Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

Zaffar Mehdi Dar; M. Arrutselvi; Chandru Muthusamy; Sundararajan Natarajan; Gianmarco Manzini; Zaffar Mehdi Dar; M. Arrutselvi; Chandru Muthusamy; Sundararajan Natarajan; Gianmarco Manzini

doi:10.3934/mine.2025005

Mathematics in Engineering

2025, Volume 7, Issue 2: 96-129. doi: 10.3934/mine.2025005

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Research article

Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India
2.
Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600036, Tamil Nadu, India
3.
T-5 Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

Received: 10 January 2025 Revised: 17 March 2025 Accepted: 20 March 2025 Published: 27 March 2025

We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the $L^2$ -norm and $H^1$ -norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.

Keywords:

Citation: Zaffar Mehdi Dar, M. Arrutselvi, Chandru Muthusamy, Sundararajan Natarajan, Gianmarco Manzini. Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes[J]. Mathematics in Engineering, 2025, 7(2): 96-129. doi: 10.3934/mine.2025005

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Abstract

1. Introduction

Colon cancer is the third leading cause of cancer-induced deaths in both women and men globally, after lung and breast cancer ^[1]. According to the American Cancer Society, in 2017, 95,520 people contracted colon cancer and 50,260 people died from it in the United States ^[2]. Several studies have reported that dietary intervention, such as the consumption of functional foods, is one of the main factors for colon cancer prevention. Results of research on the effects of functional food components on colon cancer prevention have been widely reported ^[3]. Dietary bioactive components, such as phenolic compounds and fibers, have been reported to be potential chemopreventive agents.

Black rice bran (BRB) is well-known for its high composition of antioxidants and fiber. Many bioactive components in BRB contribute to preventing colon cancer. Compared to brown and white rice, BRB has higher levels of anthocyanin and ferulic acid ^[4]. In addition, the dietary fiber in BRB is known to increase caspase enzyme activity, which marks the occurrence of apoptosis in cancer and induces changes in the morphological characteristics of cells undergoing apoptosis ^[5,6]. The dietary fiber in the digestive tract traps carcinogens that cause colon cancer and increases the fecal period, facilitating the dissolution of carcinogens in the stool ^[7].

Colon cancer cell proliferation can be detected by proliferating cell nuclear antigen (PCNA) marker expression. PCNA is a proliferation marker that is an effector in DNA synthesis and is involved in the DNA replication process ^[8]. During the development of cancer in a tissue, cells undergo a biochemical mechanism aimed at killing cancer cells (apoptosis). The enzymes involved in this apoptotic process in colon cancer are caspase-3, which acts as an effector, and caspase-8z which acts as an initiator. The qRT-PCR analysis will assess PCNA, caspase-3, and caspase-8 markers.

The potential of BRB in inhibiting colon cancer has not been studied in detail. Therefore, an in vivo research study using animal experiments with azoxymethane (AOM) and dextran sodium sulphate (DSS)-induced mice is needed. Studies aim to evaluate the effect of BRB on inhibiting colon cancer development by assessing body weight, organ weights (colon, liver, kidney, and spleen), nodule distribution in the colon, and generally expression of PCNA, caspase-8, and caspase-3.

2. Materials and methods

2.1. Preparation of BRB

BRB, the main ingredient used in this research, was prepared based on our previous study ^[9]. The "Cempo Ireng" varieties of BRB were sourced from farmers in Cigudeg, Bogor, Indonesia. It was obtained as paddy and then hulled through milling using the Yanmar Rice Huller type HW-60A (Japan) to obtain brown-black rice. The rice was then polished using Satake polisher type N-70-F (Japan), yielding two products: polished rice and BRB. Thereafter, the BRB was sieved using 20 mesh sieves, dried using a FreeZone 4.5 Liter Benchtop Freeze Dryers Labconco (USA) at −50 ℃ for 24 hours, and stored at −20 ℃ until use.

2.2. Animal studies

All procedures involving the care and use of laboratory animals were performed following approval from the Bimana Indomedical Institutional Animal Care and Use Committee, Bogor, Indonesia (approval number R.02-17-IR). Twenty-four male BALB/C mice (5–8 weeks old) were obtained from Indoani Lab (Bogor, Indonesia), classified into three groups (n = 8 per group), and housed individually in rooms with regulated air and light cycles. During a one-week acclimation, all the mice were fed a standard diet. Afterward, the BRB group alone was further adapted to a modified diet with BRB as a fiber source. Following the third week (pre-induction), the carcinogen-induced group (C+) and BRB mice were injected intraperitoneally with azoxymethane (AOM) and then with dextran sodium sulphate (DSS) (Sigma Aldrich, Merck, St. Louis, MO, USA) at a dose of 10 mg/kg body weight.

The standard diet based on the AIN-93 M standard (Table 1), should contain the following: 14% protein, 4% fat, 5% food fiber, 3.5% minerals, 1% vitamins, 10% sucrose, and carbohydrates ^[10]. The proximate analysis findings of sodium caseinate and BRB, which include water, lipid, protein, ash, dietary fiber, and carbohydrate content by difference, were used to calculate the ratio.

Table 1. Diet composition of the study.

Ingredient (g/100g)	C− group	C+ group	C+BRB group
Black rice bran	0.00	0.00	21.68
Casein	14.00	14.00	11.48
Soybean oil	4.00	4.00	1.67
Cellulose powder	5.00	5.00	0.00
Mineral mixture	3.00	3.00	1.81
Vitamin Mixture	1.00	1.00	1.00
Sucrose	10.00	10.00	10.00
Corn starch	63.00	63.00	52.35
Note: C− group: AIN-93M standard diet + 0.9% physiological NaCl (intraperitoneal, once); C+ group: AIN-93M standard diet + AOM 10 mg/kg g body weight (intraperitoneal, once) + DSS 1% (as drinking water, for four days); C+BRB group: BRB enriched diet + AOM 10 mg/1000 g body weight (intraperitoneal, once) + DSS 1% (as drinking water, for four days).

| Show Table

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The normal (C−) group was intraperitoneally administered Isotonic NaCl 0.9% (Ecosol NaCl, Braun Pharmaceutical, Karawang, Indonesia) to reduce stress levels. One week after induction, the C+ and BRB groups were given 1% dextran sulfate sodium (DSS) (MP-Bio, Solon, Ohio, USA) in their drinking water ad libitum for four consecutive days to accelerate the occurrence of colon cancer. The cancer formation protocol can be seen in Figure 1. The diet was supplied according to the daily nutrition of mice, which was 15 g/100 g of body weight with diet composition, as shown in Table 1.

Figure 1. Macroscopic photographs of colon in groups C−, C+, and C+BRB. Scale = 1 cm.

: Shape of nodules in the colon tissue which is marked by the formation of a bulge or lump.

DownLoad: Full-Size Img PowerPoint

Cage cleaning and weighing of mice were done twice a week. At the end of the 20-week study, mice in all groups were euthanized for colon, liver, kidney, and spleen tissue collection. Colonic tissue was visually observed for the presence of masses or nodules. Furthermore, the tissue was frozen at −80 ℃ for molecular evaluation. The analyses included nodule distribution evaluation using ImageJ, histopathology with lesion neoplasia analysis, and evaluation of caspase-3, caspase-8, and PCNA using real-time PCR techniques ^[11].

2.3. Quantitative RT-PCR analysis

RNA was extracted from colonic tissue that had been frozen at −80 ℃. The colonic tissue was homogenized using a grinding mortar. Then, about 30 mg of the homogenate was transferred into a sterile 1.5 mL micro-centrifugation tube containing Tryzol. The extraction procedure was performed using Direct-Zol RNA Miniprep Plus following standard protocols (Zymo Research, CA, USA). The RNA concentration was measured using Nanodrop spectrophotometer (Thermo Scientific, USA). Reverse transcription of RNA to cDNA was conducted using a standard procedure from SuperScript III Reverse Transcriptase (Invitrogen, Thermo Scientific, USA). Analysis of the relative quantification of gene expression was performed with Real-Time PCR Detection System (iQ5, Bio-Rad Laboratories Inc., Hercules, CA, USA) using SsoFast EvaGreen Supermix (Bio-Rad, CA, USA). The master mix for real-time PCR was prepared for the expression analysis of caspase-3 (CASP3), caspase-8 (CASP8), and PCNA genes, normalized toward the GAPDH gene (Glyceraldehyde 3-phosphate dehydrogenase). The primers used in this study were: CASP3 (Forward 5'-TGGGCCTGAAATACCAAGTC-3'; Reverse 5'-AAATGACCCCTTCAT CACCA-3'), CASP8 (Forward 5'-GATGAGGCAGACTTTCTGCT-3'; Reverse 5'-CATAGTTCAC GCCAGTCAGGAT-3'), PCNA (Forward 5'-CTCACGTCTCCTTGGTACAGCTTAC-3'; Reverse 5'CTTCCTCATCT TCAATCTTGGAG-3'), and GAPDH (Forward 5'GCCATCTACAAGCAGTCACA-3'; Reverse 5'CCCTTTCTTGCGGAG ATTCT-3'). The protocol for real-time PCR amplification included 42 cycles comprising enzyme activation at 98 ℃ for 2 minutes, denaturation at 98 ℃ for 5 seconds, and annealing, extension, and data collection at 55 ℃ for 10 seconds. Evaluating of gene expression was performed using the Livak method or comparison of delta threshold (∆CT) ^[12].

2.4. Data analysis

This study, which included eight replications, used a completely randomized design (CRD) in three treatments. Data are reported as averages ± SD. Experimental data were processed using MiniTab 17.0 software and displayed in one-way ANOVA to identify differences between treatment levels. Tukey's HSD test continued different results between treatment levels, and the significance value was determined based on the 5% level.

3. Results

3.1. Animal conditions during cancer development

The study found that the C−, C+, and C+BRB groups exhibited good clinical conditions and behaviors. This finding was evidenced by data showing that the body weight of these groups remained stable. According to the data (Table 2), an increase in body weight was found in the C+BRB group. All three groups were given the same feed intake, meeting the daily nutritional requirement of 15 g/100 g of body weight on average.

Table 2. Average body weight and organ weight of BALB/C mice..

Groups	Body (g)	Colon (g)	Liver (g)	Lymph (g)	Kidney (g)
C−	26.06 ± 1.96^a	0.21 ± 0.03^b	1.23 ± 0.14	0.09 ± 0.01	0.21 ± 0.02
C+	25.66 ± 1.28^a	0.25 ± 0.03^c	1.24 ± 0.15	0.09 ± 0.01	0.20 ± 0.02
C+BRB	29.10 ± 3.31^b	0.19 ± 0.04^a	1.37 ± 0.24	0.09 ± 0.03	0.22 ± 0.01
Note: All data were presented as mean±SD. Different superscripts on the same column showed a significant difference (p < 0.05).

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The analysis of the mice's body weight revealed significant differences (p < 0.05) in the C+BRB group. The mean body weight of the C+ group (25.66 ± 1.28) was significantly lower than that of the C+BRB group (29.10 ± 3.31) (p < 0.05). The weight of colon organs in the C+ group was significantly higher than in the other groups (p < 0.05). There were no significant differences in the average weight of the liver, kidney, and spleen across all groups of mice, indicating that the induction of AOM/DSS in the test group did not significantly affect the weight of these organs.

3.2. Cancers distribution in colon tissue of BALB/C mice

Macroscopic evaluation of nodules in the colon tissue was performed. Visual analysis using the ImageJ application revealed no nodules or cancers in the C− group, whereas these were found in the C+ and C+BRB groups (Figure 1).

The distribution of cancers in the C+ group was significantly higher (p < 0.05) compared to the C+BRB group (Figure 2), with more than a two-fold difference: 5.73 ± 2.93 and 1.65 ± 0.72, respectively. There was no significant difference between the C− group and the C+ BRB group. Cancer nodules in the C+ group spread to almost all parts of the colon tissue, exhibiting various shapes.

Figure 2. Distribution of nodules in the colon tissue of BALB/C mice in each test group. Data are shown in average (n = 8). Different superscripts on the same line showed a significantly different level of 5% (p < 0.05).

DownLoad: Full-Size Img PowerPoint

3.3. Histopathology of BALB/C mice colon

Histopathological analysis of the colon tissue revealed morphological differences between the C− group and the AOM/DSS treatment groups (C+ and C+BRB groups) (Figure 3). Histologically, the C− group exhibited normal colonic mucosa characterized by elongated crypts filled with goblet or epithelial cells. In contrast, the C+ group exhibited severe inflammatory cell infiltration in the lamina propia. The crypts in the C+BRB group were no longer apparent or were missing in the multifocal area, but the damaged epithelial cells were still visible on the tissue surface.

Figure 3. Microscopic images of BALB/C mice colon tissue. C− (negative control group (normal mice), C+ (group without black rice bran + AOM/DSS), C+BRB (black rice bran group + AOM/DSS).

: Inflammation of inflammatory cells:

: damaged cell.

DownLoad: Full-Size Img PowerPoint

3.4. mRNA expression level of PCNA, caspase-3, and caspase-8

Further observation of the effect of BRB on inhibiting the development of colon cancer in BALB/C mice was performed by evaluating PCNA, caspase-3, and caspase-8 markers using qRT-PCR (Table 3). The higher the value of caspase-3 and caspase-8, the higher the level of apoptotic activity in the colon tissue, indicating lower colon cancer activity. PCNA is widely used to detect cancer cells due to its involvement with enzymes in DNA replication.

Table 3. Colonic mRNA expression level by quantitative RT-PCR*.

Groups	PCNA	Caspase-3	Caspase-8
C−	1.00 + 0.21^a	1.00 + 0.18^a	1.00 + 0.51^a
C+	5.22 + 0.80^b	0.36 + 0.15^b	0.13 + 0.03^b
C+BRB	0.58 + 0.09^a	0.91 + 0.20^a	0.51 + 0.18^a
*C− (normal group + standard feed); C+ (induction of AOM/DSS + standard feed); and C+BRB (induction of AOM/DSS + feed modified black rice bran). Data are shown in average + SD (n = 8). Different superscripts on the same line showed a significantly different level of 5% (p < 0.05).

| Show Table

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The observation showed that the C+BRB group had lower PCNA gene expression (0.58 ± 0.09) than the C+ group (5.22 ± 0.80). The administration of fermented BRB extracts up to 0.2% was reported to inhibit colon cancer occurrence through a mechanism that reduces cell proliferation and lesion numbers ^[11]. At the levels of caspase-3 and caspase-8 apoptosis gene expression, the C+BRB group showed higher levels (0.91 ± 0.20 for caspase-3 and 0.51 ± 0.18 for caspase-8) compared to the C+ group. The C+ group, which received standard feed, was unable to suppress cancer development in colon tissue.

4. Discussion

4.1. Animal conditions during cancer development

The body weight result shows that the C− and C+BRB groups tend to have higher weights compared to the C+ group. This may be due to the effects of cancer development in the digestive system. This can be seen in the rate of cancer spread in the C+ group, which is more than in the C+BRB group (Figure 1). The presence of cancer in the digestive system, such as in the colon, will inhibit the digestive system from absorbing nutrients into the body, resulting in a decrease in body weight in animals with cancer ^[13]. This condition is often called cachexia—A condition of malnutrition characterized by weight loss; anemia; metabolic disorders; depression; and changes in acid-base balance, vitamin, and electrolyte levels in the body ^[14].

The higher average body weight in the C+BRB group is also attributable to the absence of a decrease in their feed intake. Feed modified with BRB can potentially inhibit the development of colon cancer in the digestive system. Therefore, the clinical conditions and body weight of the C+BRB group tend to be better. Some bioactive components in BRB actively contribute to preventing colon cancer, namely ferulic acid, anthocyanin, γ-oryzanol, β-sitosterol, tocotrienol/tocopherol, tricin, ρ-coumaric, sinapic, syringic acid, and phytic acid ^[15]. These bioactive components can potentially repair cells damaged by the induction of AOM/DSS carcinogens and suppress cancer development. They can also suppress cell damage in the cytoplasm and cancer cell proliferation ^[16]. This result is consistent with the study on Sprague-Dawley rats conditioned for colon cancer and fed BRB ^[15].

The greater weight of the colon in the C+ group is due to the distribution of nodules that spread to colon tissue. Besides being influenced by the original morphological conditions, the weight of an organ can also be affected by abnormalities such as lumps or nodules, as well as the presence of fluid or solids in the organ ^[17]. Cancer development due to AOM/DSS induction can be observed macroscopically by nodules or lumps in colon organs ^[11].

4.2. Cancer distribution in colon tissue of BALB/C mice

Nodules or cancers are tissue lumps smaller than 0.5 cm. A nodule may contain inflammation or a mixture of tissue and fluid ^[18]. Morphologically, in patients with colon cancer, the distribution of nodules/cancers in the colon will be apparent. Generally, nodules or cancers are an early sign of cancerous tissue growth. C+ group cancer nodules spread across almost all parts of the colonic tissue, exhibiting various shapes such as oval or round. BALB/C mice induced by AOM/DSS carcinogen compounds exhibited the presence of nodules or cancers in the colon ^[11]. The distribution of colonic diarrhea nodules indicates the development of colorectal cancer caused by AOM/DSS carcinogens. The spread of colonic diarrhea nodules is caused by the accumulation of inflammatory cells, resulting in clots or fluid in the colon tissue.

The data obtained from this study shows that the AIN-93M control diet was unable to suppress inflammation and cancer formation induced by the presence of AOM/DSS carcinogens. The cellulose powder component, which is the source of fiber in AIN-93M diet, is believed to be unable to suppress cancer development in mouse colon tissue. This inability may be due to cellulose's relatively low fermentation by bacteria in the colon ^[19], resulting in lower SCFA production. SCFAs are produced by the fermentation of food fiber by lactic acid bacteria (LAB) in the colon, and they can suppress the development of cancer in the colon ^[13,20].

On the other hand, this study showed that feed modified with BRB can inhibit the development of colon cancer by suppressing nodule formation in colon tissue, thus inhibiting cancer cell proliferation. Furthermore, bioactive components such as phenolic compounds in BRB can also suppress cancer cell proliferation and DNA damage in colon tissue ^[21].

4.3. Histopathology of BALB/C mice colon

Histopathological examination of colon tissue in the C+ group reveals cancer development at the dysplasia stage. Dysplasia is a stage of carcinoma characterized by abnormal cell growth, stacked cell nuclei reaching half the thickness of the epithelium, and loss of cell polarity ^[10]. The dysplasia stage is also characterized by loss of crypts and goblet cells, spread of inflammatory cells in the tissue area, visible damage to epithelial cells, and solidification of the tissue by the cell nucleus ^[18].

Histopathological examination of the BRB group showed that the cancer development rate was inhibited by substituting fiber with BRB. This is evidenced by the number of nodules in the colon area, indicating that the BRB group has a lower distribution of nodules compared to the C + group. BRB contains dietary fibers, such as β-glucans, that can be fermented by intestinal microbes, such as LAB. This fermentation can later produce SCFA compounds known to be chemopreventive agents in colon cancer. These SCFAs induce several proapoptotic genes such as caspase-3, caspase-9, and bid, while also inhibiting DNA damage in the cell nucleus ^[22]. It also plays a role in stimulating intestinal epithelial cell multiplication, as these fatty acids are a component of cell membrane phospholipids ^[21]. In addition to fiber, BRB contains some bioactive components that prevent colon cancer. Anthocyanin, the most significant antioxidant in BRB, binds to the active side of the proliferation protein, PCNA, by stopping the progression of cyclin and CDK in the cell cycle ^[23]. Furthermore, ferulic acid and tocopherol in BRB reportedly inhibit the proliferation of cancer cells by inducing apoptotic genes ^[24].

4.4. Proliferation marker (PCNA) and apoptosis markers (caspase-3 and caspase-8)

The result indicates that BRB could reduce the expression of PCNA genes during cancer development. Feed modified with BRB is believed to have an antiproliferative ability by influencing the formation of enzymes that contribute to replication and transcription. This leads to an inhibition of transcription activity in the S phase, leading to cell cycle repair and, ultimately, control of excessive proliferation ^[25]. Some bioactive components, such as ferulic acid, phytate, and oryzanol, which have the potential for antiproliferative effects in BRB, can be increased in number by fermentation. During fermentation, microbes synthesize enzymes that can break down ester bonds and release bound phenolics, thereby increasing the number of free phenolic compounds ^[26].

Caspase-3 and caspase-8 genes have not been maximally induced in the metabolism of C+ group mice, preventing suppression of inflammation and DNA damage caused by carcinogens. BRB can potentially suppress cancer occurrence and increase the expression of caspase-3 and caspase-8 apoptotic genes during cancer development in mice. The presence of phytic acid and ferulic acid in BRB has been shown to increase apoptotic genes such as caspase-3 and inhibit cancer proliferation in TRAP mice ^[27]. Also, it can induce other apoptotic genes, such as pro-caspase 8. Caspase-8, an active form of pro-caspase-8, proteolytically activates caspase-3. Caspase-3 is a caspase effector gene that can alter cell morphology, such as DNA fragmentation, thereby suppressing cancer development. Anthocyanin and ferulic acid in BRB also act as antioxidants that can neutralize free radicals, repairing DNA damage, inducing apoptosis, and suppressing transcription and translation of cancer cells in the proliferation mechanism caused by cancer-forming carcinogens ^[28]. Phenolic compounds promote apoptosis in cancer cells induced by carcinogenic compounds through intrinsic pathways (mitochondria) ^[29]. In addition, butyric acid produced from fiber-soluble food fermentation is reported to be one of the agents that can induce apoptosis in cancer cells. Butyric acid can activate apoptotic genes in the intrinsic pathway through mitochondria by activating caspase-9 and caspase-3 enzymes ^[30].

5. Conclusions

BRB-based feed can potentially inhibit colon cancer development in BALB/C mice induced with AOM/DSS. The fiber and bioactive components of BRB have been shown to inhibit cancer development by reducing cancer distribution in colon tissue, suppressing cancer cell proliferation by decreasing PCNA gene expression, and inducing apoptosis by increasing caspase-3 and caspase-8 gene expression. In vivo testing is advised on BALB/C mice fed with fermented BRB to inhibit colon cancer, alongside further confirmation of indications of decreased cancer spread and proliferation, as well as induction of apoptosis in the intrinsic pathway. Moreover, some apoptotic genes that work on intrinsic pathways, such as Bcl-2 family members, Apaf-1, and caspase 9, can be further investigated to determine their gene expression levels during colon cancer development inhibition.

Use of AI tools declaration

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

Acknowledgments

This study was supported from Ministry of Research, Technology and Higher Education of the Republic of Indonesia (Ristekdikti) and grants from the Bilateral Joint Research Projects of the Japan Society for the Promotion of Science (JSPS), Japan, and the JSPS Core-to-Core Program A (Advanced Research Networks) entitled "Establishment of international agricultural immunology research-core for a quantum improvement in food safety".

Conflict of interest

The authors declare no conflict of interest in this research.

Authors contributions

SB, LN, A, and HS: Conceptualization; YK, FNAD, LN, BPP, and US: Methodology; YK: Research; FNAD, US, NDY, S and SB: Analysis and Data Curation; SB, LN, and BPP: Supervision; SB and YK: Writing Draft Preparation; SB, A, and HS: Writing, Review and Editing. All authors have read and agreed to the published version of the manuscript.

References

[1]	G. Acosta, F. Bersetche, J. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl., 74 (2017), 784–816. https://doi.org/10.1016/j.camwa.2017.05.026 doi: 10.1016/j.camwa.2017.05.026
[2]	G. Acosta, J. P. Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), 472–495. https://doi.org/10.1137/15M1033952 doi: 10.1137/15M1033952
[3]	D. Adak, S. Natarajan, Virtual Element Method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique, Math. Comput. Simul., 172 (2020), 224–243. https://doi.org/10.1016/j.matcom.2019.12.007 doi: 10.1016/j.matcom.2019.12.007
[4]	D. Adak, S. Natarajan, E. Natarajan, Virtual Element Method for semilinear elliptic problems on polygonal meshes, Appl. Numer. Math., 145 (2019), 175–187. https://doi.org/10.1016/j.apnum.2019.05.021 doi: 10.1016/j.apnum.2019.05.021
[5]	R. Agarwal, S. Jain, R. P. Agarwal, Solution of fractional Volterra integral equation and non-homogeneous time fractional heat equation using integral transform of pathway type, Progr. Fract. Differ. Appl., 1 (2015), 145–155. https://doi.org//10.12785/pfda/010301 doi: 10.12785/pfda/010301
[6]	B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, A. Russo, Equivalent projectors for Virtual Element Methods, Comput. Math. Appl., 66 (2013), 376–391. https://doi.org/10.1016/j.camwa.2013.05.015 doi: 10.1016/j.camwa.2013.05.015
[7]	P. Antonietti, L. Beirão da Veiga, G. Manzini, The Virtual Element Method and its applications, Vol. 31, Springer, 2022. https://doi.org/10.1007/978-3-030-95319-5
[8]	P. F. Antonietti, G. Manzini, M. Verani, The fully nonconforming Virtual Element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), 387–407. https://doi.org/10.1142/S0218202518500100 doi: 10.1142/S0218202518500100
[9]	P. F. Antonietti, G. Manzini, M. Verani, The conforming Virtual Element method for polyharmonic problems, Comput. Math. Appl., 79 (2020), 2021–2034. https://doi.org/10.1016/j.camwa.2019.09.022 doi: 10.1016/j.camwa.2019.09.022
[10]	M. Arrutselvi, E. Natarajan, Virtual Element Method for nonlinear convection-diffusion-reaction equation on polygonal meshes, Int. J. Comput. Math., 98 (2020), 1852–1876. https://doi.org/10.1080/00207160.2020.1849637 doi: 10.1080/00207160.2020.1849637
[11]	B. Baeumer, M. Kovács, H. Sankaranarayanan, Higher order Grünwald approximations of fractional derivatives and fractional powers of operators, Trans. Amer. Math. Soc., 367 (2014), 813–834.
[12]	B. Bandrowski, A. Karczewska, P. Rozmej, Numerical solutions to integral equations equivalent to differential equations with fractional time, Int. J. Appl. Math. Comput. Sci., 20 (2010), 261–269. https://doi.org/10.2478/v10006-010-0019-1 doi: 10.2478/v10006-010-0019-1
[13]	L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo, Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes, ESAIM: Math. Modell. Numer. Anal., 50 (2016), 727–747. https://doi.org/10.1051/m2an/2015067 doi: 10.1051/m2an/2015067
[14]	L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo, Virtual Element Method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729–750. https://doi.org/10.1142/S0218202516500160 doi: 10.1142/S0218202516500160
[15]	L. Beirão da Veiga, K. Lipnikov, G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes, SIAM J. Numer. Anal., 49 (2011), 1737–1760. https://doi.org/10.1137/100807764 doi: 10.1137/100807764
[16]	L. Beirão da Veiga, C. Lovadina, G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM: Math. Modell. Numer. Anal., 51 (2017), 509–535. https://doi.org/10.1051/m2an/2016032 doi: 10.1051/m2an/2016032
[17]	L. Beirão da Veiga, G. Manzini, M. Putti, Post processing of solution and flux for the nodal mimetic finite difference method, Numer. Methods Part. Differ. Equ., 31 (2015), 336–363. https://doi.org/10.1002/num.21907 doi: 10.1002/num.21907
[18]	L. Beirão da Veiga, C. Lovadina, A. Russo, Stability analysis for the Virtual Element Method, Math. Models Methods Appl. Sci., 27 (2017), 2557–2594. https://doi.org/10.1142/S021820251750052X doi: 10.1142/S021820251750052X
[19]	L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini, A. Russo, Basic principles of Virtual Element Methods, Math. Models Methods Appl. Sci., 23 (2013), 199–214. https://doi.org/10.1142/S0218202512500492 doi: 10.1142/S0218202512500492
[20]	L. Beirão da Veiga, K. Lipnikov, G. Manzini, The mimetic finite difference method for elliptic problems, Vol. 11, 1 Ed., Springer, 2014. https://doi.org/10.1007/978-3-319-02663-3
[21]	S. C. Brenner, L. Y. Sung, Virtual Element Methods on meshes with small edges or faces, Math. Models Methods Appl. Sci., 28 (2017), 1291–1336. https://doi.org/10.1142/S0218202518500355 doi: 10.1142/S0218202518500355
[22]	F. Brezzi, A. Buffa, K. Lipnikov, Mimetic finite differences for elliptic problems, ESAIM: Math. Modell. Numer. Anal., 43 (2009), 277–295. https://doi.org/10.1051/m2an:2008046 doi: 10.1051/m2an:2008046
[23]	A. Cangiani, V. Gyrya, G. Manzini, The non-conforming Virtual Element Method for the Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3411–3435. https://doi.org/10.1137/15M1049531 doi: 10.1137/15M1049531
[24]	A. Cangiani, O. J. Sutton, V. Gyrya, G. Manzini, Chapter 15: Virtual element methods for elliptic problems on polygonal meshes, In: K. Hormann, N. Sukumar, Generalized barycentric coordinates in computer graphics and computational mechanics, 1 Ed., CRC Press, 2017.
[25]	A. Cangiani, G. Manzini, A. Russo, N. Sukumar, Hourglass stabilization and the Virtual Element Method, Int. J. Numer. Methods Eng., 102 (2015), 404–436. https://doi.org/10.1002/nme.4854 doi: 10.1002/nme.4854
[26]	A. Cangiani, G. Manzini, O. J. Sutton, Conforming and nonconforming Virtual Element Methods for elliptic problems, IMA J. Numer. Anal., 37 (2016), 1317–1354. https://doi.org/10.1093/imanum/drw036 doi: 10.1093/imanum/drw036
[27]	M. Caputo, Linear models of dissipation whose Q is almost frequency Ⅱ, Geophysi. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[28]	L. Chen, J. Huang, Some error analysis on Virtual Element Methods, Calcolo, 55 (2017), 5. https://doi.org/10.1007/s10092-018-0249-4 doi: 10.1007/s10092-018-0249-4
[29]	B. A. De Dios, K. Lipnikov, G. Manzini, The nonconforming Virtual Element method, ESAIM: M2AN, 50 (2016), 879–904. https://doi.org/10.1051/m2an/2015090 doi: 10.1051/m2an/2015090
[30]	Z. M. Dar, M. Arrutselvi, G. Manzini, S. Natarajan, Analytical and numerical methods for solving fractional-order partial differential equations and the virtual element method, submitted for publication. Available form: https://ssrn.com/abstract = 4913214.
[31]	Z. M. Dar, M. Chandru, A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes, Alex. Eng. J., 106 (2024), 611–619. https://doi.org/10.1016/j.aej.2024.08.050
[32]	V. J. Ervin, J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Part. Differ. Equ., 22 (2006), 558–576. https://doi.org/10.1002/num.20112 doi: 10.1002/num.20112
[33]	A. Esen, Y. Ucar, N. Yagmurlu, O. Tasbozan, A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations, Math. Modell. Anal., 18 (2013), 260–273. https://doi.org/10.3846/13926292.2013.783884 doi: 10.3846/13926292.2013.783884
[34]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Publishing Company, Inc., 2016.
[35]	B. Jin, R. Lazarov, Z. Zhou, A Petrov-Galerkin finite element method for fractional convection-diffusion equations, SIAM J. Numer. Anal., 54 (2015), 481–503. https://doi.org/10.1137/140992278 doi: 10.1137/140992278
[36]	B. Jin, B. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. https://doi.org/10.1137/16M1089320 doi: 10.1137/16M1089320
[37]	D. Kumar, S. Chaudhary, V. V. K. Srinivas Kumar, Fractional Crank-Nicolson-Galerkin finite element scheme for the time-fractional nonlinear diffusion equation, Numer. Methods Part. Differ. Equ., 35 (2019), 2056–2075. https://doi.org/10.1002/num.22399 doi: 10.1002/num.22399
[38]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[39]	D. Li, H. L. Liao, W. Sun, J. Wang, J. Zhang, Analysis of $L1$ -Galerkin FEMs for time-fractional nonlinear parabolic problems, Commun. Comput. Phys., 24 (2018), 86–103. https://doi.org/10.4208/cicp.OA-2017-0080 doi: 10.4208/cicp.OA-2017-0080
[40]	K. Lipnikov, G. Manzini, High-order mimetic methods for unstructured polyhedral meshes, J. Comput. Phys., 272 (2014), 360–385.
[41]	K. Lipnikov, G. Manzini, M. Shashkov, Mimetic finite difference method, J. Comput. Phys., 257 (2014), 1163–1227. https://doi.org/10.1016/j.jcp.2013.07.031 doi: 10.1016/j.jcp.2013.07.031
[42]	G. Manzini, K. Lipnikov, J. D. Moulton, M. Shashkov, Convergence analysis of the mimetic finite difference method for elliptic problems with staggered discretizations of diffusion coefficients, SIAM J. Numer. Anal., 55 (2017), 2956–2981. https://doi.org/10.1137/16M1108479 doi: 10.1137/16M1108479
[43]	G. Manzini, A. Mazzia, Conforming virtual element approximations of the two-dimensional Stokes problem, Appl. Numer. Math., 181 (2022), 176–203. https://doi.org/10.1016/j.apnum.2022.06.002 doi: 10.1016/j.apnum.2022.06.002
[44]	G. Manzini, A. Mazzia, A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem, J. Comput. Dyn., 9 (2022), 207–238. https://doi.org/10.3934/jcd.2021020 doi: 10.3934/jcd.2021020
[45]	G. Manzini, A. Russo, N. Sukumar, New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci., 24 (2014), 1621–1663. https://doi.org/10.1142/S0218202514400065 doi: 10.1142/S0218202514400065
[46]	L. Mascotto, III–conditioning in the virtual element method: stabilizations and bases, Numer. Methods Part. Differ. Equ., 34 (2018), 1258–1281. https://doi.org/10.1002/num.22257 doi: 10.1002/num.22257
[47]	R. Metzler, J. H. Jeon, A. G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys., 16 (2014), 24128–24164. https://doi.org/10.1039/C4CP03465A doi: 10.1039/C4CP03465A
[48]	R. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: a fractional calculus approachh, J. Chem. Phys., 103 (1995), 7180–7186. https://doi.org/10.1063/1.470346 doi: 10.1063/1.470346
[49]	K. Nishimoto, Fractional calculus: integrations and differentiations of arbitrary order, Descartes Press, 1984.
[50]	K. Oldham, J. Spanier, Chapter 3: fractional derivatives and integrals: definitions and equivalences, In: The fractional calculus theory and applications of differentiation and integration to arbitrary order, Academic Press, 111 (1974), 45–60.
[51]	I. Podlubny, Fractional differential equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Vol. 198. Elsevier, 1999.
[52]	R. Scherer, S. L. Kalla, Y. Tang, J. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902–917. https://doi.org/10.1016/j.camwa.2011.03.054 doi: 10.1016/j.camwa.2011.03.054
[53]	P. Sebah, X. Gourdon, Introduction to the gamma function, 2002. Available form: https://api.semanticscholar.org/CorpusID: 14758536.
[54]	E. Sousa, Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., 228 (2009), 4038–4054. https://doi.org/10.1016/j.jcp.2009.02.011 doi: 10.1016/j.jcp.2009.02.011
[55]	V. Thomée, Galerkin finite element methods for parabolic problems, Vol. 25, Springer, 2006. https://doi.org/10.1007/3-540-33122-0
[56]	A. M. Wazwaz, Linear and nonlinear integral equations, 1 Ed., Springer, 2011. https://doi.org/10.1007/978-3-642-21449-3
[57]	Y. Zhang, M. Feng, The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data, Comput. Math. Appl., 110 (2022), 1–18. https://doi.org/10.1016/j.camwa.2022.01.033 doi: 10.1016/j.camwa.2022.01.033
[58]	Y. Zhang, M. Feng, A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers, Appl. Math. Comput., 436 (2023), 127509. https://doi.org/10.1016/j.amc.2022.127509 doi: 10.1016/j.amc.2022.127509
[59]	J. Zhao, S. Chen, B. Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671–1687. https://doi.org/10.1142/S021820251650041X doi: 10.1142/S021820251650041X

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