenic function.
Theorem 4.1. Let p,q∈N∗, s=0,1,…,p; r=0,1,…,q.
Suppose that f∈Fp+q(¯Ω,Bn(2,αj,γij)), when 0<λ<1p+q, then for arbitrary x0∈¯Ω, ((Dλx)sf(y0))(Dλx)r is a bounded function in ¯Ω∗x0.
Proof. (i) For arbitrary x0∈¯Ω, when s=0 and r=0,1,…,q, from Theorem 3.1 and the inequality 0<λ<1p+q<1q, it follows that f(y0)(Dλx)r is a bounded function in ¯Ω∗x0.
(ii) When s=1,2,…,p, and r=1,…,q, we denote H(x)fs,0(x) by gs,0(x), we denote fs,r(x)H(x) by g′s,r(x), and let
f1,0(x)=Dxf(y0),f2,0(x)=−λf1,0(x)+Dxg1,0(x),f3,0(x)=−2λf2,0(x)+Dxg2,0(x),⋯fs,0(x)=−(s−1)λfs−1,0(x)+Dxgs−1,0(x),fs,1(x)=−sλρ−2xgs,0(x)¯H(x)+gs,0(x)Dx,fs,2(x)=−(s+1)λfs,1(x)+g′s,1(x)Dx,fs,3(x)=−(s+2)λfs,2(x)+g′s,2(x)Dx,⋯fs,r(x)=−(s+r−1)λfs,r−1(x)+g′s,r−1(x)Dx. |
As f∈Fp+q(¯Ω,Bn(2,αj,γij)), f1,0,…,fs,0,fs,1,…,fs,r are bounded functions in ¯Ω∗x0.
(a) When s=1 and r=0, by directly calculating, we can obtain
Dλxf(y0)=ρ−λxH(x)(Dxf(y0))=ρ−λxg1,0(x). |
When s=2,…,p and r=0, we suppose that s=t, where t<p,t∈N∗, and
(Dλx)tf(y0)=ρ−tλxgt,0(x), |
then
(Dλx)t+1f(y0)=Dλx(ρ−tλxgt,0(x))=ρ−λxH(x)[Dx(ρ−tλxgt,0(x))]=ρ−λxH(x)[(−tλρ−tλ−2x¯H(x))H(x)ft,0(x)+ρ−tλx(Dxgt,0(x))]=ρ−(t+1)λxH(x)(−tλft,0(x)+Dxgt,0(x))=ρ−(t+1)λxgt+1,0(x). |
According to the mathematical induction, we have
(Dλx)sf(y0)=ρ−sλxgs,0(x). | (4.1) |
For any x0∈¯Ω, when 0<λ<1p+q, we conclude that 0<λ≤1s, so (Dλ)sf(y0) is a bounded function in ¯Ω∗x0.
(b) When s=1,…,p and r=1, by Equality (4.1), we have
((Dλx)sf(y0))Dλx=[(ρ−sλxgs,0(x))Dx]ρ−λxH(x)=[−sλρ−sλ−2xgs,0(x)¯H(x)+ρ−sλx(gs,0(x)Dx)]ρ−λxH(x)=(−sλρ−2xgs,0(x)¯H(x)+gs,0(x)Dx)ρ−(s+1)λxH(x)=fs,1(x)ρ−(s+1)λxH(x)=g′s,1(x)ρ−(s+1)λx. |
When s=1,…,p and r=2,…,q, we suppose that r=l, where l<q,l∈N∗, and
((Dλx)sf(y0))(Dλx)l=g′s,l(x)ρ−(s+l)λx, |
then
((Dλx)sf(y0))(Dλx)l+1=[(g′s,l(x)ρ−(s+l)λx)Dx]ρ−λxH(x)=[−(s+l)λg′s,l(x)ρ−(s+l)λ−2x¯H(x)+ρ−(s+l)λx(g′s,l(x)Dx)]ρ−λxH(x)=(−(s+l)λfs,l(x)+g′s,l(x)Dx)ρ−(s+l+1)λxH(x)=fs,l+1(x)ρ−(s+l+1)λxH(x)=g′s,l+1(x)ρ−(s+l+1)λx. |
According to the mathematical induction, we have
((Dλx)sf(y0))(Dλx)r=g′s,r(x)ρ−(s+r)λx. |
For arbitrary x0∈¯Ω, when 0<λ<1p+q, we conclude that 0<λ<1s+r, so ((Dλx)sf(y0))(Dλx)r is a bounded function in ¯Ω∗x0, where s=1,2,…,p; r=1,…,q.
From the above, for any x0∈¯Ω, when 0<λ<1p+q, we conclude that 0<λ<1s+r, it can be concluded that ((Dλx)sf(y0))(Dλx)r is a bounded function in ¯Ω∗x0, where s=0,1,…,p; r=0,1,…,q.
Theorem 4.2. Suppose that f∈Fp+q(¯Ω,Bn(2,αj,γij)),p+q≤n,0<λ<1p+q,p,q∈N∗, for arbitrary x0∈Ω, then we have
c1(αj,γij)f(x0)=q∑r=1(−1)p+r∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)+p∑s=1(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))−(−1)p+q∫Ω∗x0Ep+q(x)[((Dλx)pf(y0))(Dλx)q]dx, | (4.2) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function f∈Fp+q(¯Ω,Bn(2,αj,γij)).
Proof. We can conclude that Theorem 4.2 holds by applying Theorem 3.2, once we prove that the following equality holds, that is,
q∑r=1(−1)p+r∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)−(−1)p+q∫Ω∗x0Ep+q(x)[((Dλx)pf(y0))(Dλx)q]dx=−(−1)p∫Ω∗x0Ep(x)((Dλx)pf(y0))dx. |
For any x0∈Ω, we know that 0+x0∈Ω, so 0∈Ω∗x0. We can choose an arbitrarily small positive number δ and make a small ball Bδ={x:|x|<δ} such that ¯Bδ is a subset of Ω∗x0.
When r=1,2,3,…,q, by Lemma 2.1 and Proposition 2.1, we have
∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)−limδ→0∫∂Bδ[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)=limδ→0∫Ω∗x0∖Bδ[((Dλx)pf(y0))(Dλx)r]Ep+r(x)dx+limδ→0∫Ω∗x0∖Bδ[((Dλx)pf(y0))(Dλx)r−1]Ep+r−1(x)dx=∫Ω∗x0[((Dλx)pf(y0))(Dλx)r]Ep+r(x)dx+∫Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]Ep+r−1(x)dx. |
By Theorem 4.1, it follows that ((Dλx)pf(y0))(Dλx)r−1 is a bounded function in ¯Ω∗x0, then |((Dλx)pf(y0))(Dλx)r−1|≤M4, hence,
|∫∂Bδ[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)|≤M4∫∂Bδ|Ep+r(x)|ρ1−λx|dσx|=M4∫∂Bδ|Cp+r|ρn−(p+r)λxρ1−λxdμ≤M5∫∂Bδ1δ−(p+r−1)λdμ1≤M6δ(p+r−1)λ+1≤M6δ2, |
where Mi>0 are positive constants, i=4,5,6, and we can conclude that
limδ→0∫∂Bδ[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)=0. |
Hence,
∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)=∫Ω∗x0[((Dλx)pf(y0))(Dλx)r]Ep+r(x)dx+∫Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]Ep+r−1(x)dx. | (4.3) |
By Equality (4.3), we can deduce that
q∑r=1(−1)p+r∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)=(−1)p+q∫Ω∗x0[((Dλx)pf(y0))(Dλx)q]Ep+q(x)dx+(−1)p+q∫Ω∗x0[((Dλx)pf(y0))(Dλx)q−1]Ep+q−1(x)dx+(−1)p+q−1∫Ω∗x0[((Dλx)pf(y0))(Dλx)q−1]Ep+q−1(x)dx+(−1)p+q−1∫Ω∗x0[((Dλx)pf(y0))(Dλx)q−2]Ep+q−2(x)dx+⋯+(−1)p+1∫Ω∗x0[((Dλx)pf(y0))Dλx]Ep+1(x)dx+(−1)p+1∫Ω∗x0((Dλx)pf(y0))Ep(x)dx=(−1)p+q∫Ω∗x0Ep+q(x)[((Dλx)pf(y0))(Dλx)q]dx−(−1)p∫Ω∗x0Ep(x)((Dλx)pf(y0))dx. |
We complete the proof.
Corollary 4.1. Suppose that f∈Fp+q(¯Ω,Bn(2,αj,γij)) is a solution of the equation ((Dλ)pf(y0))(Dλ)q=0 in ¯Ω∗x0, p+q≤n,0<λ<1p+q,p,q∈N∗, for arbitrary x0∈Ω, then we have
c1(αj,γij)f(x0)=q∑r=1(−1)p+r∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)+p∑s=1(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0)), | (4.4) |
where c1(αj,γij) is a Clifford constant. If c1(αj,γij) has a single inverse element, this formula is called the Cauchy integral formula of the (p+q)-order λ-weighted monogenic function.
Remark 4.1. Theorem 4.1 is used to prove Theorem 4.2. As p∈N∗, where N∗ is a set of positive integers, there is no direct relationship between Theorems 3.3 and 4.2. However, when p=0 in Theorem 4.2, if p∑s=1(−1)s∫∂Ω∗x0Mλs(x)dσx((Dλx)s−1f(y0))=0 in Equality (4.2), then the right end of the equality in Theorem 4.2 is reduced to the right end of the equality in Theorem 3.3. When q=0 in Theorem 4.2, if q∑r=1(−1)p+r∫∂Ω∗x0[((Dλx)pf(y0))(Dλx)r−1]dσxMλp+r(x)=0 in Equality (4.2), then the equality in Theorem 4.2 is reduced to the equality in Theorem 3.2.
In recent years, the integral representations for the solution to the higher order Dirac equation in Bn(2,αj,γij) have been studied, which generalize the integral representation in the classical Clifford algebra. In this paper, we not only prove three Cauchy-Pompeiu integral formulae for functions valued in the dependent parameter Clifford algebra, but also obtain integral representations for three different higher order λ-weighted monogenic functions.
If Bn(2,αj,γij)=Bn(2,1,0), then Corollary 3.1 in this paper is reduced to one result of Theorem 3.7 in reference [5], that is,
Theorem 5.1. [5] Suppose that Ω⊆Rn is a domain, Ω∗:={x|y0=x+x0∈Ω}, Hj(x)=Aj|x|n−jα, Aj=(−1)j−1ωnαj−1(j−1)!, 0<α<1k. If f(x+x0) is a k-monogenic function with α-weight in Ω∗, for arbitrary x0∈Ω, then we have
f(x0)=k∑j=1(−1)j−1∫∂Ω∗Hj(x)|x|−αxdσx((Dαx)j−1f(x+x0)). | (5.1) |
If Bn(2,αj,γij)=Bn(2,1,0), Corollary 4.1 in this paper is reduced to Corollary 3.5 in [9], that is,
Theorem 5.2. [9] Suppose f∈Cr(Ω,Cl0,n(R)), where r≥p+q, n≥p+q, Ω⊆Rn is a domain, Ω∗:={x|y0=x+x0∈Ω}, Hp+j(x)=Ap+j|x|n−(p+j)α, Ap+j=(−1)p+j−1ωnαp+j−1(p+j−1)!, 0<α<1p+q. If f(x+x0) is a (p,q)-monogenic function with α-weight in Ω∗, then for any x0∈Ω, we have
f(x0)=q∑j=1(−1)p+j∫∂Ω∗((Dαx)pf(x+x0))(Dαx)j−1dσx(x|x|−αHp+j(x))+p∑j=1(−1)j∫∂Ω∗Hj(x)|x|−αxdσx((Dαx)j−1f(x+x0)). | (5.2) |
With the method of the Clifford analytic approach and Newton embedding method, reference [10] proved the existence and uniqueness of solutions of the nonlinear Riemann-Hilbert problems. For a k-vector field Fk, reference [11] obtained the solution of boundary value problems for the associated with the equations (Dx)2s−1(Fk)Dx=fk, where fk∈F(Ω,B(k)m(2,1,0)), B(k)m(2,1,0) is the space of pseudo-scalars in the classical Clifford algebra Bm(2,1,0). We hope to solve the boundary value problem related to the equation (Dx)2s−1(Fk)Dx=fk in the dependent parameter Clifford algebra in our future work.
Xiaojing Du: Conceptualization, Writing-original draft, Writing-review and editing; Xiaotong Liang: Validation and Writing-review; Yonghong Xie: Supervision, Validation and Funding acquisition. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The work was supported by the Natural Science Foundation of Hebei Province (Nos. A2023205006, A2023205045, A2022208007 and A2024208005), the Key Development Foundation of Hebei Normal University (No. L2024ZD08), the National Natural Science Foundation of China (No. 12431005), and the Funding Project of Central Guidance for Local Scientific and Technological Development (No. 246Z7608G).
The authors state that there is no conflicts of interest in this paper.
[1] |
M. A. M. Mu'lla, Fractional calculus, fractional differential equations and applications, Open Access Libr. J., 7 (2020), e6244. https://doi.org/10.4236/oalib.1106244 doi: 10.4236/oalib.1106244
![]() |
[2] |
F. Mainardi, Fractional calculus: Theory and applications, Mathematics, 6 (2018), 145. https://doi.org/10.3390/math6090145 doi: 10.3390/math6090145
![]() |
[3] |
V. V. Kulish, J. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002), 803–806. https://doi.org/10.1115/1.1478062 doi: 10.1115/1.1478062
![]() |
[4] |
N. Engheta, On fractional calculus and fractional multipoles in electromagnetism, IEEE T. Antenn. Propag., 44 (1996), 554–566. https://doi.org/10.1109/8.489308 doi: 10.1109/8.489308
![]() |
[5] |
K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41 (2010), 9–12. https://doi.org/10.1016/j.advengsoft.2008.12.012 doi: 10.1016/j.advengsoft.2008.12.012
![]() |
[6] |
R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985), 918–925. https://doi.org/10.2514/3.9007 doi: 10.2514/3.9007
![]() |
[7] |
F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004 doi: 10.1016/j.cnsns.2009.05.004
![]() |
[8] |
K. A. Lazopoulos, Non-local continuum mechanics and fractional calculus, Mech. Res. Commun., 33 (2006), 753–757. https://doi.org/10.1016/j.mechrescom.2006.05.001 doi: 10.1016/j.mechrescom.2006.05.001
![]() |
[9] |
C. Lederman, J. Roquejoffre, N. Wolanski, Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames, Ann. Mat. Pura Appl., 183 (2004), 173–239. https://doi.org/10.1007/s10231-003-0085-1 doi: 10.1007/s10231-003-0085-1
![]() |
[10] | J. H. He, Nonlinear oscillation with fractional derivative and its applications, Dalian: International Conference on Vibrating Engineering, 1998,288–291. |
[11] |
E. M. Elsayed, R. Shah, K. Nonlaopon, The analysis of the fractional-order Navier-Stokes equations by a novel approach, J. Funct. Spaces, 2022, (2022), 8979447. https://doi.org/10.1155/2022/8979447 doi: 10.1155/2022/8979447
![]() |
[12] |
O.P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38 (2004), 323–337. https://doi.org/10.1007/s11071-004-3764-6 doi: 10.1007/s11071-004-3764-6
![]() |
[13] |
N. A. Shah, E. R. El-Zahar, A. Akgül, A. Khan, J. Kafle, Analysis of fractional-order regularized long-wave models via a novel transform, J. Funct. Spaces, 2022 (2022), 2754507. https://doi.org/10.1155/2022/2754507 doi: 10.1155/2022/2754507
![]() |
[14] |
A. A. Alderremy, S. Aly, R. Fayyaz, A. Khan, R. Shah, N. Wyal, The analysis of fractional-order nonlinear systems of third order KdV and Burgers equations via a novel transform, Complexity, 2022 (2022), 4935809. https://doi.org/10.1155/2022/4935809 doi: 10.1155/2022/4935809
![]() |
[15] |
K. Nonlaopon, A. M. Alsharif, A. M. Zidan, A. Khan, Y. S. Hamed, R. Shah, Numerical investigation of fractional-order Swift-Hohenberg equations via a novel transform, Symmetry, 13 (2021), 1263. https://doi.org/10.3390/sym13071263 doi: 10.3390/sym13071263
![]() |
[16] |
M. K. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. S. Abdo, Analytical investigation of noyes-field model for time-fractional belousov-zhabotinsky reaction, Complexity, 2021 (2021), 3248376. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376
![]() |
[17] |
J. T. Edwards, N. J. Ford, A. C. Simpson, The numerical solution of linear multi-term fractional differential equations: Systems of equations, J. Comput. Appl. Math., 148 (2002), 401–418. https://doi.org/10.1016/s0377-0427(02)00558-7 doi: 10.1016/s0377-0427(02)00558-7
![]() |
[18] |
R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. https://doi.org/10.3390/math6020016 doi: 10.3390/math6020016
![]() |
[19] |
S. Esmaeili, M. Shamsi, Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials, Comput. Math. Appl., 62 (2011), 918–929. https://doi.org/10.1016/j.camwa.2011.04.023 doi: 10.1016/j.camwa.2011.04.023
![]() |
[20] |
A. S. Alshehry, M. Imran, R. Shah, W. Weera, Fractional-view analysis of Fokker-Planck equations by ZZ transform with Mittag-Leffler Kernel, Symmetry, 14 (2022), 1513. https://doi.org/ 10.3390/sym14081513 doi: 10.3390/sym14081513
![]() |
[21] |
J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. A, 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
![]() |
[22] |
M. Dehghan, F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. Scr., 78 (2008), 065004. https://doi.org/10.1088/0031-8949/78/06/065004 doi: 10.1088/0031-8949/78/06/065004
![]() |
[23] |
C. M. Pappalardo, M. C. De Simone, D. Guida, Multibody modeling and nonlinear control of the pantograph/catenary system, Arch. Appl. Mech., 89 (2019), 1589–1626. https://doi.org/10.1007/s00419-019-01530-3 doi: 10.1007/s00419-019-01530-3
![]() |
[24] |
D. F. Li, C. J. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simulat., 172 (2020), 244–257. https://doi.org/10.1016/j.matcom.2019.12.004 doi: 10.1016/j.matcom.2019.12.004
![]() |
[25] |
A. S. Alshehry, M. Imran, A. Khan, R. Shah, W. Weera, Fractional view analysis of Kuramoto-Sivashinsky equations with non-singular Kernel operators, Symmetry, 14, (2022), 1463. https://doi.org/10.3390/sym14071463 doi: 10.3390/sym14071463
![]() |
[26] |
K. Z. Guan, Q. S. Wang, Asymptotic behavior of solutions of a nonlinear neutral generalized pantograph equation with impulses, Math. Slovaca, 65 (2015), 1049–1062. https://doi.org/10.1515/ms-2015-0072 doi: 10.1515/ms-2015-0072
![]() |
[27] |
P. Rahimkhani, Y. Ordokhani, E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309 (2017), 493–510. https://doi.org/10.1016/j.cam.2016.06.005 doi: 10.1016/j.cam.2016.06.005
![]() |
[28] |
C. Q. Yang, J. H. Hou, X. G. Lv, Jacobi spectral collocation method for solving fractional pantograph delay differential equations, Eng. Comput., 38 (2020), 1985–1994. https://doi.org/10.1007/s00366-020-01193-7 doi: 10.1007/s00366-020-01193-7
![]() |
[29] |
H. Dehestani, Y. Ordokhani, M. Razzaghi, Numerical technique for solving fractional generalized pantograph-delay differential equations by using fractional-order hybrid bessel functions, Int. J. Appl. Comput. Math., 6 (2020), 9. https://doi.org/10.1007/s40819-019-0756-2 doi: 10.1007/s40819-019-0756-2
![]() |
[30] |
K. Rabiei, Y. Ordokhani, Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials, Eng. Comput., 35 (2018), 1431–1441. https://doi.org/10.1007/s00366-018-0673-8 doi: 10.1007/s00366-018-0673-8
![]() |
[31] |
M. S. Hashemi, A. Atangana, S. Hajikhah, Solving fractional pantograph delay equations by an effective computational method, Math. Comput. Simulat., 177 (2020), 295–305. https://doi.org/10.1016/j.matcom.2020.04.026 doi: 10.1016/j.matcom.2020.04.026
![]() |
[32] |
L. Shi, X. H. Ding, Z. Chen, Q. Ma, A new class of operational matrices method for solving fractional neutral pantograph differential equations, Adv. Differ. Equ., 2018 (2018), 94. https://doi.org/10.1186/s13662-018-1536-8 doi: 10.1186/s13662-018-1536-8
![]() |
[33] |
B. Yuttanan, M. Razzaghi, T. N. Vo, A fractional-order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations, Math. Method. Appl. Sci., 44 (2021), 4156–4175. https://doi.org/10.1002/mma.7020 doi: 10.1002/mma.7020
![]() |
[34] |
I. Ahmed, P. Kumam, K. Shah, P. Borisut, K. Sitthithakerngkiet, M. A. Demba, Stability results for implicit fractional pantograph differential equations via ϕ-Hilfer fractional derivative with a nonlocal Riemann-Liouville fractional integral condition, Mathematics, 8 (2020), 94. https://doi.org/10.3390/math8010094 doi: 10.3390/math8010094
![]() |
[35] |
P. Sunthrayuth, R. Ullah, A. Khan, R. Shah, J. Kafle, I. Mahariq, et al., Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations, J. Funct. Spaces, 2021 (2021), 1537958. https://doi.org/10.1155/2021/1537958 doi: 10.1155/2021/1537958
![]() |
[36] |
J. P. Boyd, Spectral methods in fluid dynamics (C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang), SIAM Rev., 30 (1988), 666–668. https://doi.org/10.1137/1030157 doi: 10.1137/1030157
![]() |
[37] |
G. M. Zaslavsky, Book Review: "Theory and applications of fractional differential equations" by Anatoly A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Fractals, 15 (2007), 101–102. https://doi.org/10.1142/s0218348x07003447 doi: 10.1142/s0218348x07003447
![]() |
[38] |
A. Kadem, D. Baleanu, Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 491–501. https://doi.org/10.1016/j.cnsns.2009.05.024 doi: 10.1016/j.cnsns.2009.05.024
![]() |
[39] |
H. Jafari, M. Mahmoudi, M. H. N. Skandari, A new numerical method to solve pantograph delay differential equations with convergence analysis, Adv. Differ. Equ., 2021 (2021), 129. https://doi.org/10.1186/s13662-021-03293-0 doi: 10.1186/s13662-021-03293-0
![]() |
[40] |
T. Akkaya, S. Yalcinbas, M. Sezer, Numeric solutions for the pantograph type delay differential equation using first Boubaker polynomials, Appl. Math. Comput., 219 (2013), 9484–9492. https://doi.org/10.1016/j.amc.2013.03.021 doi: 10.1016/j.amc.2013.03.021
![]() |
[41] |
B. Benhammouda, H. Vazquez-Leal, L. Hernandez-Martinez, Procedure for exact solutions of nonlinear pantograph delay differential equations, J. Adv. Math. Comput. Sci., 4 (2014), 2738–2751. https://doi.org/10.9734/bjmcs/2014/11839 doi: 10.9734/bjmcs/2014/11839
![]() |
[42] | I. Ali, H. Brunner, T. Tang, A Spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), 254–265. |
[43] |
H. Jafari, M. Mahmoudi, M. H. N. Skandari, A new numerical method to solve pantograph delay differential equations with convergence analysis, Adv. Differ. Equ., 2021 (2021), 129. https://doi.org/10.1186/s13662-021-03293-0 doi: 10.1186/s13662-021-03293-0
![]() |
[44] |
Y. Muroya, E. Ishiwata, H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math., 152 (2003), 347–366. https://doi.org/10.1016/s0377-0427(02)00716-1 doi: 10.1016/s0377-0427(02)00716-1
![]() |
[45] |
M. Sezer, S. Yalcinbas, M. Gulsu, A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term, Int. J. Comput. Math., 85 (2008), 1055–1063. https://doi.org/10.1080/00207160701466784 doi: 10.1080/00207160701466784
![]() |
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39. | Elena E. Pohl, Olga Jovanovic, The Role of Phosphatidylethanolamine Adducts in Modification of the Activity of Membrane Proteins under Oxidative Stress, 2019, 24, 1420-3049, 4545, 10.3390/molecules24244545 | |
40. | Daniel Andrew M. Gideon, Joel James, 2021, Chapter 185-1, 978-981-15-4501-6, 1, 10.1007/978-981-15-4501-6_185-1 | |
41. | Scott J. Walmsley, Jingshu Guo, Paari Murugan, Christopher J. Weight, Jinhua Wang, Peter W. Villalta, Robert J. Turesky, Comprehensive Analysis of DNA Adducts Using Data-Independent wSIM/MS2 Acquisition and wSIM-City, 2021, 93, 0003-2700, 6491, 10.1021/acs.analchem.1c00362 | |
42. | Shunji Kato, Naoki Shimizu, Yurika Otoki, Junya Ito, Masayoshi Sakaino, Takashi Sano, Shigeo Takeuchi, Jun Imagi, Kiyotaka Nakagawa, Determination of acrolein generation pathways from linoleic acid and linolenic acid: increment by photo irradiation, 2022, 6, 2396-8370, 10.1038/s41538-022-00138-2 | |
43. | Federica Papaccio, Andrea D′Arino, Silvia Caputo, Barbara Bellei, Focus on the Contribution of Oxidative Stress in Skin Aging, 2022, 11, 2076-3921, 1121, 10.3390/antiox11061121 | |
44. | Iliana Ivanova, Radostina Toshkovska, Lyubomira Yocheva, Dayana Benkova, Vesela Yordanova, Alexandrina Nesheva, Rusina Hazarosova, Galya Staneva, Aneliya Kostadinova, 2023, Chapter 15, 978-3-031-31068-3, 147, 10.1007/978-3-031-31069-0_15 | |
45. | Mohsen Sisakht, Maryam Darabian, Amir Mahmoodzadeh, Ali Bazi, Sayed Mohammad Shafiee, Pooneh Mokarram, Zahra Khoshdel, The role of radiation induced oxidative stress as a regulator of radio-adaptive responses, 2020, 96, 0955-3002, 561, 10.1080/09553002.2020.1721597 | |
46. | Agata Los, Dana Ziuzina, Robin Van Cleynenbreugel, Daniela Boehm, Paula Bourke, Assessing the Biological Safety of Atmospheric Cold Plasma Treated Wheat Using Cell and Insect Models, 2020, 9, 2304-8158, 898, 10.3390/foods9070898 | |
47. | Ariane Schröter, Hanns-Christian Mahler, Nadia Ben Sayed, Atanas V. Koulov, Jörg Huwyler, Michael Jahn, 4-Hydroxynonenal – A Toxic Leachable from Clinically Used Administration Materials, 2021, 110, 00223549, 3268, 10.1016/j.xphs.2021.05.014 | |
48. | Tauseef Alam, Sana Rizwan, Zeba Farooqui, Subuhi Abidi, Iqbal Parwez, Farah Khan, Oral Nigella sativa oil administration alleviates arsenic-induced redox imbalance, DNA damage, and metabolic and histological alterations in rat liver, 2021, 28, 0944-1344, 41464, 10.1007/s11356-021-13493-6 | |
49. | Reyhaneh Moradi‐Marjaneh, Seyed Mahdi Hassanian, Mehraneh Mehramiz, Majid Rezayi, Gordon A. Ferns, Majid Khazaei, Amir Avan, Reactive oxygen species in colorectal cancer: The therapeutic impact and its potential roles in tumor progression via perturbation of cellular and physiological dysregulated pathways, 2019, 234, 0021-9541, 10072, 10.1002/jcp.27881 | |
50. | Kun Lu, Yun-Chung Hsiao, Chih-Wei Liu, Rita Schoeny, Robinan Gentry, Thomas B. Starr, A Review of Stable Isotope Labeling and Mass Spectrometry Methods to Distinguish Exogenous from Endogenous DNA Adducts and Improve Dose–Response Assessments, 2022, 35, 0893-228X, 7, 10.1021/acs.chemrestox.1c00212 | |
51. | Alessandro Attanzio, Ignazio Restivo, Marco Tutone, Luisa Tesoriere, Mario Allegra, Maria A. Livrea, Redox Properties, Bioactivity and Health Effects of Indicaxanthin, a Bioavailable Phytochemical from Opuntia ficus indica, L.: A Critical Review of Accumulated Evidence and Perspectives, 2022, 11, 2076-3921, 2364, 10.3390/antiox11122364 | |
52. | Nadeem G Khan, Sangavi Eswaran, Divya Adiga, S. Sriharikrishnaa, Sanjiban Chakrabarty, Padmalatha S. Rai, Shama Prasada Kabekkodu, Integrated bioinformatic analysis to understand the association between phthalate exposure and breast cancer progression, 2022, 457, 0041008X, 116296, 10.1016/j.taap.2022.116296 | |
53. | Georg Wultsch, Tahereh Setayesh, Michael Kundi, Michael Kment, Armen Nersesyan, Michael Fenech, Siegfried Knasmüller, Induction of DNA damage as a consequence of occupational exposure to crystalline silica: A review and meta-analysis, 2021, 787, 13835742, 108349, 10.1016/j.mrrev.2020.108349 | |
54. | Caroline Molinaro, Alain Martoriati, Katia Cailliau, Proteins from the DNA Damage Response: Regulation, Dysfunction, and Anticancer Strategies, 2021, 13, 2072-6694, 3819, 10.3390/cancers13153819 | |
55. | Morana Jaganjac, Lidija Milkovic, Agnieszka Gegotek, Marina Cindric, Kamelija Zarkovic, Elzbieta Skrzydlewska, Neven Zarkovic, The relevance of pathophysiological alterations in redox signaling of 4-hydroxynonenal for pharmacological therapies of major stress-associated diseases, 2020, 157, 08915849, 128, 10.1016/j.freeradbiomed.2019.11.023 | |
56. | Jacob L. Brown, Fredrick F. Peelor, Constantin Georgescu, Jonathan D. Wren, Michael Kinter, Victoria J. Tyrrell, Valerie B. O'Donnell, Benjamin F. Miller, Holly Van Remmen, Lipid hydroperoxides and oxylipins are mediators of denervation induced muscle atrophy, 2022, 57, 22132317, 102518, 10.1016/j.redox.2022.102518 | |
57. | Ruofei Du, Li Luo, Laurie G. Hudson, Sara Nozadi, Johnnye Lewis, An adjusted partial least squares regression framework to utilize additional exposure information in environmental mixture data analysis, 2022, 0266-4763, 1, 10.1080/02664763.2022.2043254 | |
58. | Rory B Conolly, Jeffry Schroeter, Julia S Kimbell, Harvey Clewell, Melvin E Andersen, P Robinan Gentry, Updating the biologically based dose-response model for the nasal carcinogenicity of inhaled formaldehyde in the F344 rat, 2023, 193, 1096-6080, 1, 10.1093/toxsci/kfad028 | |
59. | Analía Alejandra Lu-Martínez, Juan Gabriel Báez-González, Sandra Castillo-Hernández, Carlos Amaya-Guerra, José Rodríguez-Rodríguez, Eristeo García-Márquez, Studied of Prunus serotine oil extracted by cold pressing and antioxidant effect of P. longiflora essential oil, 2021, 58, 0022-1155, 1420, 10.1007/s13197-020-04653-6 | |
60. | Agnieszka Gęgotek, Anna Jastrząb, Marta Dobrzyńska, Michał Biernacki, Elżbieta Skrzydlewska, Exogenous Antioxidants Impact on UV-Induced Changes in Membrane Phospholipids and the Effectiveness of the Endocannabinoid System in Human Skin Cells, 2021, 10, 2076-3921, 1260, 10.3390/antiox10081260 | |
61. | Robert Nilsson, Ning-Ang Liu, Nuclear DNA damages generated by reactive oxygen molecules (ROS) under oxidative stress and their relevance to human cancers, including ionizing radiation-induced neoplasia part I: Physical, chemical and molecular biology aspects, 2020, 1, 26665557, 140, 10.1016/j.radmp.2020.09.002 | |
62. | Ariane Schröter, Atanas V. Koulov, Jörg Huwyler, Hanns-Christian Mahler, Michael Jahn, 4-Hydroxynonenal is An Oxidative Degradation Product of Polysorbate 80, 2021, 110, 00223549, 2524, 10.1016/j.xphs.2021.01.027 | |
63. | Shimaa I. Rakha, Mohammed A. Elmetwally, Hossam El-Sheikh Ali, Ahmed Zaky Balboula, Abdelmonem Montaser Mahmoud, Samy M. Zaabel, Lycopene Reduces the In Vitro Aging Phenotypes of Mouse Oocytes by Improving Their Oxidative Status, 2022, 9, 2306-7381, 336, 10.3390/vetsci9070336 | |
64. | Rajesh Singh Jadon, Gajanand Sharma, Neeraj K. Garg, Nikunj Tandel, Kavita R. Gajbhiye, Rajesh Salve, Virendra Gajbhiye, Ujjawal Sharma, Om Prakash Katare, Manoj Sharma, Rajeev K. Tyagi, Efficient in vitro and in vivo docetaxel delivery mediated by pH-sensitive LPHNPs for effective breast cancer therapy, 2021, 203, 09277765, 111760, 10.1016/j.colsurfb.2021.111760 | |
65. | Koraljka Gall Trošelj, Marko Tomljanović, Morana Jaganjac, Tanja Matijević Glavan, Ana Čipak Gašparović, Lidija Milković, Suzana Borović Šunjić, Brigitta Buttari, Elisabetta Profumo, Sarmistha Saha, Luciano Saso, Neven Žarković, Oxidative Stress and Cancer Heterogeneity Orchestrate NRF2 Roles Relevant for Therapy Response, 2022, 27, 1420-3049, 1468, 10.3390/molecules27051468 | |
66. | Arash Rafeeinia, Gholamreza Asadikaram, Mehrnaz Karimi-Darabi, Moslem Abolhassani, Mojtaba Abbasi-Jorjandi, Vahid Moazed, Organochlorine Pesticides, Oxidative Stress Biomarkers, and Leukemia: A Case-Control Study, 2022, 70, 1081-5589, 1736, 10.1136/jim-2021-002289 | |
67. | Nouf M. Alyami, Rafa Almeer, Hanadi M. Alyami, Role of green synthesized platinum nanoparticles in cytotoxicity, oxidative stress, and apoptosis of human colon cancer cells (HCT-116), 2022, 8, 24058440, e11917, 10.1016/j.heliyon.2022.e11917 | |
68. | Jingshu Guo, Joseph S. Koopmeiners, Scott J. Walmsley, Peter W. Villalta, Lihua Yao, Paari Murugan, Resha Tejpaul, Christopher J. Weight, Robert J. Turesky, The Cooked Meat Carcinogen 2-Amino-1-methyl-6-phenylimidazo[4,5-b]pyridine Hair Dosimeter, DNA Adductomics Discovery, and Associations with Prostate Cancer Pathology Biomarkers, 2022, 35, 0893-228X, 703, 10.1021/acs.chemrestox.2c00012 | |
69. | Vladislav Vladimirovich Tsukanov, Olga Valentinovna Smirnova, Edward Vilyamovich Kasparov, Alexander Alexandrovich Sinyakov, Alexander Viktorovich Vasyutin, Julia Leongardovna Tonkikh, Mikhail Alexandrovich Cherepnin, Dynamics of Oxidative Stress in Helicobacter pylori-Positive Patients with Atrophic Body Gastritis and Various Stages of Gastric Cancer, 2022, 12, 2075-4418, 1203, 10.3390/diagnostics12051203 | |
70. | Hong-Min Qin, Denise Herrera, Dian-Feng Liu, Chao-Qian Chen, Armen Nersesyan, Miroslav Mišík, Siegfried Knasmueller, Genotoxic properties of materials used for endoprostheses: Experimental and human data, 2020, 145, 02786915, 111707, 10.1016/j.fct.2020.111707 | |
71. | Deema Islayem, Fatima Ba Fakih, Sungmun Lee, Comparison of Colorimetric Methods to Detect Malondialdehyde, A Biomarker of Reactive Oxygen Species, 2022, 7, 2365-6549, 10.1002/slct.202103627 | |
72. | Balázs Olasz, Béla Fiser, Milán Szőri, Béla Viskolcz, Michael C. Owen, Computational Elucidation of the Solvent-Dependent Addition of 4-Hydroxy-2-nonenal (HNE) to Cysteine and Cysteinate Residues, 2022, 87, 0022-3263, 12909, 10.1021/acs.joc.2c01487 | |
73. | Josh Williamson, Gareth Davison, Targeted Antioxidants in Exercise-Induced Mitochondrial Oxidative Stress: Emphasis on DNA Damage, 2020, 9, 2076-3921, 1142, 10.3390/antiox9111142 | |
74. | Eunnara Cho, Ashley Allemang, Marc Audebert, Vinita Chauhan, Stephen Dertinger, Giel Hendriks, Mirjam Luijten, Francesco Marchetti, Sheroy Minocherhomji, Stefan Pfuhler, Daniel J. Roberts, Kristina Trenz, Carole L. Yauk, AOP report: Development of an adverse outcome pathway for oxidative DNA damage leading to mutations and chromosomal aberrations , 2022, 63, 0893-6692, 118, 10.1002/em.22479 | |
75. | Ion Alexandru Bobulescu, Laurentiu M. Pop, Chinnadurai Mani, Kala Turner, Christian Rivera, Sabiha Khatoon, Subash Kairamkonda, Raquibul Hannan, Komaraiah Palle, Renal Lipid Metabolism Abnormalities in Obesity and Clear Cell Renal Cell Carcinoma, 2021, 11, 2218-1989, 608, 10.3390/metabo11090608 | |
76. | Sarmistha Saha, Luciano Saso, Guliz Armagan, Cancer Prevention and Therapy by Targeting Oxidative Stress Pathways, 2023, 28, 1420-3049, 4293, 10.3390/molecules28114293 | |
77. | Anna Chiaramonte, Serena Testi, Caterina Pelosini, Consuelo Micheli, Aurora Falaschi, Giovanni Ceccarini, Ferruccio Santini, Roberto Scarpato, Oxidative and DNA damage in obese patients undergoing bariatric surgery: a one-year follow-up study, 2023, 00275107, 111827, 10.1016/j.mrfmmm.2023.111827 | |
78. | Amelia Rojas‐Gómez, Sara G. Dosil, Francisco J. Chichón, Nieves Fernández‐Gallego, Alessia Ferrarini, Enrique Calvo, Diego Calzada‐Fraile, Silvia Requena, Joaquin Otón, Alvaro Serrano, Rocio Tarifa, Montserrat Arroyo, Andrea Sorrentino, Eva Pereiro, Jesus Vázquez, José M. Valpuesta, Francisco Sánchez‐Madrid, Noa B. Martín‐Cófreces, Chaperonin CCT controls extracellular vesicle production and cell metabolism through kinesin dynamics, 2023, 12, 2001-3078, 10.1002/jev2.12333 | |
79. | Yulemni Morel, Jace W. Jones, Utilization of LC–MS/MS and Drift Tube Ion Mobility for Characterizing Intact Oxidized Arachidonate-Containing Glycerophosphatidylethanolamine, 2023, 1044-0305, 10.1021/jasms.3c00083 | |
80. | Karoline Felisbino, Nathalia Kirsten, Shayane da Silva Milhorini, Isabela Saragioto Marçal, Karina Bernert, Rafaela Schiessl, Leticia Nominato-Oliveira, Izonete Cristina Guiloski, Teratogenic effects of the dicamba herbicide in Zebrafish (Danio rerio) embryos, 2023, 02697491, 122187, 10.1016/j.envpol.2023.122187 | |
81. | Renan Muniz-Santos, Giovanna Lucieri-Costa, Matheus Augusto P. de Almeida, Isabelle Moraes-de-Souza, Maria Alice Dos Santos Mascarenhas Brito, Adriana Ribeiro Silva, Cassiano Felippe Gonçalves-de-Albuquerque, Lipid oxidation dysregulation: an emerging player in the pathophysiology of sepsis, 2023, 14, 1664-3224, 10.3389/fimmu.2023.1224335 | |
82. | Adam Wroński, Izabela Dobrzyńska, Szymon Sękowski, Wojciech Łuczaj, Ewa Olchowik-Grabarek, Elżbieta Skrzydlewska, Cannabidiol and Cannabigerol Modify the Composition and Physicochemical Properties of Keratinocyte Membranes Exposed to UVA, 2023, 24, 1422-0067, 12424, 10.3390/ijms241512424 | |
83. | Tao Shen, Haiyang Wang, Rongkang Hu, Yanni Lv, Developing neural network diagnostic models and potential drugs based on novel identified immune-related biomarkers for celiac disease, 2023, 17, 1479-7364, 10.1186/s40246-023-00526-z | |
84. | Krystal D. Kao, Helmut Grasberger, Mohamad El-Zaatari, The Cxcr2+ subset of the S100a8+ gastric granylocytic myeloid-derived suppressor cell population (G-MDSC) regulates gastric pathology, 2023, 14, 1664-3224, 10.3389/fimmu.2023.1147695 | |
85. | Nurbubu T. Moldogazieva, Sergey P. Zavadskiy, Dmitry V. Astakhov, Alexander A. Terentiev, Lipid peroxidation: Reactive carbonyl species, protein/DNA adducts, and signaling switches in oxidative stress and cancer, 2023, 0006291X, 149167, 10.1016/j.bbrc.2023.149167 | |
86. | Rohit Sharma, Bhawna Diwan, Lipids and the hallmarks of ageing: From pathology to interventions, 2023, 215, 00476374, 111858, 10.1016/j.mad.2023.111858 | |
87. | Nahed Nasser Eid El-Sayed, Taghreed M. Al-Otaibi, Assem Barakat, Zainab M. Almarhoon, Mohd. Zaheen Hassan, Maha I. Al-Zaben, Najeh Krayem, Vijay H. Masand, Abir Ben Bacha, Synthesis and Biological Evaluation of Some New 3-Aryl-2-thioxo-2,3-dihydroquinazolin-4(1H)-ones and 3-Aryl-2-(benzylthio)quinazolin-4(3H)-ones as Antioxidants; COX-2, LDHA, α-Glucosidase and α-Amylase Inhibitors; and Anti-Colon Carcinoma and Apoptosis-Inducing Agents, 2023, 16, 1424-8247, 1392, 10.3390/ph16101392 | |
88. | Yangling Zhang, Yuxin Song, Jiao Zhang, Lanlan Li, Lin He, Jiahui Bo, Zhihua Gong, Wenjun Xiao, L-theanine regulates the immune function of SD rats fed high-protein diets through the FABP5/IL-6/STAT3/PPARα pathway, 2023, 181, 02786915, 114095, 10.1016/j.fct.2023.114095 | |
89. | Dessislava Staneva, Neli Dimitrova, Borislav Popov, Albena Alexandrova, Milena Georgieva, George Miloshev, Haberlea rhodopensis Extract Tunes the Cellular Response to Stress by Modulating DNA Damage, Redox Components, and Gene Expression, 2023, 24, 1422-0067, 15964, 10.3390/ijms242115964 | |
90. | Junying Yuan, Dimitry Ofengeim, A guide to cell death pathways, 2023, 1471-0072, 10.1038/s41580-023-00689-6 | |
91. | Agnieszka Gęgotek, Elżbieta Skrzydlewska, Lipid peroxidation products’ role in autophagy regulation, 2024, 212, 08915849, 375, 10.1016/j.freeradbiomed.2024.01.001 | |
92. | Liuling Xiao, Miao Xian, Chuanchao Zhang, Qi Guo, Qing Yi, Lipid peroxidation of immune cells in cancer, 2024, 14, 1664-3224, 10.3389/fimmu.2023.1322746 | |
93. | Oleg M. Panasenko, Yury A. Vladimirov, Valery I. Sergienko, Free Radical Lipid Peroxidation Induced by Reactive Halogen Species, 2024, 89, 0006-2979, S148, 10.1134/S0006297924140098 | |
94. | Sydney Bartman, Giuseppe Coppotelli, Jaime M. Ross, Mitochondrial Dysfunction: A Key Player in Brain Aging and Diseases, 2024, 46, 1467-3045, 1987, 10.3390/cimb46030130 | |
95. | Adnan Moinuddin, Sophie M. Poznanski, Ana L. Portillo, Jonathan K. Monteiro, Ali A. Ashkar, Metabolic adaptations determine whether natural killer cells fail or thrive within the tumor microenvironment, 2024, 0105-2896, 10.1111/imr.13316 | |
96. | Lindalva Maria de Meneses Costa Ferreira, Poliana Dimsan Queiroz de Souza, Rayanne Rocha Pereira, Edilene Oliveira da Silva, Wagner Luiz Ramos Barbosa, José Otávio Carréra Silva-Júnior, Attilio Converti, Roseane Maria Ribeiro-Costa, Preliminary Study on the Chemical and Biological Properties of Propolis Extract from Stingless Bees from the Northern Region of Brazil, 2024, 12, 2227-9717, 700, 10.3390/pr12040700 | |
97. | Francesca Pagliari, Jeannette Jansen, Jan Knoll, Rachel Hanley, Joao Seco, Luca Tirinato, Cancer radioresistance is characterized by a differential lipid droplet content along the cell cycle, 2024, 19, 1747-1028, 10.1186/s13008-024-00116-y | |
98. | Alessandro Vacchini, Andrew Chancellor, Qinmei Yang, Rodrigo Colombo, Julian Spagnuolo, Giuliano Berloffa, Daniel Joss, Ove Øyås, Chiara Lecchi, Giulia De Simone, Aisha Beshirova, Vladimir Nosi, José Pedro Loureiro, Aurelia Morabito, Corinne De Gregorio, Michael Pfeffer, Verena Schaefer, Gennaro Prota, Alfred Zippelius, Jörg Stelling, Daniel Häussinger, Laura Brunelli, Peter Villalta, Marco Lepore, Enrico Davoli, Silvia Balbo, Lucia Mori, Gennaro De Libero, Nucleobase adducts bind MR1 and stimulate MR1-restricted T cells, 2024, 9, 2470-9468, 10.1126/sciimmunol.adn0126 | |
99. | Chenyang Fan, Xiangdong Yang, Lixiang Yan, Zhexin Shi, Oxidative stress is two‐sided in the treatment of acute myeloid leukemia, 2024, 13, 2045-7634, 10.1002/cam4.6806 | |
100. | Jacques Dupuy, Edwin Fouché, Céline Noirot, Pierre Martin, Charline Buisson, Françoise Guéraud, Fabrice Pierre, Cécile Héliès-Toussaint, A dual model of normal vs isogenic Nrf2-depleted murine epithelial cells to explore oxidative stress involvement, 2024, 14, 2045-2322, 10.1038/s41598-024-60938-2 | |
101. | Priya Borah, Hemen Deka, Polycyclic aromatic hydrocarbon (PAH) accumulation in selected medicinal plants: a mini review, 2024, 1614-7499, 10.1007/s11356-024-33548-8 | |
102. | Adrian I. Abdo, Zlatko Kopecki, Comparing Redox and Intracellular Signalling Responses to Cold Plasma in Wound Healing and Cancer, 2024, 46, 1467-3045, 4885, 10.3390/cimb46050294 | |
103. | Dioni Arrieche, Andrés F. Olea, Carlos Jara-Gutiérrez, Joan Villena, Javier Pardo-Baeza, Sara García-Davis, Rafael Viteri, Lautaro Taborga, Héctor Carrasco, Ethanolic Extract from Fruits of Pintoa chilensis, a Chilean Extremophile Plant. Assessment of Antioxidant Activity and In Vitro Cytotoxicity, 2024, 13, 2223-7747, 1409, 10.3390/plants13101409 | |
104. | Feiyi Sun, Yuyang Chen, Kristy W. K. Lam, Wutong Du, Qingqing Liu, Fei Han, Dan Li, Jacky W. Y. Lam, Jianwei Sun, Ryan T. K. Kwok, Ben Zhong Tang, Glutathione‐responsive Aggregation‐induced Emission Photosensitizers for Enhanced Photodynamic Therapy of Lung Cancer, 2024, 1613-6810, 10.1002/smll.202401334 | |
105. | Adam Wroński, Iwona Jarocka-Karpowicz, Arkadiusz Surażyński, Agnieszka Gęgotek, Neven Zarkovic, Elżbieta Skrzydlewska, Modulation of Redox and Inflammatory Signaling in Human Skin Cells Using Phytocannabinoids Applied after UVA Irradiation: In Vitro Studies, 2024, 13, 2073-4409, 965, 10.3390/cells13110965 | |
106. | Palma Fedele, Anna Natalizia Santoro, Francesca Pini, Marcello Pellegrino, Giuseppe Polito, Maria Chiara De Luca, Antonietta Pignatelli, Michele Tancredi, Valeria Lagattolla, Alessandro Anglani, Chiara Guarini, Antonello Pinto, Pietro Bracciale, Immunonutrition, Metabolism, and Programmed Cell Death in Lung Cancer: Translating Bench to Bedside, 2024, 13, 2079-7737, 409, 10.3390/biology13060409 | |
107. | Ana Valenta Šobot, Dunja Drakulić, Ana Todorović, Marijana Janić, Ana Božović, Lidija Todorović, Jelena Filipović Tričković, Gentiopicroside and swertiamarin induce non-selective oxidative stress-mediated cytotoxic effects in human peripheral blood mononuclear cells, 2024, 00092797, 111103, 10.1016/j.cbi.2024.111103 | |
108. | Arno G. Siraki, Lars-Oliver Klotz, 2024, 9780128012383, 10.1016/B978-0-323-95488-4.00062-0 | |
109. | Ismahane Abdelaziz, Abdelkader Bounaama, Bahia Djerdjouri, Zine-Charaf Amir-Tidadini, Low-dose dimethylfumarate attenuates colitis-associated cancer in mice through M2 macrophage polarization and blocking oxidative stress, 2024, 0041008X, 117018, 10.1016/j.taap.2024.117018 | |
110. | Geou-Yarh Liou, Reauxqkwuanzyiia C’lay-Pettis, Sravankumar Kavuri, Involvement of Reactive Oxygen Species in Prostate Cancer and Its Disparity in African Descendants, 2024, 25, 1422-0067, 6665, 10.3390/ijms25126665 | |
111. | Katarzyna Matusik, Katarzyna Kamińska, Aleksandra Sobiborowicz-Sadowska, Hubert Borzuta, Kasper Buczma, Agnieszka Cudnoch-Jędrzejewska, The significance of the apelinergic system in doxorubicin-induced cardiotoxicity, 2024, 1573-7322, 10.1007/s10741-024-10414-w | |
112. | Hengyu Jin, Jianxin Liu, Diming Wang, Antioxidant Potential of Exosomes in Animal Nutrition, 2024, 13, 2076-3921, 964, 10.3390/antiox13080964 | |
113. | Farha Shahabuddin, Samina Naseem, Tauseef Alam, Aijaz Ahmed Khan, Farah Khan, Chronic aluminium chloride exposure induces redox imbalance, metabolic distress, DNA damage, and histopathologic alterations in Wistar rat liver, 2024, 0748-2337, 10.1177/07482337241269784 | |
114. | Jayachithra Ramakrishna Pillai, Adil Farooq Wali, Pooja Shivappa, Sirajunisa Talath, Sabry M. Attia, Ahmed Nadeem, Muneeb U. Rehman, Evaluating the anti-cancer potential and pharmacological in-sights of Physalis angulata Root Extract as a strong candidate for future research, 2024, 22, 1687157X, 100410, 10.1016/j.jgeb.2024.100410 | |
115. | Vilma Dembitz, Hannah Lawson, Richard Burt, Sirisha Natani, Céline Philippe, Sophie C. James, Samantha Atkinson, Jozef Durko, Lydia M. Wang, Joana Campos, Aoife M. S. Magee, Keith Woodley, Michael J. Austin, Ana Rio-Machin, Pedro Casado, Findlay Bewicke-Copley, Giovanny Rodriguez Blanco, Diego Pereira-Martins, Lieve Oudejans, Emeline Boet, Alex von Kriegsheim, Juerg Schwaller, Andrew J. Finch, Bela Patel, Jean-Emmanuel Sarry, Jerome Tamburini, Jan Jacob Schuringa, Lori Hazlehurst, John A. Copland III, Mariia Yuneva, Barrie Peck, Pedro Cutillas, Jude Fitzgibbon, Kevin Rouault-Pierre, Kamil Kranc, Paolo Gallipoli, Stearoyl-CoA desaturase inhibition is toxic to acute myeloid leukemia displaying high levels of the de novo fatty acid biosynthesis and desaturation, 2024, 0887-6924, 10.1038/s41375-024-02390-9 | |
116. | Sinemyiz Atalay Ekiner, Agnieszka Gęgotek, Elżbieta Skrzydlewska, Inflammasome activity regulation by PUFA metabolites, 2024, 15, 1664-3224, 10.3389/fimmu.2024.1452749 | |
117. | Giacomo G. Rossetti, Noëlle Dommann, Angeliki Karamichali, Vasilis S. Dionellis, Ainhoa Asensio Aldave, Tural Yarahmadov, Eddie Rodriguez-Carballo, Adrian Keogh, Daniel Candinas, Deborah Stroka, Thanos D. Halazonetis, In vivo DNA replication dynamics unveil aging-dependent replication stress, 2024, 00928674, 10.1016/j.cell.2024.08.034 | |
118. | Natalia Kurhaluk, Piotr Kamiński, Halina Tkaczenko, 2024, Chapter 425, 2731-4561, 10.1007/16833_2024_425 | |
119. | Jian Gao, Linjie Yuan, Huanyu Jiang, Ganggang Li, Yuwei Zhang, Ruijun Zhou, Wenjia Xian, Yutong Zou, Quanyu Du, Xianhua Zhou, Naringenin modulates oxidative stress and lipid metabolism: Insights from network pharmacology, mendelian randomization, and molecular docking, 2024, 15, 1663-9812, 10.3389/fphar.2024.1448308 | |
120. | Ibrahim S Topiwala, Aparna Ramachandran, Meghana Shakthi. A, Ranjini Sengupta, Rajib Dhar, Arikketh Devi, Exosomes and Tumor Virus Interlink: A Complex Side of Cancer, 2024, 03440338, 155747, 10.1016/j.prp.2024.155747 | |
121. | Andrew Chancellor, Daniel Constantin, Qinmei Yang, Vladimir Nosi, José Pedro Loureiro, Rodrigo Colombo, Roman P. Jakob, Daniel Joss, Michael Pfeffer, Giulia De Simone, Aurelia Morabito, Verena Schaefer, Alessandro Vacchini, Laura Brunelli, Daniela Montagna, Markus Heim, Alfred Zippelius, Enrico Davoli, Daniel Häussinger, Timm Maier, Lucia Mori, Gennaro De Libero, The carbonyl nucleobase adduct M3Ade is a potent antigen for adaptive polyclonal MR1-restricted T cells, 2024, 10747613, 10.1016/j.immuni.2024.11.019 | |
122. | Boleslaw T. Karwowski, The Adducts Lipid Peroxidation Products with 2′-DeoxyNucleosides: A Theoretical Approach of Ionisation Potential, 2025, 15, 2076-3417, 437, 10.3390/app15010437 | |
123. | Robert Andrew Brown, 2024, Chapter 8, 978-3-031-73060-3, 247, 10.1007/978-3-031-73061-0_8 | |
124. | Çağla Zübeyde Köprü, Burcu Baba, Dilek Yonar, Zerumbone Induces Apoptosis in Non‐Small‐Cell Lung Cancer via Biomolecular Alterations: A Microscopic and Spectroscopic Study, 2025, 1864-063X, 10.1002/jbio.202400500 | |
125. | ZhV Yavroyan, AG Hovhannisyan, NR Hakobyan, ES Gevorgyan, Effect of cisplatin on lipid peroxidation in the whole blood and plasma of female rats, 2025, 2453-6725, 10.2478/afpuc-2024-0014 | |
126. | Saeed Alshahrani, Mohammad Ashafaq, Abdulmajeed M. Jali, Yosif Almoshari, Mohammad Intakhab Alam, Hamad Al Shahi, Ayed A. Alshamrani, Sohail Hussain, Nephrotoxic effect of cypermethrin ameliorated by nanocurcumin through antioxidative mechanism, 2025, 0028-1298, 10.1007/s00210-025-03825-5 | |
127. | Sara Ilari, Stefania Proietti, Francesca Milani, Laura Vitiello, Carolina Muscoli, Patrizia Russo, Stefano Bonassi, Dietary Patterns, Oxidative Stress, and Early Inflammation: A Systematic Review and Meta-Analysis Comparing Mediterranean, Vegan, and Vegetarian Diets, 2025, 17, 2072-6643, 548, 10.3390/nu17030548 | |
128. | Chonnikarn Jirasit, Panida Navasumrit, Krittinee Chaisatra, Chalida Chompoobut, Somchamai Waraprasit, Varabhorn Parnlob, Mathuros Ruchirawat, Genotoxicity and fibrosis in human hepatocytes in vitro from exposure to low doses of PBDE-47, arsenic, or both chemicals, 2025, 00092797, 111410, 10.1016/j.cbi.2025.111410 | |
129. | Daniel A. Kasal, Viviane Sena, Grazielle Vilas Bôas Huguenin, Andrea De Lorenzo, Eduardo Tibirica, Microvascular endothelial dysfunction in vascular senescence and disease, 2025, 12, 2297-055X, 10.3389/fcvm.2025.1505516 | |
130. | Teresa Catalano, Federico Selvaggi, Roberto Cotellese, Gitana Maria Aceto, The Role of Reactive Oxygen Species in Colorectal Cancer Initiation and Progression: Perspectives on Theranostic Approaches, 2025, 17, 2072-6694, 752, 10.3390/cancers17050752 | |
131. | Nirmal Manhar, Sumeet Kumar Singh, Poonam Yadav, Manish Bishnolia, Amit Khurana, Jasvinder Singh Bhatti, Umashanker Navik, Methyl Donor Ameliorates CCl4‐Induced Nephrotoxicity by Inhibiting Oxidative Stress, Inflammation, and Fibrosis Through the Attenuation of Kidney Injury Molecule 1 and Neutrophil Gelatinase‐Associated Lipocalin Expression, 2025, 39, 1095-6670, 10.1002/jbt.70188 | |
132. | B. Perez-Montero, M. L. Fermin-Rodriguez, M. Portero-Fuentes, J. Sarquis, S. Caceres, J. C. Illera del Portal, L. de Juan, G. Miro, F. Cruz-Lopez, Malondialdehyde (MDA) and 8-hydroxy-2’-deoxyguanosine (8-OHdG) levels in canine serum: establishing reference intervals and influencing factors, 2025, 21, 1746-6148, 10.1186/s12917-025-04614-1 | |
133. | Sathiyan Niranjana, Anantha Udupa Prarthana, Aiswarya Ganapathisankarakrishnan, Dhakshinamoorthy Sundaramurthi, Vellingiri Vadivel, Comparative analysis of in vitro antioxidant and wound healing activities of Indian paalai plant extracts and investigation of their phytochemical profile by GC-MS, 2025, 7, 29501997, 100202, 10.1016/j.prenap.2025.100202 | |
134. | Fabián Delgado Rodríguez, Gabriela Azofeifa, Silvia Quesada, Nien Tzu Weng Huang, Arlene Loría Gutiérrez, María Fernanda Morales Rojas, Influence of Plant Part Selection and Drying Technique: Exploration and Optimization of Antioxidant and Antibacterial Activities of New Guinea Impatiens Extracts, 2025, 14, 2223-7747, 1092, 10.3390/plants14071092 | |
135. | Asha Ashraf, Bernd Zechmann, Erica D. Bruce, Hypoxia-inducible factor 1α modulates acrolein-induced cellular damage in bronchial epithelial cells, 2025, 515, 0300483X, 154158, 10.1016/j.tox.2025.154158 | |
136. | Taiyue Jin, Seulbi Lee, Juhee Seo, Shinhee Ye, Soontae Kim, Jin-Kyoung Oh, Seyoung Kim, Byungmi Kim, Long-term ambient ozone exposure and lung cancer mortality: a nested case-control study in Korea, 2025, 02697491, 126299, 10.1016/j.envpol.2025.126299 | |
137. | Gennaro Prota, Giuliano Berloffa, Wael Awad, Alessandro Vacchini, Andrew Chancellor, Verena Schaefer, Daniel Constantin, Dene R. Littler, Rodrigo Colombo, Vladimir Nosi, Lucia Mori, Jamie Rossjohn, Gennaro De Libero, Mitochondria regulate MR1 protein expression and produce self-metabolites that activate MR1-restricted T cells, 2025, 122, 0027-8424, 10.1073/pnas.2418525122 | |
138. | Max Temnik, Sergey Gurin, Alexandr Balakin, Roman Byshovets, Olesia Kalmukova, Tetiana Vovk, Tetiana Halenova, Nataliia Raksha, Tetyana Falalyeyeva, Olexiy Savchuk, Therapeutic potential of zinc 64Zn aspartate for obesity management: impact on oxidative stress, lipid metabolism, pancreas and liver in high-calorie diet model, 2025, 16, 1663-9812, 10.3389/fphar.2025.1543166 |