An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as  Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and  Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

Analysis of Normalized Orthogonal Gradient Adaptive Algorithm Based on Spline Adaptive Filtering for Smart Communication Technology

This paper presents a general theoretical framework of spline adaptive filtering based on a normalized version of orthogonal gradient adaptive algorithm. A nonlinear spline adaptive filter normally consists of a linear combination with a memory-less function and a spline function for adaptive approach. We explain how the adaptive linear filter and spline control points are derived in a straightforward iterative gradient-based method. In order to improve the convergence characteristics, the normalized version of orthogonal gradient adaptive algorithm is introduced by the orthogonal projection along with the gradient adaptive algorithm. In addition, a simple form of adaptation algorithm is introduced how to obtain a lower bound on the excess mean square error (MSE) in a theoretical basis. Convergence and stability analysis based on the MSE criterion are proven in terms of the excess MSE. Simulation results reveal that the proposed algorithm achieves more robustness compared with the conventional spline adaptive filtering algorithm.

Numerical Analysis of the Chebyshev Collocation Method for Functional Volterra Integral Equations

The collocation method based on Chebyshev basis functions, coupled Picard iterative process, is proposed to solve a functional Volterra integral equation of the second kind. Using the Banach Fixed Point Theorem, we prove theorems on the existence and uniqueness solutions in the L2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L2-norm decay exponentially, provided that the kernel function is sufficiently smooth. Numerical results are presented and they confirm the theoretical prediction of the exponential rate of convergence.

Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.