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 improvemen

Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs

Numerical solution of fractional diffusion wave equation and fractional klein–gordon equation via two-dimensional genocchi polynomials with a ritz–galerkin method

In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples

Poly-Genocchi polynomials and its applications

In this paper, we discussed some new properties on the newly defined family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the first time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is efficient in solving various problems

B-Splines Based Finite Difference Schemes For Fractional Partial Differential Equations

Fractional partial differential equations (FPDEs) are considered to be the extended formulation of classical partial differential equations (PDEs). Several physical models in certain fields of sciences and engineering are more appropriately formulated in the form of FPDEs. FPDEs in general, do not have exact analytical solutions. Thus, the need to develop new numerical methods for the solutions of space and time FPDEs. This research focuses on the development of new numerical methods. Two methods based on B-splines are developed to solve linear and non-linear FPDEs. The methods are extended cubic B-spline approximation (ExCuBS) and new extended cubic B-spline approximation (NExCuBS). Both methods have the same basis functions but for the NExCuBS, a new approximation is used for the second order space derivative

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

In this paper, two-dimensional Genocchi polynomials and the Ritz&ndash;Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein&ndash;Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz&ndash;Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples