Wavelets operational methods for fractional differential equations and systems of fractional differential equations

In this thesis, new and effective operational methods based on polynomials and wavelets for the solutions of FDEs and systems of FDEs are developed. In particular we study one of the important polynomial that belongs to the Appell family of polynomials, namely, Genocchi polynomial. This polynomial has certain great advantages based on which an effective and simple operational matrix of derivative was first derived and applied together with collocation method to solve some singular second order differential equations of Emden-Fowler type, a class of generalized Pantograph equations and Delay differential systems. A new operational matrix of fractional order derivative and integration based on this polynomial was also developed and used together with collocation method to solve FDEs, systems of FDEs and fractional order delay differential equations. Error bound for some of the considered problems is also shown and proved. Further, a wavelet bases based on Genocchi polynomials is also constructed, its operational matrix of fractional order derivative is derived and used for the solutions of FDEs and systems of FDEs. A novel approach for obtaining operational matrices of fractional derivative based on Legendre and Chebyshev wavelets is developed, where, the wavelets are first transformed into corresponding shifted polynomials and the transformation matrices are formed and used together with the polynomials operational matrices of fractional derivatives to obtain the wavelets operational matrix. These new operational matrices are used together with spectral Tau and collocation methods to solve FDEs and systems of FDEs

Seasonal variability of anthropogenic indices of PAHs in sediment from the Kuala Selangor River, west coast Peninsular Malaysia

Rapid increase in industrialization and urbanization in the west coast of Peninsular Malaysia has led to the intense release of petroleum and products of petroleum into the environment. Surface sediment samples were collected from the Selangor River in the west coast of Peninsular Malaysia during four climatic seasons and analyzed for PAHs and biomarkers (hopanes). Sediments were soxhlet extracted and further purified and fractionated through first and second step column chromatography. A gas chromatography–mass spectrometry (GC–MS) was used for analysis of PAHs and hopanes fractions. The average concentrations of total PAHs ranged from 219.7 to 672.3 ng g−1 dw. The highest concentrations of PAHs were detected at 964.7 ng g−1 dw in station S5 in the mouth of the Selangor River during the wet inter-monsoonal season. Both pyrogenic and petrogenic PAHs were detected in the sediments with a predominance of the former. The composition of hopanes was homogeneous showing that petroleum hydrocarbons share an identical source in the study area. Diagnostic ratios of hopanes indicated that some of the sediment samples carry the crankcase oil signature

Evaluation of distribution and sources of sewage molecular marker (LABs) in selected rivers and estuaries of Peninsular Malaysia

This is the first extensive report on linear alkylbenzenes (LABs) as sewage molecular markers in surface sediments collected from the Perlis, Kedah, Merbok, Prai, and Perak Rivers and Estuaries in the west of Peninsular Malaysia. Sediment samples were extracted, fractionated, and analyzed using gas chromatography mass spectrometry (GC-MS). The concentrations of total LABs ranged from 68 to 154 (Perlis River), 103 to 314 (Kedah River), 242 to 1062 (Merbok River), 1985 to 2910 (Prai River), and 217 to 329 ng g−1 (Perak River) dry weight (dw). The highest levels of LABs were found at PI3 (Prai Estuary) due to the rapid industrialization and population growth in this region, while the lowest concentrations of LABs were found at PS1 (upstream of Perlis River). The LABs ratio of internal to external isomers (I/E) in this study ranged from 0.56 at KH1 (upstream of Kedah River) to 1.35 at MK3 (Merbok Estuary) indicating that the rivers receive raw sewage and primary treatment effluents in the study area. In general, the results of this paper highlighted the necessity of continuation of water treatment system improvement in Malaysia

Monitoring of linear alkyl benzenes (LABs) in riverine and estuarine sediments in Malaysia

The aim of this a pioneering research is to investigate linear alkylbenzenes (LABs) as biomarkers of sewage pollution in sediments collected from four rivers and estuaries of the south and east of Peninsular Malaysia. The sediment samples went through soxhlet extraction, two-step column chromatography purification, fractionation and gas chromatography–mass spectrometry (GC–MS) analysis. Principal component analysis (PCA) with multivariate linear regression (MLR) was used as well for source apportionment of LABs. The results of this study showed that total LAB concentration was 36–1196 ng g−1dw. The internal to external isomer ratios (I/E ratio) of LABs were from 0.56 to 3.12 indicated release of raw sewage and primary and secondary effluents into the environment of south and east of Peninsular Malaysia. Our research supported that continuous monitoring of sewage pollution to limit the environmental pollution in riverine and estuarine ecosystem

Assessment of Sewage Molecular Markers in Port Dickson Coast and Kim Kim River with Sediment Linear Alkylbenzenes

The present study aimed to determine linear alkylbenzenes (LABs) concentrations as organic molecular marker for sewage pollution in the sediment samples collected from Coast of Port Dickson and Kim Kim River, Peninsular Malaysia. The adverse effects of anthropogenic inputs into the rivers and coastal environment could be detected by molecular organic markers such as LABs. The sediments were processed; their sources were identified and tested by gas chromatography-mass spectrometry (GC-MS). The significance of the differences among sampling stations for LAB concentrations and distribution at p < 0.05 was performed by analysis of variance and Post Hoc Tests, LSD procedures (ANOVA) and Pearson correlation coefficient. LABs indices which include internal to external (I/E) congeners, long to short chains L/S and homologs C13/C12 were used to identify the sewage treatment and degradation levels. Results of this study are statistically uncovered that the range of RLABs concentration in the investigated locations was between 112.0; 88.3 and 256.0; 119.0 ng�g1 dw, respectively. There was significant difference (p < 0.05) of LAB homologs with high percentage of C13-LAB homologs along sampling locations. The calculated LAB ratios (I/E) were within the range between 2.0; 1.7 and 4.1, 2.0, demonstrated that, the treated effluents from primary and secondary sources were discharged to the study areas. The degradation of LABs was 40–64% and 34–38% in the studied locations. The findings of this study suggested the powerfully indicators of LABs in tracing anthropogenic sewage contamination and the necessity of continuing wastewater treatment system improvement

Assessment of Sewage Molecular Markers: Linear Alkylbenzenes in Sediments of an Industrialized Region in Peninsular Malaysia

In this study, the use of linear alkylbenzenes (LABs) was employed to pinpoint the sources of human activity that cause detrimental impacts on the coastal environment and river ecosystems. LABs were detected using GC–MS in sediment samples assembled from Kim Kim River (KKR) and the Port Dickson coast (PDC). To assess the significance of variations in the distribution and concentrations of LABs across the sampling sites, this study utilized several statistical techniques such as post hoc tests, LSD techniques, analysis of variance (ANOVA), and the Pearson correlation coefficient using a significance level of p 13 and C12, and long-to-short-chain (L/S) ratios. The results revealed that the LAB concentrations varied between 88.3 and 112 ng/g dw in KKR and 119 to 256 ng/g dw in the PDC. Most of the surveyed areas exhibited a substantial count of C13–LABs homologs that displayed a significant difference (p < 0.05). The I/E ratios ranged from 1.7 to 2.0 in KKR and from 2.0 to 4.1 in the PDC, suggesting that the effluents originated from sources associated with the physical phase and biological phase in wastewater treatment systems (WWTSs). The results revealed that the degradation of LABs varied between 34% and 38% in KKR and between 40% and 64% in the PDC. This study underscores the importance of ongoing improvements to WWTSs and emphasizes the potential of LABs as indicators for monitoring wastewater contamination