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We introduce a path following algorithm forL_{1}$-regularized generalized linear models. TheL_{1}$-regularization procedure is useful especially because it, in effect, selects variables according to the amount of penalization on theL_{1}$-norm of the coefficients, in a manner that is less greedy than forward selection-backward deletion. The generalized linear model path algorithm efficiently computes solutions along the entire regularization path by using the predictor-corrector method of convex optimization. Selecting the step length of the regularization parameter is critical in controlling the overall accuracy of the paths; we suggest intuitive and flexible strategies for choosing appropriate values. We demonstrate the implementation with several simulated and real data sets.

In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and the Caputo derivative term is discretized by the L1 discrete formula. Through the first and second derivatives of the matrix under the compact difference, we improve the precision of this scheme. Then, the existence and uniqueness are proved, and the numerical experiments are presented.

In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a new model incorporating fractional operators. In particular, by using a modified predictor–corrector method, we captured the nature of the obtained solution for different arbitrary orders. We investigated the influence of the fractional operator to present and discuss some interesting properties of the novel coronavirus infection.

In this paper, we introduce an economical technique based on a semi-implicit predictor–corrector scheme for solving fractional Benjamin–Bona–Mahony–Burgers equations, in which the Adams–Moulton schemes are used for predictor and corrector schemes. To resolve a nonlinearity of the given equations in the predictor procedure, the weighted Rubin–Graves linearization scheme is applied to convert the linearized equations at the predictor procedure. Moreover, to alleviate weak regularity at the initial time point, mixed meshes based on uniform grid are used so that it can save the computational costs by not recalculating the coefficients of Adams–Moulton methods for smaller time intervals. The convergence analysis are analytically executed to derive the convergence order and are numerically supported. Several numerical results are provided to show the efficiency of the proposed scheme.

In this article, we study some soliton-type analytical solutions of Schrödinger equation, with their numerical treatment by Galerkin finite element method. First of all, some analytical solutions to the equation are obtained for different values of parameters; thereafter, the problem of truncating infinite domain to finite interval is taken up and truncation approximations are worked out for finding out appropriate intervals so that information is not lost while reducing the domain. The benefit of domain truncation is that we do not need to introduce artificial boundary conditions to find out numerical approximations. To verify theoretical results, numerical simulations are performed by Galerkin finite element method. Crank–Nicolson method is used for the time discretization, and non-linearity is resolved using predictor corrector method, which is second order accurate and computationally efficient.

Although predictor-corrector methods have been extensively applied, they might not meet the requirements of practical applications and engineering tasks, particularly when high accuracy and efficiency are necessary. A novel class of correctors based on feedback-accelerated Picard iteration (FAPI) is proposed to further enhance computational performance. With optimal feedback terms that do not require inversion of matrices, significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts; however, the computational complexities are comparably low. These advantages enable nonlinear engineering problems to be solved quickly and accurately, even with rough initial guesses from elementary predictors. The proposed method offers flexibility, enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach. In our method, the functional formulas of FAPI are discretized into numerical forms using the collocation approach. These collocated iteration formulas can directly solve nonlinear problems, but they may require significant computational resources because of the manipulation of high-dimensional matrices. To address this, the collocated iteration formulas are further converted into finite difference forms, enabling the design of lightweight predictor-corrector algorithms for real-time computation. The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches: the modified Euler approach, the Adams-Bashforth-Moulton approach, and the implicit Runge-Kutta approach. Subsequently, the updated approaches are tested in solving strongly nonlinear problems, including the Matthieu equation, the Duffing equation, and the low-earth-orbit tracking problem. The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

Root finding is a basic example that still remnant an interest to several researchers. Several hybrid experiments are developed to obtain approximate solutions for nonlinear equations. Thus, this paper presents an analysis on numerical comparison between common method and the other methods. An evaluation iterative method MATLAB is utilized for this paper. Numerical and interpretative results prove that Dekker’s Formula is acceptably efficient, accurate, and easy to use compared with other iterative methods.

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has been conducted on the existence and uniqueness of the selected model. Also, we examined the model’s stability and the existence of equilibrium points. With the purpose of analyzing the dynamics of the Sup35 monomer and Sup35 prion population, we displayed the graphs to show the obtained solutions over time. Graphical simulations show that the behaviour of the populations can change based on fractional orders and threshold parameter values. This work may present a good example of how biological theories and data can be better understood via mathematical modelling.

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr,j=0,1,2,…,N with a≥1 and r≥1, where loga=logt0

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