We introduce the mixed-order classes C [ M ] r , s by combining mixed differentiability with the weight-matrix framework, in both the Beurling and Roumieu cases. At finite order we prove a chain rule under the Faà-di-Bruno property. The currying map is a linear injection. If r and s are finite, the class coincides with C r , s and the exponential law reduces to the mixed-order law without weights. Continuity of the inverse on the image holds when a lower weight bound is available at a single index.
This study, an algorithm for integrating the Regularized Long Wave Equation (RLWE) is developed by combining a predictor–corrector scheme with the quartic B-spline Galerkin method. The algorithm is first formulated using the four-step implicit Adams–Bashforth method as the predictor and the four-step implicit Adams–Moulton method as the corrector for time integration of the RLWE. Subsequently, the quartic B-spline Galerkin method is applied to the semi-discrete form of the RLWE. The implementation of the algorithm yields a system of algebraic equations whose solutions are used to obtain both discrete and functional approximations of the RLWE. Highly accurate results are achieved through the application of higher-order schemes to test problems involving solitary wave propagation and undulation.
In this paper, we combine the cubic B-spline Galerkin method with the predictor-corrector method to get the numerical method for solving the RLW equation(RLWE). Thus fully-integration of the RLWE is managed by employing the time integrator of the four step explicit Adams-Bashforth as predictor and the four step implicit Adams-Moulton as corrector and the cubic B-spline Galerkin method for spatial discretization. The performance of this new method is carried out studying the single solitary wave, wave undulation and wave generation and findings of error norm L∞ and numerical conservation laws are compared with exact solutions and those already published.
This paper investigates the (2 + 1)–dimensional extended Kadomtsev–Petviashvili equation (eKPE), importantly describing multidimensional nonlinear wave phenomena. Its exact traveling wave solutions are constructed using the modified rational sine–cosine method and the new extended direct algebraic method (NEDAM). The obtained solutions are expressed in diverse functional forms, including trigonometric, hyperbolic, rational, exponential, and complex-valued structures. Graphical representations of selected solutions are plotted to illustrate their physical characteristics, showing various nonlinear wave behaviors such as a singular periodic wave, a periodic wave solution, a kink soliton wave, and a singular kink-type solution. In addition, a linear stability analysis is performed to examine the modulation behavior of the model under different parameter choices. The results demonstrate that the proposed analytical approaches provide an efficient and systematic framework for generating a wide spectrum of exact solutions to the eKPE, thus enriching the understanding of its nonlinear wave dynamics.
Multiple additive models provide a robust framework for analysing families of additive models associated with the treatments of a base design. In this context, ANOVA-like procedures can be used to assess the effects of factors through corresponding components of principal estimable functions. However, the validity of the resulting F-tests is typically justified under approximate homoscedasticity and distributional assumptions that may be violated in practice. This paper studies the robustness of ANOVA-like inference in multiple additive models under non-normal error distributions. Using cumulants of order two, three and four, we characterise departures from normality in terms of variance, skewness and excess kurtosis. A simulation study evaluates the empirical type I error and power of the proposed ANOVA-like tests under normal, heavy-tailed and skewed error distributions, including Gumbel-type components. The results identify conditions under which the F approximation remains reliable and cases where non-normality leads to distorted inference. The proposed cumulant-based perspective provides both a diagnostic and methodological framework for robust inference in multiple additive models.
