In this paper, Chebyshev wavelets method is designed and presented to approximate the two-dimensional (2D) telegraph equation of hyperbolic type. Firstly, we transform the original problem into the equivalent integro-partial differential equations (integro-PDEs) containing the initial and boundary conditions. Thereafter, the operational matrices of integration and derivative of Chebyshev wavelets reduce the integro-PDEs into generalized Sylvester equations which are solved by Krylov subspace iterative method. The convergence analysis associated to the function approximation, and error estimation are also investigated. The presented procedure is found to be very effective and accurate, which is confirmed by the outcome of three test examples. The outcome of numerical results of errors achieved by the presented method is compared with some earlier works. The technique can be applied for higher dimension problems also, which is one of the key features of this technique.

In this paper, we study fractional reaction–diffusion equation using Jacobi spectral method. Reaction–diffusion equation is used as model for spatial effects in ecology. In this method, the reaction–diffusion equation is changed to the systems of equations. Convergence analysis for the proposed spectral method is established from theoretical as well as numerical point of view. Some numerical examples are solved using proposed numerical method showing the feasibility of the method. The efficiency of the spectral method is shown by listing CPU time of computation.

The decline in performance of sugarcane either due to non-availability of suitable planting material or negligence in nutrient management is increasing at an alarming rate. Therefore, a field experiment was conducted for two consecutive years in Bihar, India to study the effect of different planting material and integrated nutrient management strategies on performance and yield of sugarcane. The experiment was conducted in a factorial randomized block design replicated thrice with four planting materials (single budded sett, double budded sett, three budded sett and tissue culture plantlets) and seven integrated nutrient management strategies [control, recommended dose of fertilizer (RDF), soil test-based RDF, RDF + 25% N through pressmud + ZnSO4, soil test-based RDF+ 25% N through pressmud + ZnSO4, RDF + 25% N through FYM + ZnSO4 and soil test-based RDF+ 25% N through FYM + ZnSO4]. Crop growth attributes viz. leaf area index, plant height, tillers, total chlorophyll content was found maximum with tissue culture plantlets followed by three budded setts while the minimum in single budded sett. During both the years of the study, three budded setts increased the cane yield by 33.9 and 34.5% over single budded sett respectively. Application of soil test-based RDF + 25% N through pressmud + ZnSO4 significantly enhanced physio-agronomic performance and further, the sugar yield by 39.1, 13.3 and 38.7, 11.6 % as compared to RDF and soil test-based RDF in first and second year of the study respectively. Hence, it was concluded that three budded sett along with soil test-based RDF + 25% N through pressmud + ZnSO4 of 25 kg/ha can increase overall growth and productivity of sugarcane.

Almost complete removal (>99 % within 20 mts) of two cationic dyes Brilliant Cresyl Blue (BCB) and Fuchsin Basic (FB) from the aqueous solution has been studied using an eco-friendly and inexpensive acid activated carbon synthesized from locally available ginger waste. Scanning electron microscopy-energy dispersive X-ray analysis (SEM-EDX), X-ray diffraction analysis (XRD), and Fourier-transform infrared spectroscopy (FT-IR) techniques were used to study morphological and spectral changes in the material. Langmuir model gave the best fit for adsorption data obtained in case of BCB while data for FB dye found best fit for Freundlich model. Adsorption kinetics of the GWAC was best described with pseudo-second-order kinetics.

Abstract This paper deals with a class of Bratu’s type, Troesch’s and nonlocal elliptic boundary value problems. Due to strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems. Here we solve these classes of important equations using the Chebyshev spectral collocation method. We have provided the convergence of the proposed approximate method. The trueness of the method is shown by applying it to some illustrative examples. Results are compared with some known methods to highlight its neglectable error and high accuracy.

Acid activated carbon derived from the pulp of Citrus limetta (mosambi) was employed as completely biodegradable and highly economical adsorbent for the remediation of chromium(VI) from water. Prepared material was characterized by different techniques like PXRD, SEM, EDX and FTIR analysis. It was found that 0.1 g of activated carbon obtained from Citrus limetta was capable to almost completely adsorb hexavalent chromium (98.3 %) within 90 min at pH 2 and the adsorbent can efficiently be used up to five cycles simply by eluting with acid or alkali. Kinetic study showed that pseudo-2nd-order kinetic model gave the best fit while higher correlation factor was obtained with Freundlich isotherm model. Thermodynamic parameters confirmed endothermic and spontaneous nature of adsorption.

In this chapter, the collocation method is extended for a new class of fractional advection diffusion equation (GFADE). The GFADE is defined in terms of the generalized fractional derivatives(GFD) using scale and weight functions. The numerical solution of such GFADE is derived via a collocation method using Legendre polynomials. Convergence and error analysis of polynomial approximation are obtained. Numerical examples with homogeneous and nonhomogeneous boundary conditions are presented to confirm the theoretical findings. Through comparative study of the simulation results, we find that our proposed numerical method is simple, efficient and easy to implement.

For the past two years, the entire world has been fighting against the COVID-19 pandemic. The rapid increase in COVID-19 cases can be attributed to several factors. Recent studies have revealed that changes in environmental temperature are associated with the growth of cases. In this study, we modeled the monthly growth rate of COVID-19 cases per million infected in 126 countries using various growth curves under structural equation modeling. Moreover, the environmental temperature has been introduced as a time-varying covariate to enhance the performance of the models. The parameters of growth curve models have been estimated, and accordingly, the results are discussed for the affected countries from August 2020 to July 2021.

This paper deals with a class of Bratu's type, Troesch's, and nonlocal elliptic boundary value problems arising in the heat transfer process. Due to the strong non-linearity and presence of parameter delta, it is very difficult to solve these problems analytically as well as numerically. By using the Jacobi spectral collocation method, these problems are solved fruitfully. We have shown the numerical as well as theoretical convergence of the suggested scheme. Numerical results are presented through figures and tables, demonstrating the accuracy of the scheme. Results are compared with some known methods to highlight its neglectable error.

In this paper we introduce an efficient and new numerical algorithm for evaluating a pseudo differential operator. The proposed algorithm is time saving and fruitful. The theoretical as well as numerical error estimation of the algorithm is established, together with its stability analysis. We have provided numerical illustrations and established that the numerical findings echo the analytical findings. The proposed technique has a convergence rate of order three. CPU time of computation is also listed. Trueness of numerical findings are validated using figures.

In present paper, we solve fractional advection-dispersion equation (ADE) using Jacobi collocation method. This equation appears in the transport of solutes in ground water and soils. Using Jacobi spectral collocation method the fractional ADE is converted into systems of nonlinear algebraic equations whose solution leads the approximate solution for this equation. We provide the convergence analysis of the proposed approximate method. The applicability of proposed method is shown by testing it on some illustrative examples. Numerical results are shown using figures. Absolute error figures show the accuracy of the proposed method. CPU time is also listed in tabular form to show efficiency of the proposed method.

Computer viruses have become a serious problems as software and hardware technology are developed. Due to the complex codes of virus programs, it is difficult to detect and remove these viruses. A lot of work has been devoted to study how to remove the activities of these harmful viruses. One way to fight the propagation of these viruses is to establish defensive policies similar to those used in epidemiological studies. In this paper, we will analyze the fractional SIRA model, which is a modified version of the SIR (Susceptible-Infected-Removed) model, and describe how its parameters are connected to network characteristics. We will solve the modified fractional epidemiological model using a numerical scheme based on the approximation of fractional derivatives and discretization of the domain. The stability of proposed model is shown and numerical results are simulated. CPU time of computation is listed in tabular form.

Integration and integral transforms are playing an important role in various branches of applied mathematics and physical sciences. In the present article, first we define modified I-function of two variables and after that we will set up and prove two double integral formulas involving a product of I-function with two other special functions. In the last, we have given some integral transforms of these two double integral formulas and it is our belief that the results derived in this paper are new in literature.

This chapter describes construction of a numerical scheme to obtain the numerical solution of fractional differential equations (FDEs). The scheme is based on the generalized Taylor's series and standard Runge–Kutta method, and FDEs with fractional derivatives in the Caputo sense are considered. FDEs have wide areas of application in science and engineering, such as convection-diffusion problems, plasma physics, mechanics, polymer physics and rheology, diffraction theory, and regular variations in thermodynamics. The family of Runge–Kutta (RK) formulae is traditionally used as an important numerical solver for the initial value problem (IVP) in integer-order differential equations. This chapter proposes a development of the RK-type method suitable for finding a numerical solution to IVPs in fractional-order differential equations. The order conditions for the fractional Runge–Kutta (FRK) method are derived and a two-stage FRK method constructed. Numerical experiments governing the applications of FDEs in physical problems are given to ensure the efficiency and utility of the proposed scheme, with results compared in tabular and graphical form. The numerical solutions reported show good agreement with the exact values, and errors are found to be relatively small.

This paper deals with fractional model of Bloch equation in Nuclear Magnetic Resonance (NMR). The applications of NMR are magnetic resonance imaging (MRI) for medical diagnosis, magnetic resonance microscopy (MRM) in research settings, chemists can determine the structure of many compounds and for analysing expansive biological samples like nucleic acids, RNA, DNA and proteins. Here, we present a numerical algorithm using Chebyshev polynomial of third kind for the numerical solution of integer and fractional order Bloch equation. Entire trajectory of magnetization is shown in 3D for integer and fractional order Bloch equation. The dynamic of magnetization is also shown for integer and fractional order relaxation. Obtained numerical results are simulated with known solutions. We also compared results with known solutions. Errors tables are used to show the accuracy of the method.