Modeling the effects of insecticides and external efforts on crop production

In this paper a nonlinear mathematical model is proposed and analyzed to understand the effects of insects, insecticides and external efforts on the agricultural crop productions. In the modeling process, we have assumed that crops grow logistically and decrease due to insects, which are wholly dependent on crops. Insecticides and external efforts are applied to control the insect population and enhance the crop production, respectively. The external efforts affect the intrinsic growth rate and carrying capacity of crop production. The feasibility of equilibria and their stability properties are discussed. We have identified the key parameters for the formulation of effective control strategies necessary to combat the insect population and increase the crop production using the approach of global sensitivity analysis. Numerical simulation is performed, which supports the analytical findings. It is shown that periodic oscillations arise through Hopf bifurcation as spraying rate of insecticides decreases. Our findings suggest that to gain the desired crop production, the rate of spraying and the quality of insecticides with proper use of external efforts are much important

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors

A fourth-order arithmetic average compact finite-difference method for nonlinear singular elliptic PDEs on a 3D smooth quasi-variable grid network

The analysis of nonlinear elliptic PDEs representing stationary convection-dominated diffusion equation, Sine-Gordon equation, Helmholtz equation, and heat exchange diffusion model in a battery often lacks in closed-form solutions. For the long-term behaviour and to assess the quantitative behaviour of the model, numerical treatment is necessary. A novel numerical approach based on arithmetic average compact discretization employing a quasi-variable grid network is proposed for a wide class of nonlinear three-dimensional elliptic PDEs. The method's key benefit is that it applies to singular models and only needs nineteen-point grids with seven functional approximations. Additionally, the suggested method disseminates the truncation error across the domain, which is unrealistic for finite-difference discretization with a fixed step length of grid points. Often, small diffusion anticipates strong oscillation, and tuning the grid stretching parameter helps error dispersion over the domain. The scheme is examined for maximal error bounds and convergence property with the help of a monotone matrix and its irreducible character. The metrics of solution accuracies, mainly root-mean-squared and absolute errors alongside numerical convergence rate, are inspected by different types of variable coefficients, singular and non-singular 3D elliptic PDEs appearing in a convection-diffusion phenomenon. The performance of the numerical solution corroborates the fourth-order convergence on a quasi-variable grid network

A family of quasi-variable meshes high-resolution compact operator scheme for Burger's-Huxley, and Burger's-Fisher equation: Quasi-variable meshes compact operator scheme for Burger's type PDEs

We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize: • The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms. • The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation. • The high-order numerical scheme is computationally efficient and employs minimal data storage

An O(h4)O(h^4) accurate cubic spline TAGE method for nonlinear singular two point boundary value problems

In this paper, we propose two parameter alternating group explicit (TAGE) method for the numerical solution of Image, $u'' + \frac {\alpha}{r}\ u'$ – $\frac{\alpha} {r^2}\ u = f(r), 0 < r < 1$ using a fourth order accurate cubic spline method with specified boundary conditions at the end points. The proof of convergence of the TAGE method when the coefficient matrix is unsymmetric and real is presented. We also discuss Newton-TAGE method for the numerical solution of nonlinear singular two point boundary value problem using the cubic spline method with same accuracy of order four. Numerical results are provided to illustrate the viability of the proposed TAGE method

An O(h<SUP>4</SUP>) accurate cubic spline TAGE method for nonlinear singular two point boundary value problems

In this paper, we propose two parameter alternating group explicit (TAGE) method for the numerical solution of u"+&#945;/ru'-&#945;/r<SUP>2</SUP>u=f(r), 0&lt;r&lt;1 using a fourth order accurate cubic spline method with specified boundary conditions at the end points. The proof of convergence of the TAGE method when the coefficient matrix is unsymmetric and real is presented. We also discuss Newton-TAGE method for the numerical solution of nonlinear singular two point boundary value problem using the cubic spline method with same accuracy of order four. Numerical results are provided to illustrate the viability of the proposed TAGE method