Research article

Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear

  • In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1<q<2, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica.

    Citation: Khaled M. Saad, Manal Alqhtani. Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear[J]. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225

    Related Papers:

    [1] Fohona S. Coulibaly, Bi-Botti C. Youan . Current status of lectin-based cancer diagnosis and therapy. AIMS Molecular Science, 2017, 4(1): 1-27. doi: 10.3934/molsci.2017.1.1
    [2] Fohona S. Coulibaly, Danielle N. Thomas, Bi-Botti C. Youan . Anti-HIV lectins and current delivery strategies. AIMS Molecular Science, 2018, 5(1): 96-116. doi: 10.3934/molsci.2018.1.96
    [3] Atsushi Tsubota, Hidenori Ichijo, Kengo Homma . Mislocalization, aggregation formation and defect in proteolysis in ALS. AIMS Molecular Science, 2016, 3(2): 246-268. doi: 10.3934/molsci.2016.2.246
    [4] Mohamed A. Eldeeb . Aging: when the ubiquitin–proteasome machinery collapses. AIMS Molecular Science, 2017, 4(2): 219-223. doi: 10.3934/molsci.2017.2.219
    [5] Mohamed Amine Ben Mlouka, Thomas Cousseau, Patrick Di Martino . Application of fluorescently labelled lectins for the study of polysaccharides in biofilms with a focus on biofouling of nanofiltration membranes. AIMS Molecular Science, 2016, 3(3): 338-356. doi: 10.3934/molsci.2016.3.338
    [6] M.Bansbach Heather, H.Guilford William . Actin nitrosylation and its effect on myosin driven motility. AIMS Molecular Science, 2016, 3(3): 426-438. doi: 10.3934/molsci.2016.3.426
    [7] Isabel Saez, David Vilchez . Protein clearance mechanisms and their demise in age-related neurodegenerative diseases. AIMS Molecular Science, 2015, 1(1): 1-21. doi: 10.3934/molsci.2015.1.1
    [8] Ana Jalles, Patrícia Maciel . The disruption of proteostasis in neurodegenerative disorders. AIMS Molecular Science, 2015, 2(3): 259-293. doi: 10.3934/molsci.2015.3.259
    [9] Renata Tisi, Marco Rigamonti, Silvia Groppi, Fiorella Belotti . Calcium homeostasis and signaling in fungi and their relevance for pathogenicity of yeasts and filamentous fungi. AIMS Molecular Science, 2016, 3(4): 505-549. doi: 10.3934/molsci.2016.4.505
    [10] Mohammad Shboul, Hela Sassi, Houweyda Jilani, Imen Rejeb, Yasmina Elaribi, Syrine Hizem, Lamia Ben Jemaa, Marwa Hilmi, Susanna Gerit Kircher, Ali Al Kaissi . The phenotypic spectrum in a patient with Glycine to Serine mutation in the COL2A1 gene: overview study. AIMS Molecular Science, 2021, 8(1): 76-85. doi: 10.3934/molsci.2021006
  • In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1<q<2, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica.



    1. Introduction

    Biomineralization is found in multiple animal groups and involves the interaction between cells and secreted extracellular matrix components [1,2,3]. Calcium is transported through cells in vesicles and deposited on the extracellular material, where it forms the crystalized mineral. This process has been extensively studied, and a large number of secreted proteins have been identified, but the functional role of these proteins and how they are organized to form mineralized structures in a controlled pattern has remained elusive. Echinoderms have been used as a simple model system to study the process of biomineralization. The proteins found in the sea urchin skeleton have been most extensively studied. Using the genome of the sea urchin Strongylocentrotus purpuratus [4], proteomic studies have identified over four hundred proteins found occluded within the sea urchin skeleton [5,6,7,8]. The most abundant of these proteins are unusual C-type lectins called spicule matrix proteins. These are secreted proteins that are acidic, contain a C-type lectin domain as well as a repetitive domain with varying numbers of a six to eight amino acid repeats [9]. The exact sequence of the amino acids in the repeat vary, but all are similar in nature. Of the spicule matrix proteins examined, all are found in the organic matrix onto which calcium is precipitated.

    In a recent study we examined the conservation of these spicule matrix proteins in a related group of echinoderms, the brittle stars, or Ophiuroids, using the Caribbean brittle star Ophiocoma wendtii [10].Isolation of skeletal proteins followed by proteomic analysis revealed 44 proteins. Of these, four were C-type lectins. These proteins were secreted, had similar amino acid content and had a similar overall charge as the sea urchin spicule matrix proteins, but lacked the repetitive domains. An analysis of the relationship between the sea urchin and brittle star C-type lectin proteins indicated they likely arose independent of one another, but the confidence levels of the analysis were weak. Having only a single representative of the ophiuroids to compare to sea urchins makes it difficult to determine how conserved the C-type lectins are in this group, which is something we address in this study. We also identified a diversity of other proteins in the brittle star skeleton, some of which are also present in the sea urchin, but the importance of these proteins was not clear. Many of these proteins had not previously been implicated in skeleton formation in echinoderms. We wished to examine another brittle star to ensure the presence of these proteins was not an artifact of our analysis.

    Here we examine the skeletal proteome of a brittle star from the Pacific coast, Ophiothrix spiculata. We detect 77 proteins occluded with the skeleton, most of which were homologous to the proteins found in the Caribbean brittle star, including C-type lectins. We compare these proteins to those in both Ophiocoma wendtii and the sea urchin Strongylocentrotus purpuratus. The results indicate that the proteins we have identified are highly conserved, and suggests that C-type lectins play a role in skeleton formation. However, they indicate that the extensive gene duplication and highly repetitive regions of the sea urchin spicule matrix proteins are not required for biomineralization.


    2. Materials and methods


    2.1. Skeletal preparation

    Isolation of clean skeletal elements was as described in Seaver and Livingston [10]. Arms of 2-8 adult brittle stars were rinsed with deionized water (DI) water then stirred vigorously in cold sodium hypochlorite (5% active chlorine, Acros Organics). The resulting skeletal elements were washed 5X in cold, DI water, then ground with a mortar and pestle. Ground mineral was homogenized with homogenization buffer (4.5 M Guanidine Isothiocyanate, 0.05 M Sodium Citrate, 5% (v/v) b-mercaptoethanol, 0.5% (w/v) Sarkosyl) in a 40 mL homogenizer. Ground, homogenized skeletal elements were transferred to sterile 50 mL conical tubes and washed 5X in cold, sterile DI water. This treatment resulted in finely ground mineral, free of attached organic material.


    2.2. Purification of skeletal proteins

    Purified skeletal elements were de-mineralized in cold acetic acid (~40 mL 25% (v/v)/g mineral) under gentle agitation until mineral was dissolved. The supernatant was transferred to Spectra/Por 6 Dialysis Membrane MWCO 6-8000 (Spectrum Laboratories, Greensboro, NC) for dialysis. Dialysis was against 1% acetic acid, DI water, then 0.001 M and 0.0001 M TRIS base solutions. Dialysis solution was changed after 12 hours. After dialysis, a small amount of precipitate remained, but was removed before further processing. The soluble fraction was concentrated using Amicon Ultra-15 centrifugal filter units, followed by Amicon Ultra-0.5 centrifugal filter units (3000 NMWL) (Millipore, Bellerica, MA). Samples were dissolved in NuPAGE LDS sample buffer under reducing conditions (Invitrogen, Carlsbad, CA) and the proteins separated by SDS-PAGE using NuPAGE 4-12% Bis-Tris Polyacrylamide gels 1.0 mm, 10-well in the MES buffer system (Invitrogen, Carlsbad, CA). Running times, protocols, and reagents were used as described by the manufacturer (Invitrogen, Carlsbad, CA). Gels were incubated on an orbital shaker in gel fix (45% (v/v) methanol, 10% (v/v) acetic acid) for one hour, followed by 3X ten minute washes in DI water. Gels were stained with SYPRO RUBY Protein Gel Stain.


    2.3. Sample preparation

    Protein quantitation was performed by Qubit Fluorometry (Invitrogen, Carlsbad, CA). A total of 20 µg of material was separated on a 4-12% Bis Tris NuPage gel (Invitrogen, Carlsbad, CA) in the MOPS buffer system. The gel lane was excised into twenty equally sized segments. Using a robot (ProGest, DigiLab), gel pieces were reduced with 8 mM dithiothreitol at 60 °C followed by alkylation with 10 mM iodoacetamide at RT. Samples were then digested with sequencing grade trypsin (Promega, Madison, WI) at 37 °C for 4 h and quenched with formic acid. The supernatant was analyzed directly without further processing.


    2.4. Mass spectrometry

    For each of the 20 fractioned gel slices, LC-MS/MS was performed identically. The digested sample was analyzed by nano LC/MS/MS with a Waters NanoAcquity HPLC system interfaced to a ThermoFisher LTQ Orbitrap Velos. Peptides were loaded on a trapping column and eluted over a 75 µm analytical column at 350 nL/min; both columns were packed with Jupiter Proteo resin (Phenomenex). The mass spectrometer was operated in data-dependent mode, with MS performed in the Orbitrap at 60,000 FWHM resolution and MS/MS performed in the LTQ. The 15 most abundant ions from each precursor scan from each gel slice were selected for MS/MS. There were several thousand precursor scans in each LC-MS/MS experiment.


    2.5. Data processing

    Data were searched using a local copy of Mascot against the NCBI non-redundant protein database as well as 454-sequenced brittle star tube foot transcriptome and genomic contigs. The Contigs and Singletons database was also translated in all six frames using Proteomatic 1.2.1 (University of Munster) to visualize reading frame. The translated database was appended with common contaminants, reversed and concatenated. Trypsin was input as the digestive enzyme, mass values were monoisotopic, the peptide mass tolerance was 10 ppm, the fragment mass tolerance was 0.8 Da, and the maximum number of missed cleavages was 2. Carbamidomethylation (C) was set as a fixed modification and variable modifications included oxidation (M), acetyl (N-term), pyro-glu (N-term Q), and deamidation (NQ). Mascot DAT files were parsed into the Scaffold software for validation, filtering and to create a non-redundant list per sample. The data were filtered using a minimum protein value of 95%, a minimum peptide value of 50% (Prophet scores) and requiring at least one unique peptide per protein.


    3. Results and Discussion

    Skeletal proteins from O. spiculata were isolated and resolved into a number of major bands on SDS-PAGE gels. The proteins are clearly intact and span a large range of molecular weight (Figure 1). This is critical since SDS-PAGE was used to fractionate the proteins for LC/MS/MS analysis. The O. spiculata gel lane was divided into 20 equal slices and each slice was processed independently. The proteins in the gel slices were digested with trypsin and analyzed by LC/MS/MS. The top 15 ions for every precursor scan from liquid chromatography of each gel slice were subjected to MS/MS. The data was compared to O. spiculata nucleotide sequence databases as described for O. wendtii [9]. A total of 3959 spectra identified ninety proteins. After removal of common contaminants and peptides that mapped to short open reading frames, 73 proteins were identified.

    Figure 1. SDS PAGE gel of Ophiothrix spiculata skeletal proteins. 1 = Molecular weight markers, 2 = O. spiculata proteins. Partitioning of the gel into slices is shown at the left with the slices numbered

    When placed into functional groups the distribution of identified proteins resembled that seen in the skeletal proteome of the brittle star Ophiocoma wendtii [9]. The largest classes of proteins with defined functions observed were extracellular matrix (ECM) and transmembrane proteins (26%), proteases and protease inhibitors (19%), C-type lectins (6%) (Figure 2). There were also a number of common intracellular proteins as well as proteins of unknown function.

    Figure 2. Protein functional classes in Ophiothris spiculata skeletal proteome

    Comparison of the proteins found in the two brittle stars by reciprocal blasts showed a high degree of conservation between the two species (Figure 3). Seventy-seven percent of the proteins that were identified in O. wendtii were homologous to proteins identified in O. spiculata. These two species have been separated for at least three million years, but clearly maintain considerable homology in the proteins used for skeleton formation. Table 1 shows the major proteins that are conserved between the two brittle star species. The proteins that matched the largest percentage of the observed spectra correspond to visible bands on the SDS-PAGE gel (Figure 1), although the sensitivity of the LC/MS/MS identifies proteins spread over a wider range of the gel than would be predicted by the Molecular Weight (MW). Fibrinogen C was found in slice 11, C-type lectins in slices 12 to 15. The C-type lectins ran slightly higher than their predicted MW, but these proteins are often glycosylated. The major band in this region appears composed of a C-type lectin, a novel protein, an agrin-like protein and a matrix metalloproteinase (MMP). The ldl-receptor is found in slice 4, Cub domain containing proteins in slices 5-6, Ig-domain containing proteins in slice 8, and the syndecan/SEA protein identified in O. wendtii in slice 10 [9]. The predominant proteins found in slices 16-17 were agrin/kazal proteinase inhibitors and a matrix metalloproteinase, which also appears higher in the gel and may represent a proteolytic fragment. Most of these proteins are also found in the S. purpuratus skeletal proteome, and some have been found in vertebrate bone. We discuss these further below.

    Figure 3. Overlap between brittle star skeletal proteomes
    Table 1.Major proteins in skeletal proteome
    Protein Number in proteome
    Matrix Metalloproteinase 5
    Fibrinogen C 5
    Semaphorin 5
    C-type lectins 4
    Agrin-like 4
    Ig domains 3
    Actin 3
    Tolloid-like/Cub domain 2
    EGF_CA family 2
    Transferrin 2
    Collagen 2
    ras/rab family protein 2
    Carbonic anhydrase 1
    Syndecan/SEA protein 1
    Calmodulin 1
    Novel protease 1
    Ldl receptor-like 1
     | Show Table
    DownLoad: CSV

    Secreted matrix metalloproteases have been found in all echinoderm skeletal proteomes. This suggests that proteolytic modification of extracellular material may play a role in skeleton formation. Similarly, there is also a novel secreted protease with similarity to neurotrypsin [10] found in both sea urchin and brittle star skeletons. Agrin-like proteins are also found in all of the echinoderm skeletons studied. Agrin interacts with neurotrypsin at the neuromuscular junction and is believed to have a signaling role [11]. Such an interaction could play a role in echinoderm skeleton formation as well.

    Carbonic anhydrase was also found in all echinoderm skeletal proteomes, and has been shown to play a role in the formation of calcium carbonate skeletons in a number of invertebrates [12,13,14,15]. Recently it has been shown to be required for skeleton formation in sea urchins [16]. Calmodulin was also found, along with several other proteins with calcium binding sites. These include an Ldl receptor-like protein. Related proteins have been implicated in the formation of vertebrate bone [17]. Other proteins identified in the O. spiculata and O. wendtii skeletal proteomes that have also been found in the human bone [18] or matrix vesicle [19] proteomes include actin, collagen, agrin/scavenger receptor proteins, carbonic anhydrase, calmodulin and fibrinogen C. One protein that is unique to the brittle star skeletal proteome contains a C-terminal syndecan domain followed by a SEA domain, an agrin like domain. This combination of protein domains was previously observed in O. wendtii, but had not previously been described elsewhere. Detection of the homologous protein in O. spiculata verifies the protein is not a sequence artifact and suggests it plays a role in the brittle star skeleton.

    A fibrinogen C or ficolin-like molecule was prominent in the brittle star skeletal proteome. This is a protein with a lectin domain that binds carbohydrate in a calcium dependent manner. This lectin protein is one of the most abundant in both brittle star skeletal proteomes, and is also present in the sea urchin skeleton [4,5,6], although it is not as abundant.

    The most prevalent proteins in the sea urchin skeletal proteomes are C-type lectins called spicule matrix (SM) proteins [4,5,6]. These proteins are characterized by lectin domains as well as unusual repetitive regions [7]. The importance and evolution of these proteins is still unclear and needs to be elucidated. To further our knowledge of the role of such proteins, we have analyzed the brittle star C-type lectins in more detail. The O. spiculata skeletal proteome contained four C-type lectins. They are all in the mid-twenty kilodalton range in molecular weight (MW) as derived from their nucleotide sequence and all are acidic (Table 2). The phylogenetic relationships between the O. spiculata and O. wendtii skeletal C-type lectins is shown in Figure 4. Os SM22 and Ow SM22 seem to be direct homologues, as do O. spiculata SM23 and O. wendtii SM24. The exact relationship between the other lectins is unclear. Figure 5 shows an alignment of the C-type lectins from both O. spiculata and O. wendtii. The proteins have the typical double loop structure of C-type lectins [20,21]. There are five beta strands and two alpha-helices in each protein. The Beta 2 strand can be identified by the WIGL sequence in the O. spiculata 5681/SM22 and the O. wendtii SM22 proteins [21]. These proteins, along with O. spiciulata 46342/SM24 proteins have an additional beta-strand at the NH2 terminal of the protein. The O. spiculata 62832/SM 23 and 9997/SM 27 proteins have a potential alpha helix in the same position. All of the proteins have six conserved cysteines capable of forming disulfide bonds. The first two could potentially aid in stabilizing a hairpin at the N terminal of the proteins. This has been found in some C-type lectins [21]. The other four cysteines are in positions suggesting they play a conserved role in stabilizing the C-type lectin double loop structure [20]. There are also four conserved calcium binding sites. The second calcium binding site lies adjacent to sequences that have been implicated in binding carbohydrate. Either an EPN sequence, which binds mannose, or a QPD sequence, that binds galactose, lie adjacent to the second calcium binding site [21]. Downstream is another sequence involved in calcium binding, normally WND. Only the O. wendtii SM 20 protein contains the canonical carbohydrate binding sequences that indicate it would bind galactose. The other proteins have variations in the canonical sequences that have been implicated in binding to carbohydrates. It is not clear how these variations could affect carbohydrate binding. Interestingly, the variation in these sequences follows the evolution of these proteins, suggesting that changes in these sequences occurred before the divergence of the two species, and that the changes have been conserved. This suggest that the changes are functional.

    Table 2.Ophiothrix spiculata C-type lectins
    Identifier Domain architecture MW pI Designation
    5681 C-type lectin domain 22,584 4.77 Os SM22
    9997 C-type lectin domain 27,822 4.81 Os SM27
    62832 C-type lectin domain 23,240 5.15 Os SM23
    46342 C-type lectin domain 23,853 4.69 Os SM24
     | Show Table
    DownLoad: CSV
    Figure 4. Neighbor joining tree of brittle star C-type lectins. Os = Ophiothrix spiculata, Ow = Ophiocoma wendtii
    Figure 5. Annotated alignment of brittle star C-type lectins. Os = Ophiothrix spiculata, Ow = Ophiocoma wendtii. --- = Alpha helix, > = Beta strand, + = Calcium binding site, ccc = carbohydrate binding site

    Overall we have found a high degree of conservation of the proteins found in the skeleton of two brittle stars from opposite sides of North America. Many of these proteins are also found in sea urchins. This suggests that these proteins play a functional role in skeleton formation. Included in the proteins conserved between the two brittle stars are C-type lectins, which are the predominant proteins found in sea urchin skeletal proteomes. The brittle star C-type lectins contain protein domains characteristic of this class of protein, suggesting functional conservation of the ability to bind carbohydrates in a calcium dependent manner. There are fewer C-type lectins found in the brittle star skeleton, however, and they lack the repetitive domains characteristic of sea urchin spicule matrix proteins.


    Acknowledgements

    We would like to thank Andy Cameron (CalTech) and the Echinobase genomic database (http://www.echinobase.org/Echinobase/) for providing genomic resources. This work was supported by a grant from the California State University CSUPERB organization. RLF was supported by NIH T34 GM008074. This work was conducted all or in part by data generated and shared from NSF award #1036416 Collaborative Research: Assembling the Echinoderm Tree of Life.


    Conflict of interest

    None declared.




    [1] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499-513.
    [2] K. M. Saad, E. H. AL-Shareef, M. S. Mohamed, X. J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, Eur. Phys. J. Plus, 132 (2017), 23. doi: 10.1140/epjp/i2017-11303-6
    [3] K. M. Saad, A reliable analytical algorithm for spacetime fractional cubic isothermal autocatalytic chemical system, Pramana, 91 (2018), 51. doi: 10.1007/s12043-018-1620-3
    [4] K. M. Saad, E. H. F. AL-Shareef, A. K. Alomari, D. Baleanu, J. F. Gómez-Aguilar, On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgers equations using homotopy analysis transform method, Chinese J. Phys., 63 (2020), 149-162.
    [5] J. H. He, Variational iteration method-a kind of nonlinear analytical technique: some examples, Int. J. Non-Linear Mech., 34 (1999), 699-708. doi: 10.1016/S0020-7462(98)00048-1
    [6] K. M. Saad, E. H. F. Al-Sharif, Analytical study for time and time-space fractional Burgers equation, Adv. Differ. Equ., 2017 (2017), 300. doi: 10.1186/s13662-017-1358-0
    [7] X. C. Shi, L. L. Huang, Y. Zeng, Fast Adomian decomposition method for the Cauchy problem of the time-fractional reaction diffusion equation, Adv. Mech. Eng., 8 (2016), 1-5.
    [8] H. M. Srivastava, K. M. Saad, New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inf. Sci., 14 (2020), 1-8. doi: 10.18576/amis/140101
    [9] H. M. Srivastava, K. M. Saad, E. H. F. Al-Sharif, A new analysis of the time-fractional and space-time fractional order Nagumo equation, J. Inf. Math. Sci., 10 (2018), 545-561.
    [10] A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, Bit. Numer. Math., 54 (2014), 937-954. doi: 10.1007/s10543-014-0484-2
    [11] Y. Takeuchi, Y. Yoshimoto, R. Suda, Second order accuracy finite difference methods for space-fractional partial differential equations, J. Comput. Appl. Math., 320 (2017), 101-119. doi: 10.1016/j.cam.2017.01.013
    [12] Y. K. Yildiray, K. Aydin, The solution of the Bagley Torvik equation with the generalized Taylor collocation method, J. Franklin Inst., 347 (2010), 452-466. doi: 10.1016/j.jfranklin.2009.10.007
    [13] M. M. Khader, K. M. Saad, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos Soliton. Fract., 110 (2018), 169-177. doi: 10.1016/j.chaos.2018.03.018
    [14] K. M. Saad, New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alex. Eng. J., 59 (2020), 1909-1917. doi: 10.1016/j.aej.2019.11.017
    [15] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier (North-Holland) Science Publishers, 2006.
    [16] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, New York: Academic Press, 1999.
    [17] A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396-406. doi: 10.1016/j.chaos.2017.04.027
    [18] M. Caputo, Linear models of dissipation whose Q is almost frequency independent: II, Geophys. J. Roy. Astron. Soc., 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x
    [19] M. Caputo, M. Fabrizio, A new defnition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2 (2015), 73-85.
    [20] A. Atangana, D. Baleanu, New fractional derivative with nonlocal and non-singular kernel, Chaos Soliton. Fract., 20 (2016), 757-763.
    [21] Z. Li, Z. Liu, M. A. Khan, Fractional investigation of bank data with fractal-fractional Caputo derivative, Chaos Soliton. Fract., 131 (2020), 109528. doi: 10.1016/j.chaos.2019.109528
    [22] A. Atangana, M. A. Khan, Fatmawati, Modeling and analysis of competition model of bank data with fractal-fractional Caputo-Fabrizio operator, Alex. Eng. J., 59 (2020), 1985-1998. doi: 10.1016/j.aej.2019.12.032
    [23] W. Wanga, M. A. Khan, Analysis and numerical simulation of fractional model of bank data with fractal-fractional Atangana-Baleanu derivative, J. Comput. Appl. Math., 369 (2020), 112646. doi: 10.1016/j.cam.2019.112646
    [24] A. Atangana, A. Akgu, K. M. Owolabi, Analysis of fractal-fractional differential equations, Alex. Eng. J., 59 (2020), 1117-1134. doi: 10.1016/j.aej.2020.01.005
    [25] A. Atangana, S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Soliton. Fract., 123 (2019), 320-337. doi: 10.1016/j.chaos.2019.04.020
    [26] J. F. Gómez-Aguilar, A. Atangana, New chaotic attractors: Application of fractal-fractional differentiation and integration, Math. Meth. Appl. Sci., doi.org/10.1002/mma.6432.
    [27] K. M. Saad, M. Alqhtania, J. F. Gómez-Aguilar, Fractal-fractional study of the hepatitis C virus infection model, Results Phys., 19 (2020), 103555. doi: 10.1016/j.rinp.2020.103555
    [28] A. A. Alomari, T. Abdeljawad, D. Baleanu, K. M. Saad, Q. M. Al-Mdallal, Numerical solutions of fractional parabolic equations with generalized Mittag-Leffler kernels, Numer. Meth. Part. Differ. Equ., doi.org/10.1002/num.22699.
    [29] M. M. Khader, K. M. Saad, D. Baleanu, S. Kumar, A spectral collocation method for fractional chemical clock reactions, Comput. Appl. Math., 39 (2020), 324. doi: 10.1007/s40314-020-01377-3
    [30] J. Singh, D. Kumar, S. Kumar, An efficient computational method for local fractional transport equation occurring in fractal porous media, Comput. Appl. Math., 39 (2020), 137. doi: 10.1007/s40314-020-01162-2
    [31] S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu, et al. An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics, 8 (2020), 558. doi: 10.3390/math8040558
    [32] J. H. Merkin, D. J. Needham, S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76. doi: 10.1093/imamat/50.1.43
  • This article has been cited by:

    1. Rachel L. Flores, Brian T. Livingston, The skeletal proteome of the sea star Patiria miniata and evolution of biomineralization in echinoderms, 2017, 17, 1471-2148, 10.1186/s12862-017-0978-z
    2. Tomohisa Ogawa, Rie Sato, Takako Naganuma, Kayeu Liu, Agness Ethel Lakudzala, Koji Muramoto, Makoto Osada, Kyosuke Yoshimi, Keiko Hiemori, Jun Hirabayashi, Hiroaki Tateno, Glycan Binding Profiling of Jacalin-Related Lectins from the Pteria Penguin Pearl Shell, 2019, 20, 1422-0067, 4629, 10.3390/ijms20184629
    3. Nicola Conci, Martin Lehmann, Sergio Vargas, Gert Wörheide, Dennis Lavrov, Comparative Proteomics of Octocoral and Scleractinian Skeletomes and the Evolution of Coral Calcification, 2020, 12, 1759-6653, 1623, 10.1093/gbe/evaa162
    4. Nathalie Le Roy, Philippe Ganot, Manuel Aranda, Denis Allemand, Sylvie Tambutté, The skeletome of the red coral Corallium rubrum indicates an independent evolution of biomineralization process in octocorals, 2021, 21, 2730-7182, 10.1186/s12862-020-01734-0
    5. Yongxin Li, Akihito Omori, Rachel L. Flores, Sheri Satterfield, Christine Nguyen, Tatsuya Ota, Toko Tsurugaya, Tetsuro Ikuta, Kazuho Ikeo, Mani Kikuchi, Jason C. K. Leong, Adrian Reich, Meng Hao, Wenting Wan, Yang Dong, Yaondong Ren, Si Zhang, Tao Zeng, Masahiro Uesaka, Yui Uchida, Xueyan Li, Tomoko F. Shibata, Takahiro Bino, Kota Ogawa, Shuji Shigenobu, Mariko Kondo, Fayou Wang, Luonan Chen, Gary Wessel, Hidetoshi Saiga, R. Andrew Cameron, Brian Livingston, Cynthia Bradham, Wen Wang, Naoki Irie, Genomic insights of body plan transitions from bilateral to pentameral symmetry in Echinoderms, 2020, 3, 2399-3642, 10.1038/s42003-020-1091-1
  • Reader Comments
  • © 2021 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(3303) PDF downloads(264) Cited by(27)

Article outline

Figures and Tables

Figures(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog