
The present work is to design a novel Neuro swarm computing standards using artificial intelligence scheme to exploit the Gudermannian neural networks (GNN)accomplished with global and local search ability of particle swarm optimization (PSO) and sequential quadratic programming scheme (SQPS), called as GNN-PSO-SQPS to solve a class of the second order Lane-Emden singular nonlinear model (SO-LES-NM). The suggested intelligent computing solver GNN-PSO-SQPS using the Gudermannian kernel are unified with the configuration of the hidden layers of GNN of differential operators for solving the SO-LES-NM. An error based fitness function (FF) applying the differential form of the differential model and corresponding boundary conditions. The FF is optimized together with the combined heuristics of PSO-SQPS. Three problems of the SO-LES-NM are solved to validate the correctness, effectiveness and competence of the designed GNN-PSO-SQPS. The performance of the GNN-PSO-SQPS through statistical operators is tested to check the constancy, convergence and precision.
Citation: Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Adnène Arbi, Gilder Cieza Altamirano, Jinde Cao. Neuro-swarms intelligent computing using Gudermannian kernel for solving a class of second order Lane-Emden singular nonlinear model[J]. AIMS Mathematics, 2021, 6(3): 2468-2485. doi: 10.3934/math.2021150
[1] | Abel Cabrera Martínez, Iztok Peterin, Ismael G. Yero . Roman domination in direct product graphs and rooted product graphs. AIMS Mathematics, 2021, 6(10): 11084-11096. doi: 10.3934/math.2021643 |
[2] | Fu-Tao Hu, Xing Wei Wang, Ning Li . Characterization of trees with Roman bondage number 1. AIMS Mathematics, 2020, 5(6): 6183-6188. doi: 10.3934/math.2020397 |
[3] | Rangel Hernández-Ortiz, Luis Pedro Montejano, Juan Alberto Rodríguez-Velázquez . Weak Roman domination in rooted product graphs. AIMS Mathematics, 2021, 6(4): 3641-3653. doi: 10.3934/math.2021217 |
[4] | Mingyu Zhang, Junxia Zhang . On Roman balanced domination of graphs. AIMS Mathematics, 2024, 9(12): 36001-36011. doi: 10.3934/math.20241707 |
[5] | Jian Yang, Yuefen Chen, Zhiqiang Li . Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1. AIMS Mathematics, 2023, 8(8): 17702-17718. doi: 10.3934/math.2023904 |
[6] | Saeed Kosari, Yongsheng Rao, Zehui Shao, Jafar Amjadi, Rana Khoeilar . Complexity of signed total k-Roman domination problem in graphs. AIMS Mathematics, 2021, 6(1): 952-961. doi: 10.3934/math.2021057 |
[7] | Zhibin Du, Ayu Ameliatul Shahilah Ahmad Jamri, Roslan Hasni, Doost Ali Mojdeh . Maximal first Zagreb index of trees with given Roman domination number. AIMS Mathematics, 2022, 7(7): 11801-11812. doi: 10.3934/math.2022658 |
[8] | Bana Al Subaiei, Ahlam AlMulhim, Abolape Deborah Akwu . Vertex-edge perfect Roman domination number. AIMS Mathematics, 2023, 8(9): 21472-21483. doi: 10.3934/math.20231094 |
[9] | Chang-Xu Zhang, Fu-Tao Hu, Shu-Cheng Yang . On the (total) Roman domination in Latin square graphs. AIMS Mathematics, 2024, 9(1): 594-606. doi: 10.3934/math.2024031 |
[10] | Abel Cabrera-Martínez, Andrea Conchado Peiró . On the {2}-domination number of graphs. AIMS Mathematics, 2022, 7(6): 10731-10743. doi: 10.3934/math.2022599 |
The present work is to design a novel Neuro swarm computing standards using artificial intelligence scheme to exploit the Gudermannian neural networks (GNN)accomplished with global and local search ability of particle swarm optimization (PSO) and sequential quadratic programming scheme (SQPS), called as GNN-PSO-SQPS to solve a class of the second order Lane-Emden singular nonlinear model (SO-LES-NM). The suggested intelligent computing solver GNN-PSO-SQPS using the Gudermannian kernel are unified with the configuration of the hidden layers of GNN of differential operators for solving the SO-LES-NM. An error based fitness function (FF) applying the differential form of the differential model and corresponding boundary conditions. The FF is optimized together with the combined heuristics of PSO-SQPS. Three problems of the SO-LES-NM are solved to validate the correctness, effectiveness and competence of the designed GNN-PSO-SQPS. The performance of the GNN-PSO-SQPS through statistical operators is tested to check the constancy, convergence and precision.
In this paper, we shall only consider graphs without multiple edges or loops. Let G=(V(G),E(G)) be a graph, v∈V(G), the neighborhood of v in G is denoted by N(v). That is to say N(v)={u|uv∈E(G),u∈V(G)}. The degree of a vertex v is denoted by d(v), i.e. d(v)=|N(v)|. A graph is trivial if it has a single vertex. The maximum degree and the minimum degree of a graph G are denoted by Δ(G) and δ(G), respectively. Denote by Kn the complete graph on n vertices.
A subset D of the vertex set of a graph G is a dominating set if every vertex not in D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of G with |D|=γ(G) is called a γ-set of G.
Roman domination of graphs is an interesting variety of domination, which was proposed by Cockayne et al. [6]. A Roman dominating function (RDF) of a graph G is a function f:V(G)→{0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight w(f) of a Roman dominating function f is the value w(f)=∑u∈V(G)f(u). The minimum weight of an RDF on a graph G is called the Roman domination number γR(G) of G. An RDF f of G with w(f)=γR(G) is called a γR-function of G. The problems on domination and Roman domination of graphs have been investigated widely, for example, see list of references [8,9,10,13] and [3,7,12], respectively.
In 2016, Chellali et al. [5] introduced a variant of Roman dominating functions, called Roman {2}-dominating functions. A Roman {2}-dominating function (R{2}DF) of G is a function f:V→{0,1,2} such that ∑u∈N(v)f(u)≥2 for every vertex v∈V with f(v)=0. The weight of a Roman {2}-dominating function f is the sum ∑v∈Vf(v). The Roman {2}-domination number γ{R2}(G) is the minimum weight of an R{2}DF of G. Note that if f is an R{2}DF of G and v is a vertex with f(v)=0, then either there is a vertex u∈N(v) with f(u)=2, or at least two vertices x,y∈N(v) with f(x)=f(y)=1. Hence, an R{2}DF of G is also an RDF of G, which is also mentioned by Chellali et al [5]. Moreover, they showed that the decision problem for Roman {2}-domination is NP-complete, even for bipartite graphs.
In fact, a Roman {2}-dominating function is essentially the same as a weak {2}-dominating function, which was introduced by Brešar et al. [1] and studied in literatures [2,11,14,15].
For a mapping f:V(G)→{0,1,2}, let (V0,V1,V2) be the ordered partition of V(G) induced by f such that Vi={x:f(x)=i} for i=0,1,2. Note that there exists a 1-1 correspondence between the function f and the partition (V0,V1,V2) of V(G), so we will write f=(V0,V1,V2).
Chellali et al. [4] obtained the following lower bound of Roman domination number.
Lemma 1. (Chellali et al. [4]) Let G be a nontrivial connected graph with maximum degree Δ. Then γR(G)≥Δ+1Δγ(G).
In this paper, we generalize this result on nontrivial connected graph G with maximum degree Δ and minimum degree δ. We prove that γR(G)≥Δ+2δΔ+δγ(G). As a corollary, we obtain that 32γ(G)≤γR(G)≤2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)≤2γ{R2}(G)−1 for every graph G and there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k−1 for any integer k≥2.
Lemma 2. (Cockayne et al. [6]) Let f=(V0,V1,V2) be a γR-function of an isolate-free graph G with |V1| as small as possible. Then
(i) No edge of G joins V1 and V2;
(ii) V1 is independent, namely no edge of G joins two vertices in V1;
(iii) Each vertex of V0 is adjacent to at most one vertex of V1.
Theorem 3. Let G be a nontrivial connected graph with maximum degree Δ(G)=Δ and minimum degree δ(G)=δ. Then
γR(G)≥Δ+2δΔ+δγ(G). | (2.1) |
Moreover, if the equality holds, then
γ(G)=n(Δ+δ)Δδ+Δ+δandγR(G)=n(Δ+2δ)Δδ+Δ+δ. |
Proof. Let f=(V0,V1,V2) be a γR-function of G with V1 as small as possible. By Lemma 2, we know that N(v)⊆V0 for any v∈V1 and N(v1)∩N(v2)=∅ for any v1,v2∈V1. So we have
|V1|≤|V0|δ | (2.2) |
Since G is nontrivial, it follows that V2≠∅. Note that every vertex in V2 is adjacent to at most Δ vertices in V0; we have
|V0|≤Δ|V2| | (2.3) |
By Formulae (2.2) and (2.3), we have
|V1|≤Δδ|V2| | (2.4) |
By the definition of an RDF, every vertex in V0 has at least one neighbor in V2. So V1∪V2 is a dominating set of G. Together with Formula (2.4), we can obtain that
γ(G)≤|V1|+|V2|≤Δδ|V2|+|V2|=Δ+δδ|V2|. |
Note that f is a γR-function of G; we have
γR(G)=|V1|+2|V2|=(|V1|+|V2|)+|V2|≥γ(G)+δΔ+δγ(G)=Δ+2δΔ+δγ(G). |
Moreover, if the equality in Formula (2.1) holds, then by previous argument we obtain that |V1|=|V0|δ, |V0|=Δ|V2|, and V1∪V2 is a γ-set of G. Then we have
n=|V0|+|V1|+|V2|=|V0|+|V0|δ+|V0|Δ=Δδ+Δ+δΔδ|V0|. |
Hence, we have
|V0|=nΔδΔδ+Δ+δ,|V1|=nΔΔδ+Δ+δ, and |V2|=nδΔδ+Δ+δ. |
So
γR(G)=|V1|+2|V2|=n(Δ+2δ)Δδ+Δ+δ and γ(G)=|V1|+|V2|=n(Δ+δ)Δδ+Δ+δ |
since V1∪V2 is a γ-set of G. This completes the proof.
Now we show that the lower bound in Theorem 3 can be attained by constructing an infinite family of graphs. For any integers k≥2, δ≥2 and Δ=kδ, we construct a graph Hk from K1,Δ by adding k news vertices such that each new vertex is adjacent to δ vertices of K1,Δ with degree 1 and no two new vertices has common neighbors. Then add some edges between the neighbors of each new vertex u such that δ(Hk)=δ and the induced subetaaph of N(u) in Hk is not complete. The resulting graph Hk is a connected graph with maximum degree Δ(G)=Δ and maximum degree δ(G)=δ. It can be checked that γ(Hk)=k+1 and γR(Hk)=k+2=Δ+2δΔ+δγ(G).
For example, if k=2, δ=3 and Δ=kδ=6, then the graph H2 constructed by the above method is shown in Figure 1, where u1 and u2 are new vertices.
Furthermore, by Theorem 3, we can obtain a lower bound of the Roman domination number on regular graphs.
Corollary 4. Let G be an r-regular graph, where r≥1. Then
γR(G)≥32γ(G) | (2.5) |
Moreover, if the equality holds, then
γ(G)=2nr+2andγR(G)=3nr+2. |
Proof. Since G is r-regular, we have Δ(G)=δ(G)=r. By Theorem 3 we can obtain that this corollary is true.
For any integer n≥2, denote by G2n the (2n−2)-regular graph with 2n vertices, namely G2n is the graph obtained from K2n by deleting a perfect matching. It can be checked that γ(G2n)=2 and γR(G2n)=3=32γ(G) for any n≥2. Hence, the bound in Corollary 4 is attained.
Note that γR(G)≤2γ(G) for any graph G; we can conclude the following result.
Corollary 5. Let G be an r-regular graph, where r≥1. Then
32γ(G)≤γR(G)≤2γ(G). |
Chellali et al. [5] obtain the following bounds for the Roman {2}-domination number of a graph G.
Lemma 6. (Chellali et al. [5]) For every graph G, γ(G)≤γ{R2}(G)≤γR(G)≤2γ(G).
Lemma 7. (Chellali et al. [5]) If G is a connected graph of order n and maximum degree Δ(G)=Δ, then
γ{R2}(G)≥2nΔ+2. |
Theorem 8. For every graph G, γR(G)≤2γ{R2}(G)−1. Moreover, for any integer k≥2, there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k−1.
Proof. Let f=(V0,V1,V2) be an γ{R2}-function of G. Then γ{R2}(G)=|V1|+2|V2| and γR(G)≤2|V1|+2|V2| since V1∪V2 is a dominating set of G. If |V2|≥1, then γR(G)≤2|V1|+2|V2|=2γ{R2}(G)−2|V2|≥2γ{R2}(G)−2. If |V2|=0, then every vertex in V0 is adjacent to at least two vertices in V1. So for any vertex u∈V1, f′=(V0,{u},V1∖{u}) is an RDF of G. Then we have γR(G)≤1+2|V1∖{u}|=2|V1|−1=2γ{R2}(G)−1.
For any integer k≥2, let Ik be the graph obtained from Kk by replacing every edge of Kk with two paths of length 2. Then Δ(Ik)=2(k−1) and δ(Ik)=2. We first prove that γ{R2}(Ik)=k. Since V(Ik)=|V(Kk)|+2|E(Kk)|=k+2⋅k(k−1)2=k2, by Lemma 7 we can obtain γ{R2}(Ik)≥2|V(Ik)|Δ(Ik)+2=2k22(k−1)+2=k. On the other hand, let f(x)=1 for each x∈V(Ik) with d(x)=2(k−1) and f(y)=0 for each y∈V(Ik) with d(y)=2. It can be seen that f is an R{2}DF of Ik and w(f)=k. Hence, γ{R2}(Ik)=k.
We now prove that γR(Ik)=2k−1. Let g={V′1,V′2,V′3} be a γR-function of Ik such that |V′1| is minimum. For each 4-cycle C=v1v2v3v4v1 of Ik with d(v1)=d(v3)=2(k−1) and d(v2)=d(v4)=2, we have wg(C)=g(v1)+g(v2)+g(v3)+g(v4)≥2. If wg(C)=2, then by Lemma 2(iii) we have g(vi)∈{0,2} for any i∈{1,2,3,4}. Hence, one of v1 and v3 has value 2 and g(v2)=g(v4)=0. If wg(C)=3, then by Lemma 2(i) we have {g(v1),g(v3)}={1,2} or {g(v2),g(v4)}={1,2}. When {g(v2),g(v4)}={1,2}, let {g′(v1),g′(v2)}={1,2}, g′(v2)=g′(v4)=0 and g′(x)=g(x) for any x∈V(Ik)∖{v1,v2,v3,v4}. Then g′ is also a γR-function of Ik. If wg(C)=4, then exchange the values on C such that v1,v3 have value 2 and v2,v4 have value 0. So we obtain that Ik has a γR-function h such that h(y)=0 for any y∈V(Ik) with degree 2. Note that any two vertices of Ik with degree 2(k−1) belongs to a 4-cycle considered above; we can obtain that there is exactly one vertex z of Ik with degree 2(k−1) such that h(z)=1. Hence, γR(Ik)=w(h)=2k−1.
Note that the graph Ik constructed in Theorem 8 satisfies that γ(Ik)=k=γ{R2}(Ik). By Theorem 8, it suffices to prove that γ(Ik)=k. Let A={v:v∈V(Ik),d(v)=2(k−1)} and B=V(Ik)∖A. We will prove that Ik has a γ-set containing no vertex of B. Let D be a γ-set of Ik. If D contains a vertex u∈B. Since the degree of u is 2, let u1 and u2 be two neighbors of u in Ik. Then d(u1)=d(u2)=2(k−1) and, by the construction of Ik, u1 and u2 have two common neighbors u,u′ with degree 2. Hence, at least one of u′,u1, and u2 belongs to D. Let D′=(D∖{u,u′})∪{u1,u2}. Then D′ is also a γ-set of Ik. Hence, we can obtain a γ-set of Ik containing no vertex of B by performing the above operation for each vertex v∈D∩B. So A is a γ-set of Ik and γ(Ik)=|A|=k.
By Lemma 6 and Theorem 8, we can obtain the following corollary.
Corollary 9. For every graph G, γ{R2}(G)≤γR(G)≤2γ{R2}(G)−1.
Theorem 10. For every graph G, γR(G)≤γ(G)+γ{R2}(G)−1.
Proof. By Lemma 6 we can obtain that γR(G)≤2γ(G)≤γ(G)+γ{R2}(G). If the equality holds, then γR(G)=2γ(G) and γ(G)=γ{R2}(G). So γR(G)=2γ{R2}(G), which contradicts Theorem 8. Hence, we have γR(G)≤γ(G)+γ{R2}(G)−1.
In this paper, we prove that γR(G)≥Δ+2δΔ+δγ(G) for any nontrivial connected graph G with maximum degree Δ and minimum degree δ, which improves a result obtained by Chellali et al. [4]. As a corollary, we obtain that 32γ(G)≤γR(G)≤2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)≤2γ{R2}(G)−1 for every graph G and the bound is achieved. Although the bounds in Theorem 3 and Theorem 8 are achieved, characterizing the graphs that satisfy the equalities remain a challenge for further work.
The author thanks anonymous referees sincerely for their helpful suggestions to improve this work. This work was supported by the National Natural Science Foundation of China (No.61802158) and Natural Science Foundation of Gansu Province (20JR10RA605).
The author declares that they have no conflict of interest.
[1] | H. J. Lane, On the theoretical temperature of the Sun, under the Hypothesis of a gaseous Mass maintaining its Volume by its internal Heat and depending on the laws of gases as known to terrestrial Experiment, Am. J. Sci., 148 (1870), 57-74. |
[2] | R. Emden, Gaskugeln Teubner. Leipzig und Berlin, 1907. |
[3] |
Z. Sabir, H. A. Wahab, H. Umr, M. G. Sakar, M. A. Z. Raja, Novel design of Morlet wavelet neural network for solving second order Lane-Emden equation, Math. Comput. Simul., 172 (2020), 1-14. doi: 10.1016/j.matcom.2020.01.005
![]() |
[4] |
D. Baleanu, S. S. Sajjadi, A, Jajarmi, J. H. Asad, New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator. Eur. Phys. J. Plus, 134 (2019), 181. doi: 10.1140/epjp/i2019-12561-x
![]() |
[5] |
T. Luo, Z. Xin, H. Zeng, Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities, Comm. Math. Phys, 347 (2016), 657-702. doi: 10.1007/s00220-016-2753-1
![]() |
[6] |
J. A. Khan, M. A. Z. Raja, M. M. Rashidi, M. I. Syam, A. M. Wazwaz, Nature-inspired computing approach for solving non-linear singular Emden-Fowler problem arising in electromagnetic theory, Connect. Sci., 27 (2015), 377-396. doi: 10.1080/09540091.2015.1092499
![]() |
[7] |
M. Ghergu, V. Rădulescu, On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math., 344 (2007), 163-168. doi: 10.1016/j.crma.2006.12.008
![]() |
[8] |
R Rach, J. S. Duan, A. M. Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method, J. Math. Chem., 52 (2014), 255-267. doi: 10.1007/s10910-013-0260-6
![]() |
[9] | A. H. Bhrawy, A. S. Alofi, R. A. Van Gorder, An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry, Abst. Appl. Anal., 2014 (2014). |
[10] |
M. Dehghan, F. Shakeri, Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method, Prog. Electromagn. Res., 78 (2008), 361-376. doi: 10.2528/PIER07090403
![]() |
[11] |
D. Flockerzi, K. Sundmacher, On coupled Lane-Emden equations arising in dusty fluid models, J. Phys.: Conference Series, 268 (2011), 012006. doi: 10.1088/1742-6596/268/1/012006
![]() |
[12] |
V. Rădulescu, D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. Theor. Methods Appl., 75 (2012), 1524-1530. doi: 10.1016/j.na.2011.01.037
![]() |
[13] |
W. Adel, Z. Sabir, Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model via Bernoulli collocation method, Eur. Phys. J. Plus, 135 (2020), 427. doi: 10.1140/epjp/s13360-020-00449-x
![]() |
[14] |
S. Mall, S. Chakraverty, Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev Neural Network method, Neurocomputing, 149 (2015), 975-982. doi: 10.1016/j.neucom.2014.07.036
![]() |
[15] | S. Mall, S. Chakraverty, Chebyshev neural network based model for solving Lane-Emden type equations, Appl. Math. Comput., 247 (2014), 100-114. |
[16] | S. Mall, S. Chakraverty, Regression-based neural network training for the solution of ordinary differential equations, Int. J. Math. Modell. Numer. Optim., 4 (2013), 136-149. |
[17] | Z. Sabir, M. Umar, J. L. G. Guirao, M. Shoaib, M. A. Z. Raja, Integrated intelligent computing paradigm for nonlinear multi-singular third-order Emden-Fowler equation, Neural Comput. Appl., (2020). Available from: https://doi.org/10.1007/s00521-020-05187-w. |
[18] |
I Ahmad, H. Ilyas, A. Urooj, M. S. Aslam, M. Shoaib, M. A. Z. Raja, Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels, Neural Comput. Appl., 31 (2019), 9041-9059. doi: 10.1007/s00521-019-04203-y
![]() |
[19] | Z. Sabir, F. Amin, D. Pohl, J. L. G. Guirao, Intelligence computing approach for solving second order system of Emden-Fowler model, J. Intell. Fuzzy Syst., In press. |
[20] |
Z. Sabir, S. Saoud, M. A. Z. Raja, H. A. Wahab, A. Arbi, Heuristic computing technique for numerical solutions of nonlinear fourth order Emden-Fowler equation, Math. Comput. Simul., 178 (2020), 534-548. doi: 10.1016/j.matcom.2020.06.021
![]() |
[21] | M. A. Z. Raja, J. Mehmood, Z. Sabir, A. K. Nasab, M. A. Manzar, Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput. Appl., 31 (2019), 793-812. |
[22] |
S. U. I. Ahmed, F. Faisal, M. Shoaib, M. A. Z. Raja, A new heuristic computational solver for nonlinear singular Thomas-Fermi system using evolutionary optimized cubic splines, European Phys. J. Plus, 135 (2020), 1-29. doi: 10.1140/epjp/s13360-019-00059-2
![]() |
[23] |
Z. Sabir, M. A. Z. Raja, M. Umar, M. Shoaib, Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden-Fowler equation, Eur. Phys. J. Plus, 135 (2020), 1-17. doi: 10.1140/epjp/s13360-019-00059-2
![]() |
[24] |
M. Umar, M. A. Z. Raja, Z. Sabir, A. S. Alwabli, M. Shoaib, A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment, Eur. Phys. J. Plus, 135 (2020), 1-23. doi: 10.1140/epjp/s13360-019-00059-2
![]() |
[25] |
A. H. Bukhari, M. Sulaiman, M. A. Z. Raja, S. Islam, M. Shoaib, P. Kumam, Design of a hybrid NAR-RBFs neural network for nonlinear dusty plasma system, Alex. Eng. J., 59 (2020), 3325-3345. doi: 10.1016/j.aej.2020.04.051
![]() |
[26] |
Z. Sabir, M. A. Z. Raja, M. Umar, M. Shoaib, Neuro-swarm intelligent computing to solve the second-order singular functional differential model, Eur. Phys. J. Plus, 135 (2020), 474. doi: 10.1140/epjp/s13360-020-00440-6
![]() |
[27] | Z Sabir, H. A. Wahab, M. Umar, F. Erdoğan, Stochastic numerical approach for solving second order nonlinear singular functional differential equation, Appl. Math. Comput., 363 (2019), 124605. |
[28] |
M. A. Z. Raja, F. H. Shah, M. Tariq, I. Ahmad, Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch's problem arising in plasma physics, Neural Comput. Appl., 29 (2018), 83-109. doi: 10.1007/s00521-016-2530-2
![]() |
[29] |
Z. Sabir, M. A. Manzar, M. A. Z. Raja, M. Sheraz, A. M. Wazwaz, Neuro-heuristics for nonlinear singular Thomas-Fermi systems, Appl. Soft Comput., 65 (2018), 152-169. doi: 10.1016/j.asoc.2018.01.009
![]() |
[30] |
M. Umar, Z. Sabir, F. Amin, J. L. G. Guirao, M. A. Z. Raja, Stochastic numerical technique for solving HIV infection model of CD4+ T cells, Eur. Phys. J. Plus, 135 (2020), 403. doi: 10.1140/epjp/s13360-020-00417-5
![]() |
[31] |
M. Umar, Z. Sabir, M. A. Z. Raja, Intelligent computing for numerical treatment of nonlinear prey-predator models, Appl. Soft Comput., 80 (2019), 506-524. doi: 10.1016/j.asoc.2019.04.022
![]() |
[32] | Z. Sabir, M. A. Z. Raja, J. L. G. Guirao, M. Shoaib, A neuro-swarming intelligence based computing for second order singular periodic nonlinear boundary value problems, (2020), Pre-print. |
[33] |
M. A. Z. Raja, U. Farooq, N. I. Chaudhary, A. M. Wazwaz, Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes, Appl. Soft Comput., 38 (2016), 561-586. doi: 10.1016/j.asoc.2015.10.015
![]() |
[34] |
A. Mehmood, A. Zameer, S. H. Ling, M. A. Z. Raja, Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery-Hamel flow, J. Taiwan Institute Chem. Eng., 91 (2018), 57-85. doi: 10.1016/j.jtice.2018.05.046
![]() |
[35] |
M. Umar, F. Amin, H. A. Wahab, D. Baleanu, Unsupervised constrained neural network modeling of boundary value corneal model for eye surgery, Appl. Soft Comput., 85 (2019), 105826. doi: 10.1016/j.asoc.2019.105826
![]() |
[36] |
Z. Sabir, M. A. Z. Raja, M. Shoaib, J. F. G. Aguilar, FMNEICS: Fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane-Emden system, Comput. Appl. Math., 39 (2020), 1-18. doi: 10.1007/s40314-019-0964-8
![]() |
[37] |
M. A. Z. Raja, A. Zameer, A. U. Khan, A. M. Wazwaz, A new numerical approach to solve Thomas-Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming, Springer Plus, 5 (2016), 1400. doi: 10.1186/s40064-016-3093-5
![]() |
[38] | Y. Shi, R. C. Eberhart, Empirical study of particle swarm optimization, Proceedings of the 1999 congress on evolutionary computation-CEC99 (Cat. No. 99TH8406), 3, 1945-1950. IEEE, 1999. |
[39] | Y. Shi, Particle swarm optimization: Developments, applications and resources, In: Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No. 01TH8546), 1, 81-86. IEEE, 2001. |
[40] | A. P., Engelbrecht, Computational intelligence: an introduction. John Wiley & Sons, 2007. |
[41] | S. Kefi, N. Rokbani, P. Krö mer, A. M. Alimi, Ant supervised by PSO and 2-opt algorithm, AS-PSO-2Opt, applied to traveling salesman problem. In: 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC) (004866-004871), 2016, IEEE. |
[42] | A. Taieb, A. Ferdjouni, A new design of fuzzy logic controller optimized by PSO-SCSO applied to SFO-DTC induction motor drive, Int. J. Elect. Comput. Eng., 10 (2020), 2088-8708. |
[43] |
Y. Ding, W. Zhang, L. Yu, K. Lu, The accuracy and efficiency of GA and PSO optimization schemes on estimating reaction kinetic parameters of biomass pyrolysis, Energy, 176 (2019), 582-588. doi: 10.1016/j.energy.2019.04.030
![]() |
[44] |
E. Keybondorian, A. Taherpour, A. Bemani, T. Hamule, Application of novel ANFIS-PSO approach to predict asphaltene precipitation, Petrol. Sci. Technol., 36 (2018), 154-159. doi: 10.1080/10916466.2017.1411948
![]() |
[45] |
N. Ghorbani, A. Kasaeian, A. Toopshekan, L. Bahrami, A. Maghami, Optimizing a hybrid wind-PV-battery system using GA-PSO and MOPSO for reducing cost and increasing reliability, Energy, 154 (2018), 581-591. doi: 10.1016/j.energy.2017.12.057
![]() |
[46] |
G. Wang, J. Guo, Y. Chen, Y. Li, Q. Xu, A PSO and BFO-based learning strategy applied to faster R-CNN for object detection in autonomous driving, IEEE Access, 7 (2019), 18840-18859. doi: 10.1109/ACCESS.2019.2897283
![]() |
[47] |
K. Long, X. Wang, X. Gu, Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm, Eng. Optimiz., 50 (2018), 2091-2107. doi: 10.1080/0305215X.2017.1417401
![]() |
[48] | S. Sun, Geometric optimization of radiative enclosures using sequential quadratic programming algorithm, ES Energy Environ., 6 (2019), 57-68. |
[49] | G. Torrisi, S. Grammatico, R. S. Smith, M. Morari, A variant to sequential quadratic programming for nonlinear model predictive control. In: 2016 IEEE 55th Conference on Decision and Control (CDC), 2814-2819. IEEE, 2016. |
[50] |
G. Singh, M. Rattan, S. S. Gill, N. Mittal, Hybridization of water wave optimization and sequential quadratic programming for cognitive radio system, Soft Comput., 23 (2019), 7991-8011. doi: 10.1007/s00500-018-3437-x
![]() |
[51] |
R. Hult, M. Zanon, G. Frison, S. Gros, P. Falcone, Experimental validation of a semi‐distributed sequential quadratic programming method for optimal coordination of automated vehicles at intersections, Optim. Contr. Appl. Met., 41 (2020), 1068-1096. doi: 10.1002/oca.2592
![]() |
[52] |
M. Umar, Z. Sabir, M. A. Z. Raja, M. Shoaib, M. Gupta, Y. G. Sánchez, A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics, Symmetry, 12 (2020), 1628. doi: 10.3390/sym12101628
![]() |
[53] |
T. N. Cheema, M. A. Z. Raja, I. Ahmad, S. Naz, H. Ilyas, M. Shoaib, Intelligent computing with Levenberg-Marquardt artificial neural networks for nonlinear system of COVID-19 epidemic model for future generation disease control, Eur. Phys. J. Plus, 135 (2020), 1-35. doi: 10.1140/epjp/s13360-019-00059-2
![]() |
[54] |
M. Umar, Z. Sabir, I. Ali, H. A. Wahab, M. Shoaib, M. A. Z. Raja, The 3-D flow of Casson nanofluid over a stretched sheet with chemical reactions, velocity slip, thermal radiation and Brownian motion, Therm. Sci., 24 (2020), 2929. doi: 10.2298/TSCI190625339U
![]() |
[55] |
M. Shoaib, M. A. Z. Raja, M. T. Sabir, S. Islam, Z. Shah, P. Kumam, H. Alrabaiah, Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet, Sci. Rep., 10 (2020), 1-15. doi: 10.1038/s41598-019-56847-4
![]() |
[56] |
Z. Shah, M. A. Z. Raja, Y. M. Chu, W. A. Khan, M. Waqas, M. Shoaib, et al. Design of neural network based intelligent computing for neumerical treatment of unsteady 3D flow of Eyring-Powell magneto-nanofluidic model, J. Mater. Res. Technol., 9 (2020), 14372-14387. doi: 10.1016/j.jmrt.2020.09.098
![]() |
1. | Chang-Xu Zhang, Fu-Tao Hu, Shu-Cheng Yang, On the (total) Roman domination in Latin square graphs, 2024, 9, 2473-6988, 594, 10.3934/math.2024031 | |
2. | Sakander Hayat, Raman Sundareswaran, Marayanagaraj Shanmugapriya, Asad Khan, Venkatasubramanian Swaminathan, Mohamed Hussian Jabarullah, Mohammed J. F. Alenazi, Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling, 2024, 16, 2073-8994, 663, 10.3390/sym16060663 | |
3. | Tatjana Zec, On the Roman domination problem of some Johnson graphs, 2023, 37, 0354-5180, 2067, 10.2298/FIL2307067Z | |
4. | Jian Yang, Yuefen Chen, Zhiqiang Li, Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1, 2023, 8, 2473-6988, 17702, 10.3934/math.2023904 |