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A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives

  • In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo–Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo–Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.

    Citation: Abdon Atangana, Seda İğret Araz. A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives[J]. AIMS Mathematics, 2023, 8(11): 27309-27327. doi: 10.3934/math.20231397

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  • In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo–Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo–Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.



    Defined as any degree of glucose intolerance with onset during pregnancy, gestational diabetes mellitus (GDM) is a temporary condition that predisposes pregnant women to type 2 diabetes [1],[2]. Prenatal screenings are used to diagnose GDM, a condition that is present when blood glucose levels are above normal but below diagnostic value for diabetes [3]. According to International Diabetes Federation's statistics, 21.3 million of live births (16.2%) had some form of hyperglycaemia in pregnancy [4], and globally GDM accounts for up to 90% of cases of hyperglycaemia during pregnancy (4,5). Collectively, low- and middle-income countries accounted for more than 90% of cases of hyperglycaemia in pregnancy [5]. Middle-Eastern and North African regions recorded highest prevalence of GDM (12.9%), followed by South-East Asian and Western Pacific regions (11.7% respectively) [6].

    Most common risk factors for GDM were older age, obesity, excessive pregnancy weight gain, family history of diabetes, history of GDM and previous history of poor obstetric outcomes (macrosomia and congenital anomalies) [7]. In addition, a recently published systematic review and meta-analysis reported pregnancy-induced hypertension, polycystic ovarian syndrome, history of abortion, and preterm delivery or abortion to be associated with the risk for GDM [8].

    With every 1 in 7 live births is affected by GDM [4], its consequences are of public health concern. Although GDM-caused proportions of maternal and perinatal deaths, and obstructed births are unknown [7], GDM increases the risk of various adverse outcomes for the mother and child during pregnancy, childbirth and post-delivery [9]. Women with GDM have higher risks for pregnancy-related hypertension, pre-eclampsia spontaneous abortion, preterm labour and caesarean section [10],[11]. Studies have also shown these women to be at a higher risk of developing type 2 diabetes, metabolic syndrome, and cardiovascular, renal and ophthalmic diseases [2],[6],[11]. Short-term adverse outcomes for infants born to women with GDM include macrosomia, hypoglycaemia, polycythemia, cardiac complications, neurological impairment [12]. Risks of future development of obesity, type 2 diabetes and metabolic syndrome also increase in offspring of women with GDM [7],[13].

    Evidence from varying populations suggest that 70–85% of women diagnosed with GDM can achieve normal glycaemic levels with lifestyle modification alone [14]. The value of lifestyle modification as the first-line strategy for the prevention and management of GDM is therefore, well-recognised. Thus, measures to improve diet and physical activity are important prior to, during and post-delivery in GDM pregnancies [15].

    Medical Nutrition Therapy (MNT) is commonly described as the ‘cornerstone’ of GDM management. Recent American Diabetes Association Standards of Medical Care in Diabetes (2019) recommends that a diagnosis of GDM is followed by a treatment that ‘starts with MNT, physical activity, and weight management’. The goal of this treatment is to (i) support maternal, placental, and foetal metabolic needs [16] while achieving targets recommended for maternal weight gain, glycaemic control and foetal development and (ii) to prevent long-term complications [14],[17]. The benefits of MNT in managing GDM pregnancies cannot be understated since effective management ensures the long-term health for these mothers and their infants.

    Generally, MNT for GDM mothers includes an individualized diet plan that maintains adequate nutrition to facilitate appropriate weight gain and optimises carbohydrate consumption to manage maternal glycaemia within an acceptable range [15],[18]. The former goal is achieved ideally through ensuring a daily calorie intake in the range of 1800–2000 kcal. The latter is attempted through achieving a macronutrient distribution to account for approximately 50–60%, 15–20%, and 25–30% of daily energy intake from carbohydrate, protein, and fat intakes respectively [15]. Dietary quality indicators such as adequate intake of fruits, green leafy vegetables, poultry, fish and nuts are also known to be beneficial. Additionally, reducing postprandial glycaemia in GDM pregnancies with low glycaemic index (GI) is also associated with reduced the prevalence of maternal insulin use and reduced central adiposity in women born to GDM mothers [15],[18]. Therefore, current MNT practice ensures appropriate dietary quantity and quality through individualised diet counselling during pregnancy [15].

    Interestingly, while MNT has long been recognised as a key to GDM treatment, experts lament that current MNT advice lacks sufficient scientific substantiation and it is feared to be “non–evidence-based, fragmented, and inconsistent” [16]. Therefore, the growing momentum to build evidence in this area is more than justified. A recent meta-analysis on the effectiveness of MNT in the GDM management found that available evidence is limited by the small sample size and short duration of the intervention trials in this target population. The authors emphasised the need for holistic evaluation of nutrient quality and quantity, and dietary patterns in GDM management. Specifically, an urgent need for well-designed and sufficiently powered dietary randomised-controlled trials in low and middle-income countries, where the long-term and inter-generational consequences of GDM create the heaviest burden, was identified [19]. Furthermore, there is very little specific advice for women at risk for GDM or for women with prior GDM. Current recommendations for these populations are built on the premise of body weight management and its beneficial effect in preventing the deterioration of glucose tolerance.

    This AIMS Medical Science special issue on “Diet in Gestational Diabetes” synthesises some interesting evidence in this much needed area of maternal nutrition. Our final selection of papers for this Special Issue presents three interesting reviews, giving the readers an opportunity to be aware of possible areas of research ranging from epidemiology to clinical nutrition to functional foods. Misra et al evaluate the existing evidence for dietary patterns, diet quality and micronutrient composition in GDM prevention and management across populations that vary in their socio-economic status. Mahadzir et al in their review evaluate the evidence for lifestyle interventions in improving maternal and foetal outcomes. Gulati et al explore the potential for development of novel functional foods from mushroom for prevention and treatment of GDM. Collectively these reviews showcase the scope for further research in the area of nutrition in the prevention and management of GDM. We believe that these articles will be useful to readers who are working in the area of prevention, treatment and management of acute and long-term complications of GDM.



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    [6] E. A. Coddington, N. Levinson, T. Teichmann, Theory of ordinary differential equations, New York: McGraw-Hill, 1956. https://doi.org/10.1063/1.3059875
    [7] L. Euler, Institutiones calculi integrals, Legare Street Press, 1768.
    [8] Euler, Foundations of differential calculus, New York: Springer-Verlag, 2000.
    [9] B. Ghanbari, K. S. Nisar, Some effective numerical techniques for chaotic systems involving fractal-fractional derivatives with different laws, Front. Phys., 8 (2020), 192. https://doi.org/10.3389/fphy.2020.00192 doi: 10.3389/fphy.2020.00192
    [10] E. Hairer, G. Wanner, S. P. Nørsett, Solving ordinary differential equations Ⅰ: Nonstiff problems, Berlin: Springer, 1993. https://doi.org/10.1007/978-3-540-78862-1
    [11] X. Zheng, H. Wang, H. Fu, Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative, Chaos Soliton. Fract., 138 (2020), 109966. https://doi.org/10.1016/j.chaos.2020.109966 doi: 10.1016/j.chaos.2020.109966
    [12] X. Guo, X. Zheng, Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: Regularity analysis and uniqueness of determining variable order, Z. Angew. Math. Phys., 74 (2023), 64. https://doi.org/10.1007/s00033-023-01959-1 doi: 10.1007/s00033-023-01959-1
    [13] D. W. Jordan, P. Smith, Nonlinear ordinary differential equations: Problems and solutions: A sourcebook for scientists and engineers, Oxford: Oxford University Press, 2007.
    [14] Q. Lai, A. Akgul, C. Li, G. Xu, Ü. Çavuşoğlu, A new chaotic system with multiple attractors: Dynamic analysis, circuit realization and S-Box design, Entropy, 20 (2018), 12. https://doi.org/10.3390/e20010012 doi: 10.3390/e20010012
    [15] J. Loustau, Numerical differential equations: Theory and technique, ODE methods, finite differences, finite elements and collocation, World Scientific, 2016. https://doi.org/10.1142/9770
    [16] W. Romberg, Vereinfachte numerische integration, Det Kongelige Norske Videnskabers Selskabs, 28 (1955), 30–36.
    [17] L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. Royal Soc. A, 210 (1911), 307–357. https://doi.org/10.1098/rsta.1911.0009 doi: 10.1098/rsta.1911.0009
    [18] E. Süli, D. F. Mayers, An introduction to numerical analysis, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511801181
    [19] L. W. Neustadt, On the solutions of certain integral-like operator equations, Existence, uniqueness and dependence theorems, Arch. Ration. Mech. Anal., 38 (1970), 131–160. https://doi.org/10.1007/BF00249976 doi: 10.1007/BF00249976
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