Research article Special Issues

A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order

  • The main objective of the investigation is to broaden the description of Caputo fractional derivatives (in short, CFDs) (of order 0<α<r) considering all relevant permutations of entities involving t1 equal to 1 and t2 (the others) equal to 2 via fuzzifications. Under gH-differentiability, we also construct fuzzy Elzaki transforms for CFDs for the generic fractional order α(r1,r). Furthermore, a novel decomposition method for obtaining the solutions to nonlinear fuzzy fractional partial differential equations (PDEs) via the fuzzy Elzaki transform is constructed. The aforesaid scheme is a novel correlation of the fuzzy Elzaki transform and the Adomian decomposition method. In terms of CFD, several new results for the general fractional order are obtained via gH-differentiability. By considering the triangular fuzzy numbers of a nonlinear fuzzy fractional PDE, the correctness and capabilities of the proposed algorithm are demonstrated. In the domain of fractional sense, the schematic representation and tabulated outcomes indicate that the algorithm technique is precise and straightforward. Subsequently, future directions and concluding remarks are acted upon with the most focused use of references.

    Citation: M. S. Alqurashi, Saima Rashid, Bushra Kanwal, Fahd Jarad, S. K. Elagan. A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order[J]. AIMS Mathematics, 2022, 7(8): 14946-14974. doi: 10.3934/math.2022819

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  • The main objective of the investigation is to broaden the description of Caputo fractional derivatives (in short, CFDs) (of order 0<α<r) considering all relevant permutations of entities involving t1 equal to 1 and t2 (the others) equal to 2 via fuzzifications. Under gH-differentiability, we also construct fuzzy Elzaki transforms for CFDs for the generic fractional order α(r1,r). Furthermore, a novel decomposition method for obtaining the solutions to nonlinear fuzzy fractional partial differential equations (PDEs) via the fuzzy Elzaki transform is constructed. The aforesaid scheme is a novel correlation of the fuzzy Elzaki transform and the Adomian decomposition method. In terms of CFD, several new results for the general fractional order are obtained via gH-differentiability. By considering the triangular fuzzy numbers of a nonlinear fuzzy fractional PDE, the correctness and capabilities of the proposed algorithm are demonstrated. In the domain of fractional sense, the schematic representation and tabulated outcomes indicate that the algorithm technique is precise and straightforward. Subsequently, future directions and concluding remarks are acted upon with the most focused use of references.



    The coastal regions often face water scarcity despite the proximity to abundant seawater resources due to challenges in obtaining clean water [1]. Seawater, with an average salinity of 3.5% and a salinity measurement of around 35 psu, contains NaCl rendering and unsuitable for direct human consumption [2]. The salinity measurement and Total Dissolved Solids (TDS) ranging from 20.000 to 50.000 ppm in the Gulf of Prigi is needed for desalination processes [3,4,5]. Desalination offers a viable solution, often utilizing adsorption methods involving membrane technology, which efficiently reduce NaCl levels in seawater [6]. The demand for clean water remains a challenge, necessitating innovative solutions.

    The exploration of calcium-based membranes for salt adsorption was aimed to optimize desalination efficiency [7]. The membranes synthesized from coral skeletons, predominantly composed of calcium carbonate (CaCO3), are augmented with chitosan and sodium alginate, providing a cost-effective yet efficient solution for desalination [8,9]. The membrane synthesis process involves the crosslinking of sodium alginate and calcium, forming calcium alginate membranes capable of adsorbing NaCl from seawater [10,11].

    Chitosan, a biopolymer with active NH2 and OH- functional groups, holds potential for membrane manufacturing, specifically in adsorption applications [12]. Combination chitosan with alginate potentially enhances the membrane's properties, potentially forming pores and binding with metals [13,14]. The coral skeleton is a significant resource for calcium-based membrane production [9]. The extraction of calcium oxide (CaO) from coral skeletons through the calcination process further improves the membrane's functional properties [15]. Through future desalination, power material membranes aim to transform mineral-rich seawater into clean water in coastal areas where water supply remains insufficient [16].

    Here, the synthesis of calcium alginate membranes involve the utilization of abundant coral skeletons, primarily composed of CaCO3, and crosslinked with sodium alginate and calcium to form membranes capable of binding salt from seawater [17,18]. The combined coral skeletons and polymer materials like chitosan and sodium alginate in membrane synthesis provided a significant step toward creating efficient and cost-effective desalination methods [19]. We explored the addition of chitosan to calcium-alginate membranes for enhanced seawater salt adsorption, further advancing the development of viable desalination processes.

    The materials were coral skeletons, chitosan powder, sodium alginate, 37% HCl solution, NaCl solid, 1% K2CrO4 solution, AgNO3 solid, distilled water, NaOH solution, and filter paper.

    Corals taken from Prigi Beach undergo a series of preparation stages, such as cleaning with demineralized water, drying at 60 ℃ until dryness, grinding and sieving (200 mesh) until a fine powder was obtained. The resulting coral powder was calcined at 800 ℃ for 2 hours to produce high purity CaCO3 [20,21].

    This CaCO3 was processed into a homogeneous CaCl2 solution, starting with diluting 83.3 mL of 37% HCl to obtain a 1 M HCl solution. Five grams of CaCO3 was combined with 100 mL of 1 M HCl solution, producing a CaCl2 solution after homogenization and filtration.

    A total of 1 and 2 g of sodium alginate were individually weighed and each added to 100 mL of distilled water. The solutions were stirred using a hotplate magnetic stirrer at 300 rpm at 80 ℃ and left for 24 hours. Chitosan was prepared in concentrations of 1%, 2%, and 3%, respectively. A total of 1, 2, and 3 g of chitosan powder (w/v) were added to 100 mL of acetic acid solution with a concentration corresponding to the chitosan mass (1%, 2%, and 3% acetic acid). The solutions were then stirred using a hotplate magnetic stirrer at 300 rpm at 80 ℃ and left for 24 hours.

    The synthesis of Ca-alginate-chitosan membranes was carried out by mixing the polymer solutions (sodium alginate and chitosan) in a volume of 40 mL with a chitosan-to-alginate ratio of 1:2, resulting in a chitosan solution volume of 16.13 mL and a sodium alginate solution volume of 26.6 mL with the variation concentration ratio of chitosan to alginate (1:1, 1:2, 1:3, 1:4). After 24 hours, the alginate-chitosan mixture was added with 10 mL of 0.5 M CaCl2 and left for 8 hours, followed by lifting and drying at room temperature for 24 hours. The resulting beads and transparent membranes were then characterized by measuring pH and washing with distilled water.

    The process of optimizing the concentration of the alginate solution involved adsorption experiments with NaCl solutions using different chitosan-alginate compositions. This aims to determine the optimum alginate composition that produces the highest Cl- adsorption concentration. Further optimization was focused on determining the ideal chitosan concentration for maximum Cl- adsorption. The characterized membranes were applied by FT-IR analysis to identify functional groups and SEM analysis for microstructural observation.

    The initial characterization of Calcium- Alginate-Chitosan Membrane Synthesis involved employing a Fourier transform infrared spectrometer (FT-IR, Shimadzu IR Prestige 21) to identify the compounds' functional groups. Subsequently, X-ray fluorescence (PANalytical) determined the elemental composition of a material/sample, while Scanning Electron Microscopy and Energy Dispersive X-ray (SEM EDX) can be used to characterize a material and its morphology.

    In laboratory scale applications, the membrane was tested by stirring a 0.1 N NaCl solution with varying contact times. The resulting solution underwent Na+ concentration analysis using AAS (Thermo scientific ICE3000) after appropriate dilution. In addition, analysis of Cl- concentration was carried out via Mohr's argentometric titration, which involves titration with a 0.1 N AgNO3 standard solution.

    After adsorption, the filtrate was analyzed for Na+ and Cl- concentrations using specific techniques. The Na+ concentration was determined using AAS after adequate dilution, while the Cl- concentration was determined via Mohr's argentometry titration involving titration with a 0.1 N AgNO3 standard solution, both carried out in duplicate for accuracy. Blank titration was carried out as a reference for calculating Cl- levels.

    The sieved coral skeleton was calcined at 800 ℃ to obtain high purity calcium oxide. Calcination functions remove CO2, dirt, and minerals other than calcium. During this process, heat was applied to the surface of the coral particles, thereby encouraging the released CO2 gas to migrate to the surface and disperse into the calcined sample. The resulting calcined coral as the natural sources for material CaCl2 productions. XRF analysis of corals determines the content of calcium and various other elements. The elemental composition of coral, both before and after calcination is listed in Tables 1 and 2.

    Table 1.  XRF Analyses of coral before alcination.
    No Elements Wt (%) Oxide Wt (%)
    1 Ca 90.82 CaO 90.86
    2 Si 1.30 SiO2 2.30
    3 Sr 4.40 SrO 3.42
    4 Fe 1.70 Fe2O3 1.62
    5 Ti 0.15 TiO2 0.16
    6 Mo 0.89 MoO3 1.10
    7 Cu 0.03 CuO 0.03
    8 Mn 0.07 MnO 0.06
    9 Ba 0.10 BaO 0.10
    10 Yb 0.47 Yb2O3 0.35
    11 Lu 0.09 Lu2O3 0.06

     | Show Table
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    Table 2.  XRF analyses of coral after calcination.
    No Elements Wt (%) Oxide Wt (%)
    1 Ca 92.58 CaO 93.41
    2 Si - SiO2 -
    3 Sr 4.54 SrO 3.56
    4 Fe - Fe2O3 1.43
    5 Ti 0.11 TiO2 0.12
    6 Mo 1.10 MoO3 1.30
    7 Cu 0.04 CuO 0.03
    8 Mn 0.06 MnO 0.05
    9 Ba 0.10 BaO 0.08
    10 Yb - Yb2O3 -
    11 Lu - Lu2O3 -

     | Show Table
    DownLoad: CSV

    The XRF measurements showed that the coral was composed mainly of calcium oxide, which has the potential to be used as a component of Ca2+ in the membrane. The percentage of CaO before and after calcination was 90.82 and 93.41%, respectively.

    The process of dissolving sodium alginate in water using a hotplate magnetic stirrer is recommended at 80 ℃ with a speed of 300 rpm [22]. Chitosan powder was dissolved in high-concentration acetate acid or organic acid solvents with a pH range of 4–6 [22]. Furthermore, chitosan was dissolved at a relatively high temperature, around 80°–100℃. The solubility process of chitosan occurs through a protonation reaction, where the amine group (-NH2) in chitosan accepts H+ released by acetic acid, resulting in a positive charge (-NH3+). The addition of Ca2+ ion from CaCl2 reacted as a cross-link with sodium alginate and chitosan. CaCl2 functions as a crosslinking bound sodium alginate, with Ca2+ replacing Na+ to form calcium alginate. That cross link is essential for the formation of the gel structure. The formation of Ca-alginate-chitosan involves a drip technique, where a vacuum pump ensures a consistent drip frequency of the sodium alginate solution into the calcium chloride solution.

    During Ca-alginate-chitosan beads production, the NaCl was removed through washing, achieved by soaking the beads in distilled water. The process requires a 300 mL sodium alginate solution and 150 mL of calcium chloride solution to release 100 grams of calcium alginate granules. This method is widely used in various scientific studies and industrial applications, and offers a controlled and efficient means of producing calcium alginate beads for diverse uses.

    Figure 1.  Ca-alginate-chitosan beads.

    The calcium alginate-chitosan membrane was analyzed using FTIR to determine the functional group content present in the membrane. FTIR spectrum analysis was performed using the Origin application. The FTIR characterization is shown in Figure 2.

    Figure 2.  FTIR Spectra of Calcium Alginate Membrane and Calcium Alginate-Chitosan Membrane.

    The infrared (IR) spectrum shows characteristic features of calcium alginate and calcium alginate-chitosan membranes. In the calcium alginate spectrum, the prominent peak at 1683.85 cm-1 indicates the characteristics of the carboxyl functional group (C = O). The range between 1594-1418 cm-1 depicts the C-O symmetric stretch of the carboxyl group, and the presence of a C-O asymmetric stretch was identified at 1180.43 cm-1 [23]. The peak at 2945.3 cm-1 indicates intramolecular hydrogen bonds involving O-H and -COO- along with Ca2+ ions, while the peak at 3647.39 cm-1 represents O-H stretching vibrations especially in calcium alginate [24].

    In the alginate-chitosan spectrum, there was a shift and the emergence of new functional groups due to the addition of chitosan. An important shift occurs at 3630.15 cm-1, which indicates O-H stretching vibrations and N-H stretching vibrations, consistent with previous research by Venkatesan et al. [25]. The wider band at 2943.37 cm-1 indicates intramolecular bonds between O-H, -COO-, and Ca2+ ions together with the NH2 group. The peak at 1186.22 and 1750.10 cm-1 indicates asymmetric C-O and C = O stretching. The characteristic peak at 1649.13 cm-1 represents the ketone C = O stretching vibration on primary NH2, which indicates the presence of N-H bending which indicates interaction with the protonated amino group on chitosan. The wavenumber shift in the -OH group indicates an increase in energy level [26].

    The calcium-based alginate-chitosan membrane that has been formed is analyzed for its microstructure using SEM (Scanning Electron Microscope) to examine the surface and pore diameter of the membrane. The SEM results obtained under optimal conditions can be observed in Figure 3.

    Figure 3.  SEM analysis of a calcium alginate membrane at (a) 1000x (b) 5000x magnification.

    Under 5000x magnification, the surface of the calcium alginate particles shows a considerable surface area, displaying a particle diameter of 138 nm. Smaller particle size correlates with expanded surface area, leading to increase adsorption rates. These characteristics place calcium alginate in the classification of microporous materials, as indicated by its particle size and is described in detail in this study [27]. Figure 4 illustrates the comparative analysis between various calcium alginate membranes.

    Figure 4.  SEM analysis of calcium alginate-chitosan membrane at (a) 1000x, (b) 5000x magnification.

    SEM analysis findings at 5000x magnification show the inner surface structure of the membrane, which is characterized by roughness and porosity, with a diameter of 185.96 nm. This size classification aligns them with microporous membranes [27]. These membrane pores play an important role as adsorbents during NaCl adsorption, facilitating movement within the pore walls in the adsorption process.

    Sodium alginate functions as the base material in gel formation along with Ca2+ from CaCl2, which cross-links with sodium alginate and chitosan. The optimization process focuses on identifying the most effective concentration of sodium alginate for adsorption purposes. This optimization phase involves testing membrane configurations utilizing sodium alginate solutions at concentrations of 1 and 2% w/v, combined with 1% w/v of chitosan. The quantities utilized consist of 26.6 mL of Na-Alginate solution, 13.3 mL of chitosan solution, and 10 mL of 0.5 M CaCl2. The optimized condition of ideal composition is shown in Table 3, as follows:

    Table 3.  Optimized sodium alginate solution concentration form maximum adsorption.
    Na-Alginate (ml; w/v) Chitosan (vol; w/v) Concentration (ppm) %Adsorption
    [Cl-] Initial [Cl-] Residue [Cl-] Adsorbed
    26.6; 1% 13.3; 1 % 3543 2290 1253 34.8
    26.6; 2% 13.3; 1 % 3543 2130 1413 40.0
    26.6; 3% 13.3; 1 % 3543 2330 1213 34.2
    26.6; 4% 13.3; 1 % 3543 2370 1173 33.1

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    As listed in Table, the membrane containing a 2%w/v alginate concentration demonstrates a higher capacity for adsorbing Cl- ions compared to the 1% w/v alginate concentration membrane. The concentration of sodium alginate plays a crucial role in gel formation, influencing the density of particles within the solution. Higher concentrations lead to a denser presence of charges or particles, facilitating increased electrostatic interactions between more functional groups and the -NH3+ groups in chitosan. Consequently, this interaction enhances the membrane's capacity for substance adsorption [28].

    The optimization of chitosan solution concentration is performed to determine the best membrane composition for further membrane application with contact time. The optimization data is listed in Table 4.

    Table 4.  Optimized Chitosan Solution for adsorption maximum.
    Na-Alginate (ml; w/v) Chitosan (ml; w/v) Concentration (ppm) %Adsorption
    [Cl-] Initial [Cl-]Residue [Cl-] Adsorbed
    26.6; 2% 13.3; 0 % 3542 2662 880 24.8
    26.6; 2% 13.3; 1 % 3542 2130 1412 39.8
    26.6; 2% 13.3; 2 % 3542 1846.0 1696 48.4
    26.6; 2% 13.3; 3 % 3542 2378.5 1164 32.3

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    Table 4 shows the membrane incorporating chitosan exhibits superior performance due to the incorporation of active cross-linking agents, which leads to an increase in membrane pore size, thereby enhancing its capacity for adsorption. The alginate-chitosan composite membrane was presumed to demonstrate superior absorption of chloride ions (Cl-) up to 48.4%, and then decreasing of 32.3% with variation 2:3 between Na-Alginate and Chitosan composition.

    The decrease in adsorption level at the 2:3 ratio of Na-alginate to chitosan may be due to the saturation of the chitosan solution, resulting in a decrease in the availability of active amino groups for cross-linking with alginate. This reduced the effective surface area for ion adsorption. Additionally, excess chitosan content possibly causes the formation of chitosan aggregates, which can hinder the formation of a uniform and stable membrane. The phenomenon of influenced chemical factors are related to the balance of ionic interactions between chitosan and alginate, as well as the availability of functional groups for cross-linking. An abundance of chitosan can disrupt this optimal interaction balance, resulting in a decrease in adsorption efficiency [29,30].

    The process of combination between chitosan solution with alginate reveals in an interaction between the positively charged amino groups (-NH3+) of chitosan and the -COO- groups of alginates through ionic interactions, establishing cross-linking bonds.

    The utilization of the calcium alginate and chitosan as the membrane was assessed by adsorbing NaCl in the ranged from 10 to 50 minutes in the batch system. The outcomes of the membrane's work were measured in terms of the maximal percentage adsorption of Na+ and Cl-. The data of adsorption test of calcium alginate with addition chitosan in the membrane for Na+ adsorption is presented in Table 5.

    Table 5.  Analysis of the Na+ ion adsorption level by membrane using AAS.
    Contact Time (Minutes) Concentration (ppm) %Adsorption
    [Na+] Initial [Na+] Residue [Na+] Adsorbed
    10 1105 668 437 39.5
    20 1105 693 411 37.2
    30 1105 688 416 37.7
    40 1105 657 447 40.5
    50 1105 745 359 32.5

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    As listed in Table 5, the optimal time analysis towards the maxima Na+ ion adsorption is 40 minutes. The particle aggregation on the membrane potentially affected the adsorption of Na+ ions. These ions are typically bound by -COO- groups originating from alginate, crucial in the NaCl adsorption process. The presence of -NH3+ groups from chitosan is suspected to contribute to this aggregation.

    Figure 5 illustrates the correlation between contact time and the percentage of Na+ adsorption. This reveals a notable decline at the 50-minute. This implies a reduction in the membrane's adsorption capacity, potentially linked to a decrease in chitosan content or a decline in NH2 groups within the membrane. Chitosan's efficacy in binding metals stems from its free amino groups and integral for ion exchange capabilities. The nitrogen content in chitosan with its high polymer chain correlates with its capacity to bind metals.

    Figure 5.  Relationship between contact time and the percentage of Na+ and Cl- ion adsorption.

    As listed in Table 6, the ion Cl- adsorption test was utilized using the Mohr method. As listed in the table, it becomes evident that the optimal duration for Cl- ion adsorption is 40 minutes, the same with its pattern of Na+ ion. The contact time plays a vital role in the adsorption process, directly impacting the amount of adsorbed substance. The optimized contact allows for more solute molecules to be adsorbed, facilitating a well-functioning diffusion process [31]. It significantly influences adsorption until equilibrium is attained. Once equilibrium is reached, further contact time does not affect the process or possibly desorption.

    Table 6.  Analysis of the Cl- adsorption levels in the NaCl 0.1 N solution using the Mohr Method Argentometric titration.
    Contact Time (Minute) Concentration (ppm)
    [Cl-] Initial [Cl-] Residue [Cl-] Adsorbed %Adsorption
    10 3543 1988 1554 43.9
    20 3543 1952 1590 44.9
    30 3543 1864 1661 47.4
    40 3543 1828 1696 48.4
    50 3543 1881 1661 46.9

     | Show Table
    DownLoad: CSV

    Figure 5 indicates a substantial increase in the Na+ and Cl- ions adsorption process at 40 minutes, revealing them as the optimal time for both ions. Conversely, there is a decline in observing them at 50 minutes. Prolonging the process beyond 60 minutes could potentially lead to desorption, releasing the adsorbate back into the solution. Figure 5 depicts the graphical representation of the relationship between contact time and the percentage of Na+ and Cl- ions adsorption.

    The possible mechanism reaction between Ca-alginate and chitosan during adsorption of Na+ and Cl- as shown in Figure 6, as follows:

    Figure 6.  The possible mechanism reaction between Ca-alginate chitosan with NaCl during the adsorption process.

    The initial mechanism reaction started a reaction between Ca-alginate and chitosan to form a Ca-alginate-chitosan complex, causing the cross linking of the polymer to involve hydrogel formation for potentially adsorption application. During the adsorption, the first step involved the displacement of Ca2+ changing with Na+ ion due to the higher affinity of Na+ ion to alginate. The first product in the adsorption process was the dissolution of Ca complex and trapped Na-alginate chitosan in the membrane. In the second product, when the Cl- ion was applied in the adsorption process, chitosan could possibly interact with Cl- ions through electrostatic interactions, hydrogen bonding, and its positively charged amino groups attracted Cl- ions. The residue of this process adsorption formed a Ca(OH)2 complex. In the future prospect, the specific condition of pH and temperature between Ca-alginate chitosan and NaCl should be considered to determine the best capability membrane as adsorbent.

    The inclusion of chitosan in calcium alginate had an impact on membrane formation, which was reflected in FTIR characterization by observing shifts in the wave numbers of certain functional groups. On the calcium alginate membrane, the absorption of the O-H group at 3647.39 shifted to 3437 cm-1, which indicated the existence of intramolecular bonds with the N-H group of chitosan. In addition, C = O, CO, and N-H groups were involved in NaCl binding. SEM analysis showed a particle size diameter of 185.96 nm, indicating rough and porous surface characteristics, indicating the micro-porous nature of the membrane, which is capable of absorbing NaCl. The calcium alginate-chitosan based membrane showed a Na+ and Cl- adsorption peak of 40.5% and 48.4%, respectively. This maximum adsorption level was achieved using a membrane composition consisting of 13.3 mL of 2% chitosan (w/v) and 26.6 mL. The optimized contact time for this Ca-alginate addition with chitosan was detected at 40 minutes to adsorb of Na+ and Cl- ions.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research was funded by Penelitian Dasar grant No. 0162/E5.4/DT.05.00/2023 on DRPM 2023

    The authors state that there are no conflicts of interest for this project.



    [1] M. Nazeer, F. Hussain, M. I. Khan, A. ur Rehman, E. R. El-Zahar, Y. M. Chu, et al., Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel, Appl. Math. Comput., 420 (2022), 126868. https://doi.org/10.1016/j.amc.2021.126868 doi: 10.1016/j.amc.2021.126868
    [2] Y. M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. Ijaz Khan, S. U. Khan, Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface, Appl. Math. Comput., 419 (2022), 126883. https://doi.org/10.1016/j.amc.2021.126883 doi: 10.1016/j.amc.2021.126883
    [3] Y. M. Chu, U. Nazir, M. Sohail, M. M. Selim, J. R. Lee, Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract., 5 (2021), 119. https://doi.org/10.3390/fractalfract5030119 doi: 10.3390/fractalfract5030119
    [4] T. H. Zhao, M. I. 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
    [5] Y. M. Chu, S. Bashir, M. Ramzan, M. Y. Malik, Model-based comparative study of magnetohydrodynamics unsteady hybrid nanofluid flow between two infinite parallel plates with particle shape effects, Math. Methods Appl. Sci., 2022. https://doi.org/10.1002/mma.8234 doi: 10.1002/mma.8234
    [6] 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
    [7] 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
    [8] 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), 2240051. https://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
    [9] S. Rashid, A. Khalid, S. Sultana, F. Jarad, K. M. Abualnaja, Y. S. Hamed, Novel numerical investigation of the fractional oncolytic effectiveness model with M1 virus via generalized fractional derivative with optimal criterion, Results Phys., 37 (2022), 105553. https://doi.org/10.1016/j.rinp.2022.105553 doi: 10.1016/j.rinp.2022.105553
    [10] T. H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, et al., A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176.
    [11] 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 Phys., 35 (2022), 105304. https://doi.org/10.1016/j.rinp.2022.105304 doi: 10.1016/j.rinp.2022.105304
    [12] 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
    [13] T. H. Zhao, M. K. Wang, Y. M. Chu, On the bounds of the perimeter of an ellipse, Acta Math. Sci., 42 (2022), 491–501. https://doi.org/10.1007/s10473-022-0204-y doi: 10.1007/s10473-022-0204-y
    [14] T. H. Zhao, M. K. Wang, G. J. Hai, Y. M. Chu, Landen inequalities for Gaussian hypergeometric function, RACSAM, 116 (2022), 53. https://doi.org/10.1007/s13398-021-01197-y doi: 10.1007/s13398-021-01197-y
    [15] H. H. Chu, T. H. Zhao, Y. M. Chu, Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means, Math. Slovaca, 70 (2020), 1097–1112. https://doi.org/10.1515/ms-2017-0417 doi: 10.1515/ms-2017-0417
    [16] K. S. Miller, B. Ross, Introduction to the fractional calculus and fractional differential equations, New York: John Wiley & Sons, 1993.
    [17] I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1998.
    [18] S. Rashid, E. I. Abouelmagd, S. Sultana, Y. M. Chu, New developments in weighted n-fold type inequalities via discrete generalized ĥ-proportional fractional operators, Fractals, 30 (2022), 2240056. https://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
    [19] S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y. M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 2240026. https://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
    [20] 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), 2240110. https://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
    [21] H. M. Srivastava, A. K. N. Alomari, K. M. Saad, W. M. Hamanah, Some dynamical models involving fractional-order derivatives with the Mittag-Leffler type kernels and their applications based upon the Legendre spectral collocation method, Fractal Fract., 5 (2021), 131. https://doi.org/10.3390/fractalfract5030131 doi: 10.3390/fractalfract5030131
    [22] H. M. Srivastava, K. M. Saad, Numerical Simulation of the fractal-fractional Ebola virus, Fractal Fract., 4 (2020), 49. https://doi.org/10.3390/fractalfract4040049 doi: 10.3390/fractalfract4040049
    [23] S. Rashid, S. Sultana, N. Idrees, E. Bonyah, On analytical treatment for the fractional-order coupled partial differential equations via fixed point formulation and generalized fractional derivative operators, J. Funct. Spaces, 2022 (2022), 3764703. https://doi.org/10.1155/2022/3764703 doi: 10.1155/2022/3764703
    [24] M. Al Qurashi, S. Rashid, S. Sultana, F. Jarad, A. M. Alsharif, Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory, AIMS Math., 7 (2022), 12587–12619. https://doi.org/10.3934/math.2022697 doi: 10.3934/math.2022697
    [25] M. Sharifi, B. Raesi, Vortex theory for two dimensional Boussinesq equations, Appl. Math. Nonliner Sci., 5 (2020), 67–84. https://doi.org/10.2478/amns.2020.2.00014 doi: 10.2478/amns.2020.2.00014
    [26] T. A. Sulaiman, H. Bulut, H. M. Baskonus, On the exact solutions to some system of complex nonlinear models, Appl. Math. Nonliner Sci., 6 (2020), 29–42. https://doi.org/10.2478/amns.2020.2.00007 doi: 10.2478/amns.2020.2.00007
    [27] S. Rashid, Y. G. Sánchez, J. Singh, K. M. Abualnaja, Novel analysis of nonlinear dynamics of a fractional model for tuberculosis disease via the generalized Caputo fractional derivative operator (case study of Nigeria), AIMS Math., 7 (2022), 10096–10121. https://doi.org/10.3934/math.2022562 doi: 10.3934/math.2022562
    [28] M. Caputo, Elasticita e dissipazione, Zanichelli, Bologna, 1969.
    [29] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
    [30] D. Li, W. Sun, C. Wu, A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions, Numer. Math. Theor. Methods Appl., 14 (2021), 355–376. https://doi.org/10.4208/nmtma.OA-2020-0129 doi: 10.4208/nmtma.OA-2020-0129
    [31] M. She, D. Li, H. W. Sun, A transformed L1 method for solving the multi-term time-fractional diffusion problem, Math. Comput. Simulat., 193 (2022), 584–606. https://doi.org/10.1016/j.matcom.2021.11.005 doi: 10.1016/j.matcom.2021.11.005
    [32] H. Qin, D. Li, Z. Zhang, A novel scheme to capture the initial dramatic evolutions of nonlinear sub-diffusion equations, J. Sci. Comput., 89 (2021), 65. https://doi.org/10.1007/s10915-021-01672-z doi: 10.1007/s10915-021-01672-z
    [33] M. El-Borhamy, N. Mosalam, On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic, Appl. Math. Nonlinear Sci., 5 (2020), 93–108. https://doi.org/10.2478/amns.2020.1.00010 doi: 10.2478/amns.2020.1.00010
    [34] R. A. de Assis, R. Pazim, M. C. Malavazi, P. P. da C. Petry, L. M. E. da Assis, E. Venturino, A mathematical model to describe the herd behaviour considering group defense, Appl. Math. Nonlinear Sci., 5 (2020), 11–24. https://doi.org/10.2478/amns.2020.1.00002 doi: 10.2478/amns.2020.1.00002
    [35] J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solition. Fract., 140 (2020), 110127. https://doi.org/10.1016/j.chaos.2020.110127 doi: 10.1016/j.chaos.2020.110127
    [36] P. A. Naik, Z. Jain, K. M. Owolabi, Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, Chaos Solition. Fract., 138 (2020), 109826. https://doi.org/10.1016/j.chaos.2020.109826 doi: 10.1016/j.chaos.2020.109826
    [37] A. Atangana, E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Adv. Differ. Equ., 2013 (2013), 94. https://doi.org/10.1186/1687-1847-2013-94 doi: 10.1186/1687-1847-2013-94
    [38] H. Günerhan, E. Çelik, Analytical and approximate solutions of fractional partial differential-algebraic equations, Appl. Math. Nonlinear Sci., 5 (2020), 109–120. https://doi.org/10.2478/amns.2020.1.00011 doi: 10.2478/amns.2020.1.00011
    [39] F. Evirgen, S. Uçar, N. Özdemir, System analysis of HIV infection model with CD4+T under non-singular kernel derivative, Appl. Math. Nonlinear Sci., 5 (2020), 139–146. https://doi.org/10.2478/amns.2020.1.00013 doi: 10.2478/amns.2020.1.00013
    [40] M. R. R. Kanna, R. P. Kumar, S. Nandappa, I. N. Cangul, On solutions of fractional order telegraph partial differential equation by Crank-Nicholson finite difference method, Appl. Math. Nonliner Sci., 5 (2020), 85–98. https://doi.org/10.2478/amns.2020.2.00017 doi: 10.2478/amns.2020.2.00017
    [41] M. A. Alqudah, R. Ashraf, S. Rashid, J. Singh, Z. Hammouch, T. Abdeljawad, Novel numerical investigations of fuzzy Cauchy reaction-diffusion models via generalized fuzzy fractional derivative operators, Fractal Fract., 5 (2021), 151. https://doi.org/10.3390/fractalfract5040151 doi: 10.3390/fractalfract5040151
    [42] S. Rashid, M. K. A. Kaabar, A. Althobaiti, M. S. Alqurashi, Constructing analytical estimates of the fuzzy fractional-order Boussinesq model and their application in oceanography, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.01.003 doi: 10.1016/j.joes.2022.01.003
    [43] S. Rashid, R. Ashraf, Z. Hammouch, New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography, J. Ocean Eng. Sci., 2021. https://doi.org/10.1016/j.joes.2021.11.004 doi: 10.1016/j.joes.2021.11.004
    [44] Z. Li, C. Wang, R. P. Agarwal, R. Sakthivel, Hyers-Ulam-Rassias stability of quaternion multidimensional fuzzy nonlinear difference equations with impulses, Iran. J. Fuzzy Syst., 18 (2021), 143–160.
    [45] A. Kandel, W. J. Byatt, Fuzzy differential equations, Proceedings of the International Conference Cybernetics and Society, Tokyo, Japan, 1978.
    [46] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal.: Theory Methods Appl., 72 (2010), 2859–2862. https://doi.org/10.1016/j.na.2009.11.029 doi: 10.1016/j.na.2009.11.029
    [47] K. Nemati, M. Matinfar, An implicit method for fuzzy parabolic partial differential equations, J. Nonlinear Sci. Appl., 1 (2008), 61–71.
    [48] T. Allahviranloo, M. Afshar Kermani, Numerical methods for fuzzy partial differential equations under new definition for derivative, Iran. J. Fuzzy Syst., 7 (2010), 33–50.
    [49] O. A. Arqub, M. Al-Smadi, S. Momani, T. Hayat, Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems, Soft Comput., 21 (2017), 7191–7206. https://doi.org/10.1007/s00500-016-2262-3 doi: 10.1007/s00500-016-2262-3
    [50] O. A. Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput. Applic., 28 (2017), 1591–1610. https://doi.org/10.1007/s00521-015-2110-x doi: 10.1007/s00521-015-2110-x
    [51] T. M. Elzaki, S. M. Ezaki, Application of new transform "Elzaki transform" to partial differential equations, Global J. Pure Appl. Math., 7 (2011), 65–70.
    [52] S. Rashid, K. T. Kubra, S. U. Lehre, Fractional spatial diffusion of a biological population model via a new integral transform in the settings of power and Mittag-Leffler nonsingular kernel, Phys. Scr., 96 (2021), 114003.
    [53] S. Rashid, Z. Hammouch, H. Aydi, A. G. Ahmad, A. M. Alsharif, Novel computations of the time-fractional Fisher's model via generalized fractional integral operators by means of the Elzaki transform, Fractal Fract., 5 (2021), 94. https://doi.org/10.3390/fractalfract5030094 doi: 10.3390/fractalfract5030094
    [54] S. Rashid, R. Ashraf, A. O. Akdemir, M. A. Alqudah, T. Abdeljawad, S. M. Mohamed, Analytic fuzzy formulation of a time-fractional Fornberg-Whitham model with power and Mittag-Leffler kernels, Fractal Fract., 5 (2021), 113. https://doi.org/10.3390/fractalfract5030113 doi: 10.3390/fractalfract5030113
    [55] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. https://doi.org/10.1016/0022-247X(88)90170-9 doi: 10.1016/0022-247X(88)90170-9
    [56] G. Adomian, R. Rach, On composite nonlinearities and the decomposition method, J. Math. Anal. Appl., 113 (1986), 504–509. https://doi.org/10.1016/0022-247X(86)90321-5 doi: 10.1016/0022-247X(86)90321-5
    [57] S. S. L. Chang, L. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern., 2 (1972), 30–34. https://doi.org/10.1109/TSMC.1972.5408553 doi: 10.1109/TSMC.1972.5408553
    [58] R. Goetschel Jr., W. Voxman, Elementary fuzzy calculus, Fuzzy. Sets. Syst., 18 (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6 doi: 10.1016/0165-0114(86)90026-6
    [59] A. Kaufmann, M. M. Gupta, Introduction to fuzzy arithmetic, New York: Van Nostrand Reinhold Company, USA, 1991.
    [60] B. Bede, J. Fodor, Product type operations between fuzzy numbers and their applications in geology, Acta Polytech. Hung., 3 (2006), 123–139.
    [61] A. Georgieva, Double fuzzy Sumudu transform to solve partial Volterra fuzzy integro-differential equations, Mathematics, 8 (2020), 692. https://doi.org/10.3390/math8050692 doi: 10.3390/math8050692
    [62] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119–141. https://doi.org/10.1016/j.fss.2012.10.003 doi: 10.1016/j.fss.2012.10.003
    [63] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151 (2005), 581–599. https://doi.org/10.1016/j.fss.2004.08.001 doi: 10.1016/j.fss.2004.08.001
    [64] Y. Chalco-Cano, H. Román-Flores, On new solutions of fuzzy differential equations, Chaos Solition. Fract., 38 (2008), 112–119. https://doi.org/10.1016/j.chaos.2006.10.043 doi: 10.1016/j.chaos.2006.10.043
    [65] H. C. Wu, The improper fuzzy Riemann integral and its numerical integration, Inf. Sci., 111 (1998), 109–137. https://doi.org/10.1016/S0020-0255(98)00016-4 doi: 10.1016/S0020-0255(98)00016-4
    [66] A. H. Sedeeg, A coupling Elzaki transform and homotopy perturbation method for solving nonlinear fractional heat-like equations, Am. J. Math. Comput. Model., 1 (2016), 15–20
    [67] A. Georgieva, A. Pavlova, Fuzzy Sawi decomposition method for solving nonlinear partial fuzzy differential equations, Symmetry, 13 (2021), 1580. https://doi.org/10.3390/sym13091580 doi: 10.3390/sym13091580
    [68] R. Henstock, Theory of integration, Butterworth, London, 1963.
    [69] Z. T. Gong, L. L. Wang, The Henstock-Stieltjes integral for fuzzy-number-valued functions, Inf. Sci., 188 (2012), 276–297. https://doi.org/10.1016/j.ins.2011.11.024 doi: 10.1016/j.ins.2011.11.024
    [70] L. Jäntschi, The Eigenproblem translated for alignment of molecules, Symmetry, 11 (2019), 1027. https://doi.org/10.3390/sym11081027 doi: 10.3390/sym11081027
    [71] L. Jäntschi, D. Bálint, S. D. Bolboacǎ, Multiple linear regressions by maximizing the likelihood under assumption of generalized Gauss-Laplace distribution of the error, Comput. Math. Methods Med., 2016 (2016), 8578156. https://doi.org/10.1155/2016/8578156 doi: 10.1155/2016/8578156
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