Research article Special Issues

A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay

  • Received: 20 July 2022 Revised: 31 July 2022 Accepted: 02 August 2022 Published: 05 September 2022
  • Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system's equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order δ with constant fractal-dimension ϖ, δ with changing ϖ, and δ with changing both δ and ϖ. White noise concentration has a significant impact on how bacterial infections are treated.

    Citation: Maysaa Al Qurashi, Saima Rashid, Fahd Jarad. A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 12950-12980. doi: 10.3934/mbe.2022605

    Related Papers:

    [1] Tariq Mahmood, Liaqat Ali, Muhammad Aslam, Ghulam Farid . On commutativity of quotient semirings through generalized derivations. AIMS Mathematics, 2023, 8(11): 25729-25739. doi: 10.3934/math.20231312
    [2] Liaqat Ali, Yaqoub Ahmed Khan, A. A. Mousa, S. Abdel-Khalek, Ghulam Farid . Some differential identities of MA-semirings with involution. AIMS Mathematics, 2021, 6(3): 2304-2314. doi: 10.3934/math.2021139
    [3] Saba Al-Kaseasbeh, Madeline Al Tahan, Bijan Davvaz, Mariam Hariri . Single valued neutrosophic (m,n)-ideals of ordered semirings. AIMS Mathematics, 2022, 7(1): 1211-1223. doi: 10.3934/math.2022071
    [4] Pakorn Palakawong na Ayutthaya, Bundit Pibaljommee . On n-ary ring congruences of n-ary semirings. AIMS Mathematics, 2022, 7(10): 18553-18564. doi: 10.3934/math.20221019
    [5] Abdelghani Taouti, Waheed Ahmad Khan . Fuzzy subnear-semirings and fuzzy soft subnear-semirings. AIMS Mathematics, 2021, 6(3): 2268-2286. doi: 10.3934/math.2021137
    [6] Rukhshanda Anjum, Saad Ullah, Yu-Ming Chu, Mohammad Munir, Nasreen Kausar, Seifedine Kadry . Characterizations of ordered h-regular semirings by ordered h-ideals. AIMS Mathematics, 2020, 5(6): 5768-5790. doi: 10.3934/math.2020370
    [7] Gurninder S. Sandhu, Deepak Kumar . A note on derivations and Jordan ideals of prime rings. AIMS Mathematics, 2017, 2(4): 580-585. doi: 10.3934/Math.2017.4.580
    [8] Gurninder S. Sandhu, Deepak Kumar . Correction: A note on derivations and Jordan ideals in prime rings. AIMS Mathematics, 2019, 4(3): 684-685. doi: 10.3934/math.2019.3.684
    [9] Faiza Shujat, Faarie Alharbi, Abu Zaid Ansari . Weak (p,q)-Jordan centralizer and derivation on rings and algebras. AIMS Mathematics, 2025, 10(4): 8322-8330. doi: 10.3934/math.2025383
    [10] Kaiqing Huang, Yizhi Chen, Miaomiao Ren . Additively orthodox semirings with special transversals. AIMS Mathematics, 2022, 7(3): 4153-4167. doi: 10.3934/math.2022230
  • Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system's equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order δ with constant fractal-dimension ϖ, δ with changing ϖ, and δ with changing both δ and ϖ. White noise concentration has a significant impact on how bacterial infections are treated.



    Semirings have significant applications in theory of automata, optimization theory, and in theoretical computer sciences (see [1,2,3]). A group of Russian mathematicians was able to create novel probability theory based on additive inverse semirings, called idempotent analysis (see[4,5]) having interesting applications in quantum physics. Javed et al. [6] identified a proper subclass of semirings known as MA-Semirings. The development of commutator identities and Lie type theory of semirings [6,7,8,9,10] and derivations [6,7,8,11,12] make this class quite interesting for researchers. To investigate commuting conditions for rings through certain differential identities and certain ideals are still interesting problems for researchers in ring theory (see for example [13,14,15,16,17,18,19]) and some of them are generalized in semirings (see [6,8,9,10,11,20]). In this paper we investigate commuting conditions of prime MA-semirings through certain differential identities and Jordan ideals (Theorems 2.5–2.8) and also study differential identities with the help of Jordan ideals (Theorem 2.3, Theorem 2.4, Theorem 2.10). In this connection we are able to generalize a few results of Oukhtite [21] in the setting of semirings. Now we present some necessary definitions and preliminaries which will be very useful for the sequel. By a semiring S, we mean a semiring with absorbing zero '0' in which addition is commutative. A semiring S is said to be additive inverse semiring if for each sS there is a unique sS such that s+s+s=s and s+s+s=s, where s denotes the pseudo inverse of s. An additive inverse semiring S is said to be an MA-semiring if it satisfies s+sZ(S),sS, where Z(S) is the center of S. The class of MA-semirings properly contains the class of distributive lattices and the class of rings, we refer [6,8,11,22] for examples. Throughout the paper by semiring S we mean an MA-semiring unless stated otherwise. A semiring S is prime if aSb={0} implies that a=0 or b=0 and semiprime if aSa={0} implies that a=0. S is 2-torsion free if for sS,2s=0 implies s=0. An additive mapping d:SS is a derivation if d(st)=d(s)t+sd(t). The commutator is defined as [s,t]=st+ts. By Jordan product, we mean st=st+ts for all s,tS. The notion of Jordan ideals was introduced by Herstein [23] in rings which is further extended canonically by Sara [20] for semirings. An additive subsemigroup G of S is called the Jordan ideal if sjG for all sS,jG. A mapping f:SS is commuting if [f(s),s]=0, sS. A mapping f:SS is centralizing if [[f(s),s],r]=0, s,rS. Next we include some well established identities of MA-semirings which will be very useful in the sequel. If s,t,zS and d is a derivation of S, then [s,st]=s[s,t], [st,z]=s[t,z]+[s,z]t, [s,tz]=[s,t]z+t[s,z], [s,t]+[t,s]=t(s+s)=s(t+t), (st)=st=st, [s,t]=[s,t]=[s,t], s(t+z)=st+sz, d(s)=(d(s)). To see more, we refer [6,7].

    From the literature we recall a few results of MA-semirings required to establish the main results.

    Lemma 1. [11] Let G be a Jordan ideal of an MA-semiring S. Then for all jG (a). 2[S,S]GG (b). 2G[S,S]G (c). 4j2SG (d). 4Sj2G (e). 4jSjG.

    Lemma 2. [11] Let S be a 2-torsion free prime MA-semiring and G a Jordan ideal of S. If aGb={0} then a=0 or b=0.

    In view of Lemma 1 and Lemma 2, we give some very useful remarks.

    Remark 1. [11]

    a). If r,s,tS,uG, then 2[r,st]uG.

    b). If aG={0} or Ga={0}, then a=0.

    Lemma 3. [12] Let G be a nonzero Jordan ideal and d be a derivation of a 2-torsion free prime MA-semiring S such that for all uG, d(u2)=0. Then d=0.

    Lemma 4. Let G be a nonzero Jordan ideal of a 2-torsion free prime MA-semiring S. If aS such that for all gG, [a,g2]=0. Then [a,s]=0,sS and hence aZ(S).

    Proof. Define a function da:SS by da(s)=[a,s], which is an inner derivation. As every inner derivation is derivation, therefore in view of hypothesis da is derivation satisfying da(g2)=[a,g2]=0,gG. By Lemma 3, da=0, which implies that da(s)=[a,s]=0, for all sS. Hence aZ(S).

    Lemma 5. Let S be a 2-torsion free prime MA-semiring and G a nonzero Jordan ideal of S. If S is noncommutative such that for all u,vG and rS

    a[r,uv]b=0, (2.1)

    then a=0 or b=0.

    Proof. In (2.1) replacing r by ar and using MA-semiring identities, we obtain

    aa[r,uv]b+a[a,uv]rb=0 (2.2)

    Using (2.1) again, we get a[a,uv]Sb=0. By the primeness of S, we have either b=0 or a[a,uv]=0. Suppose that

    a[a,uv]=0 (2.3)

    In view of Lemma 1, replacing v by 2v[s,t] in (2.3) and using 2-torsion freeness of S, we get 0=a[a,uv[s,t]]=auv[a,[s,t]]+a[a,uv][s,t]. Using (2.3) again auv[a,[s,t]]=0 and therefore auG[a,[s,t]]={0}. By the Lemma 2, we have either aG={0} or [a,[s,t]]=0. By Remark 1, aG={0} implies a=0. Suppose that

    [a,[s,t]]=0 (2.4)

    In (2.4) replacing s by sa, we get [a,s[a,t]]+[a,[s,t]a]=0 and therefore [a,s[a,t]]+[a,[s,t]]a=0. Using (2.4) again, we get [a,s][a,t]=0. By the primeness of S, [a,s]=0 and therefore aZ(S). Hence from (2.2), we can write aS[r,uv]b={0}. By the primeness of S, we obtain a=0 or

    [r,uv]b=0 (2.5)

    In (2.5) replacing r by rs and using (2.5) again, we get [r,uv]Sb={0}. By the primeness of S, we have either b=0 or [r,uv]=0. Suppose that

    [r,uv]=0 (2.6)

    In (2.6) replacing y by 2v[s,t] and using (2.6) again, we obtain 2[r,uv[s,t]]=0. As S is 2-torsion free, [r,uv[s,t]]=0 which further gives uG[r,[s,t]]={0}. As G{0}, by Lemma 2 [r,[s,t]]=0 which shows that S is commutative, a contradiction. Hence we conclude that a=0 or b=0.

    Theorem 1. Let S be a 2-torsion free prime MA-semiring and G a nonzero Jordan ideal of S. If d1 and d2 are derivations of S such that for all uG,

    d1d2(u)=0 (2.7)

    then either d1=0 or d2=0.

    Proof. Suppose that d20. We will show that d1=0. In view of Lemma 1, replacing u by 4u2v,vG in (2.7), we obtain d1d2(4u2v)=0 and by the 2-torsion freeness of S, we have d1d2(u2v)=0. Using (2.7) again, we obtain

    d2(u2)d1(v)+d1(u2)d2(v)=0 (2.8)

    By lemma 1, replacing v by 2[r,jk]v,j,kG in (2.8), we get

    d2(u2)d1(2[r,jk]v)+d1(u2)d2(2[r,jk]v)=0

    and

    2d2(u2)[r,jk]d1(v)+2d2(u2)d1([r,jk])v+2d1(u2)[r,jk]d2(v)+2d1(u2)d2([r,jk])v=0

    Using (2.8) again and hence by the 2-torsion freeness of S, we obtain

    d2(u2)[r,jk]d1(v)+d1(u2)[r,jk]d2(v)=0 (2.9)

    In (2.9), replacing v by 4v2t,tS and using (2.9) again, we obtain

    4d2(u2)[r,jk]v2d1(t)+4d1(u2)[r,jk]v2d2(t)=0

    As S is 2-torsion free, therefore

    d2(u2)[r,jk]v2d1(t)+d1(u2)[r,jk]v2d2(t)=0 (2.10)

    In (2.10), taking t=d2(g),gG and using (2.7), we obtain

    d1(u2)[r,jk]v2d2(d2(g))=0 (2.11)

    In (2.11) writing a for d1(u2) and b for v2d2(d2(g)), we have a[r,jk]b=0,rS,j,kG.

    Firstly suppose that S is not commutative. By Lemma 5, we have a=0 or b=0. If d1(u2)=a=0, then by Lemma 3, d1=0. Secondly suppose that S is commutative. In (2.7) replacing u by 2u2, we obtain 0=d1d2(2u2)=2d1d2(u2)=4d1(ud2(u))=4(d1(u)d2(u)+ud1d2(u)). Using (2.7) and the 2-torsion freeness of S, we obtain d1(u)d2(u)=0. By our assumption S is commutative, therefore d1(u)Sd2(u)={0}. By the primeness of S, we have either d1(G)={0} or d2(G)={0}. By Theorem 2.4 of [11], we have d1=0 or d2=0. But d20. Hence d1=0 which completes the proof.

    Theorem 2. Let S be a 2-torsion free prime MA-semiring and G a nonzero Jordan ideal of S. If d1 and d2 are derivations of S such that for all uG

    d1(d2(u)+u)=0, (2.12)

    then d1=0.

    Proof. Firstly suppose that S is commutative. Replacing u by 2u2 in (2.12) and using (2.12) again, we obtain d1(u)d2(u)=0 which further implies d1(u)Sd2(u)={0}. In view of Theorem 2.4 of [11], by the primeness of S we have d1=0 or d2=0. If d2=0, then from (2.12), we obtain d1(u)=0,uG and hence by Lemma 3, we conclude d1=0. Secondly suppose that S is noncommutative. Further suppose that d20. We will show that d1=0. In (2.12) replacing u by 4u2v,vG, and using (2.12) again, we obtain 2(d2(u2)d1(v)+d1(u2)d2(v))=0. As S is 2-torsion free, therefore

    d2(u2)d1(v)+d1(u2)d2(v)=0 (2.13)

    In (2.13) replacing v by 2[r,jk]v,rS,j,k,vG, we obtain

    d2(u2)d1(2[r,jk])v+2d2(u2)[r,jk]d1(v)+d1(u2)d2(2[r,jk])v+2d1(u2)[r,jk]d2(v)=0

    As by MA-semiring identities, 2[r,jk]=2j[r,k]+2[r,j]k, by Lemma 1 2[r,jk]G. Therefore using (2.13) again and the 2-torsion freeness of S, we obtain

    d2(u2)[r,jk]d1(v)+d1(u2)[r,jk]d2(v)=0 (2.14)

    In (2.14) replacing v by 4v2t,tS and using (2.14) again, we get

    d2(u2)[r,jk]v2d1(t)+d1(u2)[r,jk]v2d2(t)=0 (2.15)

    In (2.15) taking t=t(d2(w)+w),wG, we get

    d2(u2)[r,jk]v2d1(t(d2(w)+w))+d1(u2)[r,jk]v2d2(t(d2(w)+w))=0

    and therefore

    d2(u2)[r,jk]v2d1(t)(d2(w)+w)+d2(u2)[r,jk]v2td1((d2(w)+w))

    +d1(u2)[r,jk]v2d2(t)(d2(w)+w)+d1(u2)[r,jk]v2td2(d2(w)+w)=0

    Using (2.12) and (2.15) in the last expression, we obtain

    (d1(u2))[r,jk](v2td2(d2(w)+w))=0 (2.16)

    Applying Lemma 5 on (2.15), we get either d1(u2)=0 or v2td2(d2(w)+w)=0. If d1(u2)=0 then by Lemma 3, d1=0. If v2Sd2(d2(w)+w)={0}, the by the primeness of S, we have v2=0 or d2(d2(w)+w)=0. If v2=0,vG, then G={0}, a contradiction. Suppose that for all wG

    d2(d2(w)+w)=0 (2.17)

    In (2.17)replacing w by 4z2u,z,uG, and using (2.17) again, we obtain

    d2(z2)d2(u)=0 (2.18)

    In (2.18), replacing u by 4xz2,xG and using (2.18) again, we obtain d2(z2)Gd2(z2)={0}. By Lemma 2, d2(z2)=0 and hence by Lemma 3, we conclude that d2=0. Taking d2=0 in the hypothesis to obtain d1(u)=0 and hence by Theorem 2.4 of [11], we have d1=0.

    Theorem 3. Let G be a nonzero Jordan ideal of a 2-torsion free prime MA-semiring S and d1 and d2 be derivations of S such that for all u,vG

    [d1(u),d2(v)]+[u,v]=0 (2.19)

    Then S is commutative.

    Proof. If d1=0 or d2=0, then from (2.19), we obtain [G,G]={0}. By Theorem 2.3 of [11] S is commutative. We assume that both d1 and d2 are nonzero. In (2.19) replacing u by 4uw2 and using MA-semiring identities and 2-torsion freeness of S, we get

    d1(u)[2w2,d2(v)]+([d1(u),d2(v)]+[u,v])2w2+u([d1(2w2),d2(v)]

    +[2w2,v])+[u,d2(v)]d1(2w2)=0

    Using (2.19) again, we get

    d1(u)[2w2,d2(v)]+[u,d2(v)]d1(2w2)=0

    and by the 2-torsion freeness of S, we have

    d1(u)[w2,d2(v)]+[u,d2(v)]d1(w2)=0 (2.20)

    Replacing u by 2u[r,jk] in (2.20) and using it again, we obtain

    d1(u)[r,jk][w2,d2(v)]+[u,d2(v)][r,jk]d1(w2)=0 (2.21)

    In (2.21) replacing u by 4su2 and using (2.21) again, we obtain

    d1(s)u2[r,jk][w2,d2(v)]+[s,d2(v)]u2[r,jk]d1(w2)=0 (2.22)

    In (2.22) replacing s by d2(v)s and then using commutator identities, we get

    d1d2(v)su2[r,jk][w2,d2(v)]=0 (2.23)

    Therefore d1d2(v)Su2[r,jk][w2,d2(v)]={0}. By the primeness of S, we obtain either d1d2(v)=0 or u2[r,jk][w2,d2(v)]=0. Consider the sets

    G1={vG:d1d2(v)=0}

    and

    G2={vG:u2[r,jk][w2,d2(v)=0}

    We observe that G=G1G2. We will show that either G=G1 or G=G2. Suppose that v1G1G2 and v2G2G1. Then v1+v2G1+G2G1G2=G. We now see that 0=d1d2(v1+v2)=d1d2(v2), which shows that v2G1, a contradiction. On the other hand 0=u2[r,jk][w2,d2(v1+v2)]=u2[r,jk][w2,d2(v1)], which shows that v1G2, a contradiction. Therefore either G1G2 or G2G1, which respectively show that either G=G1 or G=G2. Therefore we conclude that for all vG, d1d2(v)=0 or u2[r,jk][w2,d2(v)]=0. Suppose that d1d2(v)=0,vG. then by Lemma 2.1, d1=0 or d2=0. Secondly suppose that u2[r,jk][w2,d2(v)]=0,u,v,w,j,kG,rS. By Lemma 5, we have either u2=0 or [w2,d2(v)]=0. But u2=0 leads to G={0} which is not possible. Therefore [w2,d2(v)]=0 and employing Lemma 4, [d2(v),s]=0,sS. Hence by Theorem 2.2 of [22], S is commutative.

    Theorem 4. Let G be a nonzero Jordan ideal of a 2-torsion free prime MA-semiring S and d1 and d2 be derivations of S such that for all u,vG

    d1(u)d2(v)+[u,v]=0 (2.24)

    Then d1=0 or d2=0 and thus S is commutative.

    Proof. It is quite clear that if at least one of d1 and d2 is zero, then we obtain [G,G]={0}. By Theorem 2.3 of [11] and Theorem 2.1 of [22], S is commutative. So we only show that at least one of the derivations is zero. Suppose that d20. In (2.24), replacing v by 4vw2,wG, we obtain d1(u)d2(4vw2)+[u,4vw2]=0 and therefore using MA-semirings identities, we can write

    4d1(u)vd2(w2)+4d1(u)d2(v)w2+4v[u,w2]+4[u,v]w2=0

    In view of Lemma 1 as 2w2G, using (2.24) and the 2-torsion freeness of S, we obtain

    d1(u)vd2(w2)+v[u,w2]=0 (2.25)

    In (2.25) replacing v by 2[s,t]v, s,tS and hence by the 2-torsion freeness of S, we get

    d1(u)[s,t]vd2(w2)+[s,t]v[u,w2]=0 (2.26)

    Multiplying (2.25) by [s,t] from the left, we get

    [s,t]d1(u)vd2(w2)+[s,t]v[u,w2]=0

    and since S is an MA-semiring, therefore

    [s,t]d1(u)vd2(w2)=[s,t]v[u,w2] (2.27)

    Using (2.27) into (2.26), we obtain d1(u)[s,t]vd2(w2)+[s,t]d1(u)vd2(w2)=0. By MA-semirings identities, we further obtain [d1(u),[s,t]]Gd2(w2)={0}. By Lemma 2, we obtain either [d1(u),[s,t]]=0 or d2(w2)=0. If d2(w2)=0, then by Lemma 3, d2=0. On the other hand, if

    [d1(u),[s,t]]=0 (2.28)

    In (2.28) replacing t by st, we get [d1(u),s[s,t]]=0 and using (2.23) again [d1(u),s][s,t]=0 and therefore [d1(u),s]S[s,t]={0} and by the primeness of S, we get [S,S]={0} and hence S is commutative or [d1(u),s]=0. In view of Theorem 2.2 of [22] from [d1(u),s]=0 we have [S,S]={0} which further implies S is commutative. Hence (2.19)becomes d1(u)d2(v)=0. As above we have either d1=0 or d2=0.

    Theorem 5. Let S be a 2-torsion free prime MA-semiring and G a nonzero Jordan ideal of S. If d1, d2 and d3 be nonzero. derivations such that for all u,vG either

    1). d3(v)d1(u)+d2(u)d3(v)=0 or

    2). d3(v)d1(u)+d2(u)d3(v)+[u,v]=0.

    Then S is commutative and d1=d2.

    Proof. 1). By the hypothesis for the first part, we have

    d3(v)d1(u)+d2(u)d3(v)=0 (2.29)

    which further implies

    d3(v)d1(u)=d2(u)d3(v) (2.30)

    In (2.29) replacing u by 4uw2, we obtain

    4d3(v)d1(u)w2+4d3(v)ud1(w2)+4d2(u)w2d3(v)+4ud2(w2)d3(v)=0

    and therefore by the 2-torsion freeness of S, we have

    d3(v)d1(u)w2+d3(v)ud1(w2)+d2(u)w2d3(v)+ud2(w2)d3(v)=0 (2.31)

    Using (2.30) into (2.31), we obtain

    d2(u)[d3(v),w2]+[d3(v),u]d1(w2)=0 (2.32)

    In (2.32) replacing u by 2u[r,jk],rS,j,kG, and using (2.32) again, we get

    d2(u)[r,jk][d3(v),w2]+[d3(v),u][r,jk]d1(w2)=0 (2.33)

    In (2.33) replacing u by 4tu2,tS and using 2-torsion freeness and (2.33) again, we get

    d2(t)u2[r,jk][d3(v),w2]+[d3(v),t]u2[r,jk]d1(w2)=0 (2.34)

    Taking t=d3(v)t in (2.34) and using (2.34) again we obtain

    d2d3(v)tu2[r,jk][d3(v),w2]=0 (2.35)

    We see that equation (2.35) is similar as (2.23) of the previous theorem, therefore repeating the same process we obtain the required result.

    2). By the hypothesis, we have

    d3(v)d1(u)+d2(u)d3(v)+[u,v]=0 (2.36)

    For d3=0, we obtain [G,G]={0} and by Theorem 2.3 of [11] this proves that S is commutative. Assume that d30. From (2.36), using MA-semiring identities, we can write

    d3(v)d1(u)=d2(u)d3(v)+[u,v] (2.37)

    and

    d3(v)d1(u)+[u,v]=d2(u)d3(v) (2.38)

    In (2.36), replacing u by 4uz2, we obtain

    4(d3(v)ud1(z2)+d3(v)d1(u)z2+d2(u)z2d3(v)+ud2(z2)d3(v)+u[z2,v])+[u,v]z2)=0

    and using (2.37) and (2.38) and then 2-torsion freeness of S, we obtain

    [d3(v),u]d1(z2)+d2(u)[d3(v),z2]=0 (2.39)

    We see that (2.39) is same as (2.32) of the previous part of this result. This proves that [S,S]={0} and hence S is commutative. Also then by the hypothesis, since d30, d1=d2.

    Theorem 6. Let G be nonzero Jordan ideal of a 2-torsion free prime MA-semiring S and d1 and d2 be nonzero derivations of S such that for all u,vG

    [d2(v),d1(u)]+d1[v,u]=0 (2.40)

    Then S is commutative.

    In (2.40) replacing u by 4uw2,wG and using 2-torsion freeness and again using(2.40), we obtain

    [d2(v)+v,u]d1(w2)+d1(u)[d2(v)+v,w2]=0 (2.41)

    In (2.41) replacing u by 2u[r,jk],j,kG,rS, we obtain

    u[d2(v)+v,2[r,jk]]d1(w2)+2[d2(v)+v,u][r,jk]d1(w2)

    +ud1(2[r,jk])[d2(v)+v,w2]+2d1(u)[r,jk][d2(v)+v,w2]=0

    Using 2-torsion freeness and (2.41) again, we get

    [d2(v)+v,u][r,jk]d1(w2)+d1(u)[r,jk][d2(v)+v,w2]=0 (2.42)

    In(2.42) replacing u by 4tu2,tSand using (2.42) again, we get

    [d2(v)+v,t]u2[r,jk]d1(w2)+d1(t)u2[r,jk][d2(v)+v,w2]=0 (2.43)

    In (2.43) taking t=(d2(v)+v)t and using MA-semirings identities, we obtain

    (d2(v)+v)[d2(v)+v,t]u2[r,jk]d1(w2)+d1(d2(v)+v)tu2[r,jk][d2(v)+v,w2]

    +(d2(v)+v)d1(t)u2[r,jk][d2(v)+v,w2]=0

    and using (2.43) again, we obtain

    d1(d2(v)+v)tu2[r,jk][d2(v)+v,w2]=0 (2.44)

    By the primeness we obtain either d1(d2(v)+v)=0 or u2[r,jk][d2(v)+v,w2]=0. If d1(d2(v)+v)=0, then by Theorem 2 we have d1=0, which contradicts the hypothesis. Therefore we must suppose u2[r,jk][d2(v)+v,w2]=0. By Lemma 5, we have either u2=0 or [d2(v)+v,w2]=0. But u2=0 implies G={0} which is not possible. On the other hand applying Lemma 5, we have [d2(v)+v,r]=0,rS and therefore taking r=v,vG and [d2(v),v]+[v,v]=0 and using MA-semiring identities, we get

    [d2(v),v]+[v,v]=0 (2.45)

    As [v,v]=[v,v], from (2.45), we obtain [d2(v),v]+[v,v]=0 and therefore

    [d2(v),v]=[v,v] (2.46)

    Using (2.46) into (2.45), we get 2[d2(v),v]=0 and by the 2-torsion freeness of S, we get [d2(v),v]=0. By Theorem 2.2 of [22], we conclude that S is commutative.

    Corollary 1. Let G be nonzero Jordan ideal of a 2-torsion free prime MA-semiring S and d be a nonzero derivation of S such that for all u,vG d[v, u] = 0. Then S is commutative

    Proof. In theorem (6) taking d2=0 and d1=d, we get the required result.

    Theorem 7. Let G be a nonzero Jordan ideal of a 2-torsion free prime MA-semiring and d2 be derivation of S. Then there exists no nonzero derivation d1 such that for all u,vG

    d2(v)d1(u)+d1(vu)=0 (2.47)

    Proof. Suppose on the contrary that there is a nonzero derivation d1 satisfying (2.47). In (2.47) replacing u by 4uw2,wG and using (2.47) again, we obtain

    d1(u)[w2,d2(v)+v]+[u,d2(v)]d1(w2)+ud1(vw2)+(uv)d1(w2)+ud1[v,w2]=0 (2.48)

    In (2.48), replacing u by u[r,jk],rS,j,kG and using (2.48) again, we get

    d1(u)[r,jk][w2,d2(v)+v]+[u,d2(v)+v][r,jk]d1(w2)=0 (2.49)

    In (2.49) replacing u by 4tu2,tS and using (2.49) again, we obtain

    d1(t)u2[r,jk][w2,d2(v)+v]+td1(u2)[r,jk][w2,d2(v)+v]

    +t[u2,d2(v)+v][r,jk]d1(w2)+[t,d2(v)+v]u2[r,jk]d1(w2)=0

    and using2-torsion freeness and (2.49) again, we obtain

    d1(t)u2[r,jk][w2,d2(v)+v]+[t,d2(v)+v]u2[r,jk]d1(w2)=0 (2.50)

    In (2.50) taking t=(d2(v)+v)t and using MA-semirings identities, we get d1(d2(v)+v)tu2[r,jk][w2,d2(v)+v]+(d2(v)+v)d1(t)u2[r,jk][w2,d2(v)+v]

    +(d2(v)+v)[t,d2(v)+v]u2[r,jk]d1(w2)=0

    Using (2.50) again, we obtain

    d1(d2(v)+v)tu2[r,jk][w2,d2(v)+v]=0 (2.51)

    that is d1(d2(v)+v)Su2[r,jk][w2,d2(v)+v]=0. Therefore by the primeness following the same process as above, we have either d1(d2(v)+v)=0 or u2[r,jk][w2,d2(v)+v]=0 for all u,v,j,k,wG,rS. If d1(d2(v)+v)=0. As d10, therefore d2(v)+v=0. Secondly suppose that u2[r,jk][w2,d2(v)+v]=0. By Lemma 5, we have either u2=0 or [w2,d2(v)+v]=0. But u2=0 implies that G={0}, a contradiction. Therefore we consider the case when [w2,d2(v)+v]=0, which implies, by Lemma 4, that [d2(v)+v,r]=0,rS and taking in particular t=vG, we have

    [d2(v),v]+[v,v]=0 (2.52)

    Also by definition of MA-semirings, we have [v,v]=[v,v]. Therefore [d2(v),v]+[v,v]=0 and therefore

    [d2(v),v]=[v,v] (2.53)

    Using (2.53) into (2.52) and then using 2-torsion freeness of S, we obtain [d(v),v]=0. By Theorem 2.2 of [22], we conclude that S is commutative. Therefore (2.47) will be rewritten as 2d1(u)d2(v)+2(d1(v)u+vd1(u))=0 and hence by the 2-torsion freeness of S, we obtain

    d1(u)d2(v)+d1(v)u+vd1(u)=0 (2.54)

    In (2.54) replacing u by 2uw and using 2-torsion freeness of S, we get

    d1(u)wd2(v)+ud1(w)d2(v)+d1(v)uw+vd1(u)w+vud1(w)=0

    and therefore

    w(d1(u)d2(v)+d1(v)u+vd1(u))+ud1(w)d2(v)+vud1(w)=0

    Using (2.54) again, we obtain

    ud1(w)d2(v)+vud1(w)=0 (2.55)

    In (2.55) replacing v by 2vz, we get

    ud1(w)d2(v)z+ud1(w)vd2(z)+vzud1(w)=0

    and therefore

    z(ud1(w)d2(v)+vud1(w))+ud1(w)vd2(z)=0

    and using (2.55) again, we get d1(w)uGd2(z)={0}. By the above Lemma 2, we have either d1(w)u=0 or d2(z)=0 and therefore by Remark 1, we have either d1(w)=0 or d2(z)=0. As d10, therefore d2=0. Therefore our hypothesis becomes d1(uv)=0 and therefore d1(u2)=0, uG. By Lemma 3, d1=0 a contraction to the assumption. Hence d1 is zero.

    We have proved the results of this paper for prime semirings and it would be interesting to generalize them for semiprime semirings, we leave it as an open problem.

    Taif University Researchers Supporting Project number (TURSP-2020/154), Taif University Taif, Saudi Arabia.

    The authors declare that they have no conflict of interest.



    [1] S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol., 247 (2007), 23–35. https://doi.org/10.1016/j.jtbi.2007.02.017 doi: 10.1016/j.jtbi.2007.02.017
    [2] R. M. Ribeiro, A. Lo, A. S. Perelson, Dynamics of hepatitis B virus infection, Microb. Infect., 4 (2002), 829–835. https://doi.org/10.1016/S1286-4579(02)01603-9 doi: 10.1016/S1286-4579(02)01603-9
    [3] D. H. Kim, H. S. Kang, K.-H. Kim, Roles of hepatocyte nuclear factors in hepatitis B virus infection, World J Gastroenterol., 22 (2016), 7017–7029. https://doi.org/10.3748/wjg.v22.i31.7017 doi: 10.3748/wjg.v22.i31.7017
    [4] I. S. Oh, S. H. Park, Immune-mediated liver injury in hepatitis B virus infection, Immun. Netw, 15 (2015), 191. https://doi.org/10.4110/in.2015.15.4.191 doi: 10.4110/in.2015.15.4.191
    [5] C. A. Janeway, J. P. Travers, M. Walport, M. J. Sholmchik, Immunobiology: The Immune System in Health and Disease 5th edition, New York, Garland Science, 2001.
    [6] R. Kapoor, S. Kottilil, Strategies to eliminate HBV infection, Future Virol., 9 (2014). https://doi.org/10.2217/fvl.14.36 doi: 10.2217/fvl.14.36
    [7] K. Hattaf, N. Yousfi, A generalized HBV model with diffusion and two delays, Comput. Math. Appl. 69 (2015), 31–40. https://doi.org/10.1016/j.camwa.2014.11.010 doi: 10.1016/j.camwa.2014.11.010
    [8] K. Manna, S. P. Chakrabarty, Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids, Comput. Appl. Math, 36 (2017), 525–536. https://doi.org/10.1007/s40314-015-0242-3 doi: 10.1007/s40314-015-0242-3
    [9] T. Luzyanina, G. Bocharov, Stochastic modeling of the impact of random forcing on persistent hepatitis B virus infection, Math. Comput. Simul., 96 (2014), 54–65. https://doi.org/10.3934/mbe.2021034 doi: 10.3934/mbe.2021034
    [10] X. Wang, Y. Tan, Y. Cai, K. Wang, W. Wang, Dynamics of a stochastic HBV infection model with cell-to-cell transmission and immune response, Math. Biosci. Eng., 18 (2021), 616–642. https://doi.org/10.3934/mbe.2021034 doi: 10.3934/mbe.2021034
    [11] C. Ji, The stationary distribution of hepatitis B virus with stochastic perturbation, Appl. Math. Lett., 100 (2020), 106017. https://doi.org/10.1016/j.aml.2019.106017 doi: 10.1016/j.aml.2019.106017
    [12] D. Kiouach, Y. Sabbar, Ergodic stationary distribution of a stochastic hepatitis B epidemic model with interval-valued parameters and compensated poisson process, Comput. Math. Meth. Med., 2020 (2020). https://doi.org/10.1155/2020/9676501 doi: 10.1155/2020/9676501
    [13] H. Hui, L. F. Nie, Analysis of a stochastic HBV infection model with nonlinear incidence rate, J. Bio. Syst., 27 (2019), 399–421. https://doi.org/10.1142/S0218339019500177 doi: 10.1142/S0218339019500177
    [14] C. Ji, The stationary distribution of hepatitis B virus with stochastic perturbation, Appl. Math. Lett., 100 (2020), 106017. https://doi.org/10.1016/j.aml.2019.106017 doi: 10.1016/j.aml.2019.106017
    [15] Y. Wang, K. Qi, D. Jiang, An HIV latent infection model with cell-to-cell transmission and stochastic perturbation, Chaos Soliton. Fract., 151 (2021), 111215. https://doi.org/10.1016/j.chaos.2021.111215 doi: 10.1016/j.chaos.2021.111215
    [16] A. Din, Y. Li, A. Yusuf, Delayed hepatitis B epidemic model with stochastic analysis, Chaos Soliton. Fract., 146 (2021), 110839. https://doi.org/10.1016/j.chaos.2021.110839 doi: 10.1016/j.chaos.2021.110839
    [17] J.Sun, M. Gao, D. Jiang, Threshold dynamics of a Non-linear stochastic viral model with Time Delay and CTL responsiveness, Life, 11 (2021), 766. https://doi.org/10.3390/life11080766 doi: 10.3390/life11080766
    [18] F. A. Rihan, H. J. Alsakaji, Analysis of a stochastic HBV infection model with delayed immune response, Math. Biosci. Eng, 18 (2021), 5194–5220. https://doi.org/10.3934/mbe.2021264 doi: 10.3934/mbe.2021264
    [19] T.-H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, Y.-M. Chu, A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math, 20 (2021), 160–176.
    [20] S.-W Yao, M. Farman, M. Amin, M. Inc, A. Akgül, A. Ahmad, Fractional order COVID-19 model with transmission rout infected through environment, AIMS Math., 7 (2022), 5156–5174. https://doi.org/10.3934/math.2022288 doi: 10.3934/math.2022288
    [21] Z. Ul. A. Zafar, H. Rezazadeh, M. Inc, K. S. Nisar, T. A. Sulaiman, et al., Fractional order heroin epidemic dynamics, Alexandria Eng. J., 60 (2021), 5157–5165. https://doi.org/10.1016/j.aej.2021.04.039 doi: 10.1016/j.aej.2021.04.039
    [22] I. Podlubny, Fractional differential equations, San Diego: Academic Press, (1999).
    [23] T.-H. Zhao, M. Ijaz Khan, Y.-M. Chu, Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks, Math. Methods Appl. Sci., (2021). https://doi.org/10.1002/mma.7310 doi: 10.1002/mma.7310
    [24] K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam, Y.-M. Chu, Almost sectorial operators on Ψ-Hilfer derivative fractional impulsive integro-differential equations, Math. Methods Appl. Sci, (2021). https://doi.org/10.1002/mma.7954 doi: 10.1002/mma.7954
    [25] S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y.-M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), Article ID 2240026. https://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
    [26] S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y.-M. Chu, On multi-step methods for singular fractional q-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
    [27] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl., 2 (2015), 73–85.
    [28] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 396 (2017), 102. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
    [29] A. Atangana, S. Jain, A new numerical approximation of the fractal ordinary differential equation, Eur. Phys. J. Plus., 133 (2018), 37. https://doi.org/10.1140/epjp/i2018-11895-1 doi: 10.1140/epjp/i2018-11895-1
    [30] F. Jin, Z.-S. Qian, Y.-M. Chu, M. ur Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. https://doi.org/10.11948/20210357 doi: 10.11948/20210357
    [31] S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y.-M. Chu, Some recent developments on dynamical -discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels, Fractals, 30 (2022), Article ID 2240110. https://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
    [32] F.-Z. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y.-M. Chu, Numerical solution of traveling waves in chemical kinetics: time-fractional fishers equations, Fractals, 30 (2022), Article ID 2240051. https://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
    [33] S. Rashid, R. Ashraf, F. Jarad, Strong interaction of Jafari decomposition method with nonlinear fractional-order partial differential equations arising in plasma via the singular and nonsingular kernels, AIMS Math., 7 (2022), 7936–7963. https://doi.org/10.3934/math.2022444 doi: 10.3934/math.2022444
    [34] S. Rashid, F. Jarad, A. G. Ahmad, K. M. Abualnaja, New numerical dynamics of the heroin epidemic model using a fractional derivative with Mittag-Leffler kernel and consequences for control mechanisms, Results Phy., 35 (2022). https://doi.org/10.1016/j.rinp.2022.105304 doi: 10.1016/j.rinp.2022.105304
    [35] S. Rashid, E. I. Abouelmagd, S. Sultana, Y.-M. Chu, New developments in weighted n-fold type inequalities via discrete generalized ˆh-proportional fractional operators, Fractals, 30 (2022), Article ID 2240056. https://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
    [36] S. A. Iqbal, M. G. Hafez, Y.-M. Chu, C. Park, Dynamical Analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative, J. Appl. Anal. Comput., 12 (2022), 770–789. https://doi.org/10.11948/20210324 doi: 10.11948/20210324
    [37] A. N. Shiryaev, Essentials of Stochastic Finance, Facts, Models and Theory. World Scientific, Singapore, (1999). https://doi.org/10.1142/3907
    [38] K. X. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582–1591. https://doi.org/10.1002/mma.3169 doi: 10.1002/mma.3169
    [39] M. Kerboua, A. Debbouche, D. Baleanu, Approximate controllability of Sobolev-type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), Article ID 262191. https://doi.org/10.1155/2013/262191 doi: 10.1155/2013/262191
    [40] B. Pei, Y. Xu, On the non-Lipschitz stochastic differntial equations driven by fractional Brownian motion, Adv. Differ. Equ., 2016 (2016), 194. https://doi.org/10.1186/s13662-016-0916-1 doi: 10.1186/s13662-016-0916-1
    [41] A. Atangana, S. I. Araz, Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe, Adv. Differ. Eqs., 2021 (2021), 1–107. https://doi.org/10.1186/s13662-021-03213-2 doi: 10.1186/s13662-021-03213-2
    [42] B. S. T. Alkahtani, I. Koca, Fractional stochastic SIR model, Results Phy., 24 (2021), 104124. https://doi.org/10.1016/j.rinp.2021.104124 doi: 10.1016/j.rinp.2021.104124
    [43] S. Rashid, M. K. Iqbal, A. M. Alshehri, R. Ahraf, F. Jarad, A comprehensive analysis of the stochastic fractal-fractional tuberculosis model via Mittag-Leffler kernel and white noise, Results Phy., 39 (2022), 105764. https://doi.org/10.1016/j.rinp.2022.105764 doi: 10.1016/j.rinp.2022.105764
    [44] X. Zhang, H. Peng, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020). https://doi.org/10.1016/j.aml.2019.106095 doi: 10.1016/j.aml.2019.106095
    [45] F. A. Rihan, H. J. Alsakaji, C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 1–20. https://doi.org/10.1186/s13662-019-2438-0 doi: 10.1186/s13662-019-2438-0
    [46] Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8–15. https://doi.org/10.1016/j.aml.2017.04.021 doi: 10.1016/j.aml.2017.04.021
    [47] O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface., 7 (2010), 873–885. https://doi.org/10.1098/rsif.2009.0386 doi: 10.1098/rsif.2009.0386
    [48] A. Atangana, Mathematical model of survival of fractional calculus, critics and their impact: How singular is our world? Adv. Diff. Equ., 2021 (2021), 403. https://doi.org/10.1186/s13662-021-03494-7 doi: 10.1186/s13662-021-03494-7
    [49] P. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
    [50] P. Baldi, L. Mazliak, P. Priouret, Pierre, Martingales and Markov Chains, Chapman and Hall., ISBN 978-1-584-88329-6, (1991).
    [51] D. Wodarz, J. P. Christensen, A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends Immunol., 23 (2002), 194–200. https://doi.org/10.1016/S1471-4906(02)02189-0 doi: 10.1016/S1471-4906(02)02189-0
    [52] B. Berrhazi, M. E. Fatini, T. G. Caraballo, R. Pettersson, A stochastic SIRI epidemic model with levy noise, Discret. Contin. Dyn. Syst. Ser. B., 23 (2018), 3645–3661.
    [53] K. Hattaf, A new generalized definition of fractional derivative with non-singular kernel, Computation, 8 (2020), 4. https://doi.org/10.3390/computation8020049 doi: 10.3390/computation8020049
    [54] J. M. Heffernan, R. J. Smith, L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interf., 2 (2005), 281–293. https://doi.org/10.1098/rsif.2005.0042 doi: 10.1098/rsif.2005.0042
    [55] H. Dahari, A. Lo, R. M. Ribeiro, A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371–381. https://doi.org/10.1016/j.jtbi.2007.03.006 doi: 10.1016/j.jtbi.2007.03.006
    [56] J. Reyes-Silveyra, A. R. Mikler, Modeling immune response and its effect on infectious disease outbreak dynamics, Theor. Biol. Med. Model., 13 (2016), 1–21. https://doi.org/10.1186/s12976-016-0033-6 doi: 10.1186/s12976-016-0033-6
    [57] D. Wodarz, Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743–1750. https://doi.org/10.1099/vir.0.19118-0 doi: 10.1099/vir.0.19118-0
  • This article has been cited by:

    1. Tariq Mahmood, Liaqat Ali, Muhammad Aslam, Ghulam Farid, On commutativity of quotient semirings through generalized derivations, 2023, 8, 2473-6988, 25729, 10.3934/math.20231312
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2242) PDF downloads(80) Cited by(10)

Figures and Tables

Figures(13)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog