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Approximate solution of painlevé equation i by natural decomposition method and laplace decomposition method

Amir Muhammad, Haider Jamil Abbase, Ahmad Shahbaz, Ashraf Asifa, Gul Sana

Acta Mechanica et Automatica

2023

Vol. 17, no. 3

417--422

The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply. Novelty: One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.

Development and validation of RP HPLC method for the estimation of methyl paraben sodium and propyl paraben sodium in iron protein succinylate syrup

Javed Muhammad Faisal, Zahra Manzar, Javed Iqra, Ahmad Shahbaz, Jabeen Tayyaba, Ahmad Muhammad

Acta Chromatographica

2023

Vol. 35, no. 1

52--59

A new precise, selective and reliable reversed phase high performance liquid chromatographic (RP-HPLC) method has been developed and validated for the determination of Methyl paraben sodium (MPS) and Propyl paraben sodium (PPS) (preservatives) in Iron protein succinylate syrup. Optimized conditions were; Methanol: Water (65: 35) as mobile phase, UV/Vis detector at the wavelength of 254 nm and flow rate was set at 1.3 ml min⁻¹. By applying the set of conditions, separation of components was carried out in less than 7 min for both the analytes. The method was validated according to International conference of Harmonization (ICH) guidelines and the analytical characteristic parameters of validation included specificity, limit of detection (LOD), limit of quantification, linearity, accuracy, precision and robustness were evaluated. The calibration curve was found to be linear in the range of 0.045 mg mL⁻¹ to 0.075 mg mL⁻¹ for Methyl paraben sodium and 0.015 mg mL⁻¹ to 0.025 mg mL⁻¹ for propyl paraben sodium with a correlation coefficient r2 > 0.999. Accuracy; reported as percentage recovery was found to be in the range of 98.71%–101.64% for Methyl paraben sodium and 99.85%–101.47% for Propyl paraben sodium at 80%, 100% and 120% concentration for both the analytes. The proposed method was found to be precise and robust when evaluated by variations in wavelength, mobile phase composition, temperature and analyst. The limit of detection (LOD) was found 0.001 mg mL⁻¹ (3 ppm) for Methyl paraben sodium and 0.001 mg mL⁻¹ (1 ppm) for Propyl paraben sodium.