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The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean…
This paper applies He's new amplitude-frequency relationship recently established by Ji-Huan He (Int J Appl Comput Math 3 1557-1560, 2017) to study periodic solutions of strongly nonlinear systems with odd nonlinearities. Some examples are…
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a…
First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on…
With the immense computing power at our disposal, the numerical solution of partial differential equations (PDEs) is becoming a day-to-day task for modern computational scientists. However, the complexity of real-life problems is such that…
We established a new eighth-order iterative method, consisting of three steps, for solving nonlinear equations. Per iteration the method requires four evaluations (three function evaluations and one evaluation of the first derivative).…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the…
We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which…
Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
We introduce a higher order phase averaging method for nonlinear oscillatory systems. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. Phase averaging is…
We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous…
In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level…
In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou \& Wu, {\textit{J. Comput. Phys.}}, 1997] and [Efendiev, Hou \& Wu, \textit{SIAM J. Numer.…
In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved.…
Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the…