A mesh-free method for the numerical solution of the KdV–Burgers equation

AbstractThis paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the nonlinear dispersive and dissipative KdV–Burgers’ (KdVB) equation. The computed results show implementation of the method to nonlinear partial differential equations. This method has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. Accuracy of the method is assessed in terms of error norms L2,L∞, number of nodes in the domain of influence, parameter dependent RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving nonlinear dispersive and dissipative problems

(R1969) On the Approximation of Eventual Periodicity of Linearized KdV Type Equations using RBF-PS Method

Water wave propagation phenomena still attract the interest of researchers from many areas and with various objectives. The dispersive equations, including a large body of classes, are widely used models for a great number of problems in the fields of physics, chemistry and biology. For instance, the Korteweg-de Vries (KdV) equation is one of the famous dispersive wave equation appeared in the theories of shallow water waves with the assumption of small wave-amplitude and large wave length, also its various modifications serve as the modeling equations in several physical problems. Another interesting qualitative characteristic of solutions of some dispersive wave equations indicated through experiments that are connected with their large-time behavior termed as Eventual Time Periodicity which is exhibited by solutions of initial-boundary-value problems (IBVPs henceforth). Laboratory experiments in a channel with a flap-type or piston-type wave maker mounted at one end of a channel exposed this interesting phenomena. Here in this study we numerically investigate the solutions periodicity for linearized KdV type equations on a finite (bounded) domain with periodic boundary conditions using meshfree technique known as Radial basis function pseudo spectral (RBF-PS) method

(R1992) RBF-PS Method for Eventual Periodicity of Generalized Kawahara Equation

In engineering and mathematical physics, nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations evidenced by findings link belonging to their long-term actions named as eventual time periodicity discovered over solutions to IBVPs (initial-boundary-value problems). Here we investigate the solution’s eventual periodicity for generalized fifth order Kawahara equation (IBVP) on bounded domain in combination with periodic boundary conditions numerically exploiting mesh-free technique called as Radial basis function pseudo spectral (RBF-PS) method

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Method

PCA-based dimensionality reduction for face recognition

In this paper, we conduct a comprehensive study on dimensionality reduction (DR) techniques and discuss the mostly used statistical DR technique called principal component analysis (PCA) in detail with a view to addressing the classical face recognition problem. Therefore, we, more devotedly, propose a solution to either a typical face or individual face recognition based on the principal components, which are constructed using PCA on the face images. We simulate the proposed solution with several training and test sets of manually captured face images and also with the popular Olivetti Research Laboratory (ORL) and Yale face databases. The performance measure of the proposed face recognizer signifies its superiority

Ensemble machine learning-based recommendation system for effective prediction of suitable agricultural crop cultivation

Agriculture is the most critical sector for food supply on the earth, and it is also responsible for supplying raw materials for other industrial productions. Currently, the growth in agricultural production is not sufficient to keep up with the growing population, which may result in a food shortfall for the world’s inhabitants. As a result, increasing food production is crucial for developing nations with limited land and resources. It is essential to select a suitable crop for a specific region to increase its production rate. Effective crop production forecasting in that area based on historical data, including environmental and cultivation areas, and crop production amount, is required. However, the data for such forecasting are not publicly available. As such, in this paper, we take a case study of a developing country, Bangladesh, whose economy relies on agriculture. We first gather and preprocess the data from the relevant research institutions of Bangladesh and then propose an ensemble machine learning approach, called K-nearest Neighbor Random Forest Ridge Regression (KRR), to effectively predict the production of the major crops (three different kinds of rice, potato, and wheat). KRR is designed after investigating five existing traditional machine learning (Support Vector Regression, Naïve Bayes, and Ridge Regression) and ensemble learning (Random Forest and CatBoost) algorithms. We consider four classical evaluation metrics, i.e., mean absolute error, mean square error (MSE), root MSE, and R2, to evaluate the performance of the proposed KRR over the other machine learning models. It shows 0.009 MSE, 99% R2 for Aus; 0.92 MSE, 90% R2 for Aman; 0.246 MSE, 99% R2 for Boro; 0.062 MSE, 99% R2 for wheat; and 0.016 MSE, 99% R2 for potato production prediction. The Diebold–Mariano test is conducted to check the robustness of the proposed ensemble model, KRR. In most cases, it shows 1% and 5% significance compared to the benchmark ML models. Lastly, we design a recommender system that suggests suitable crops for a specific land area for cultivation in the next season. We believe that the proposed paradigm will help the farmers and personnel in the agricultural sector leverage proper crop cultivation and production