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A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a…
Differential equations are important tools to portray dynamic problems, and are widely used in finance, engineering and biology. Here, multiple dynamic differential models were built innovatively, and discretized with the Runge-Kutta…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this…
We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that under certain…
Recently, the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), has been proposed for the efficient solution of Hamiltonian problems, as well as for other types of conservative problems. In…
Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a…
This paper is devoted to examining the stability of Runge-Kutta methods for solving nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
In this paper, a recently published method [Hussain, Ismail, Senua, Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method, J. Comput. Appl. Math. 306 (2016) 179-199] for solving fourth-order…
We study the application of the generalized convolution quadrature (gCQ) based on Runge--Kutta methods to approximate the solution of an important class of sectorial problems. The gCQ generalizes Lubich's original convolution quadrature…
A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals…
In the last few decades, numerical simulation for nonlinear oscillators has received a great deal of attention, and many researchers have been concerned with the design and analysis of numerical methods for solving oscillatory problems. In…
As quantum hardware rapidly advances toward the early fault-tolerant era, a key challenge is to develop quantum algorithms that are not only theoretically sound but also hardware-friendly on near-term devices. In this work, we propose a…
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other…
The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of…
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for…
We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK)…
Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to…
A recently published method for solving the neutrino evolution equation with constant matter density is further refined and used to lay out an exact algorithm for computing oscillation probabilities, which is moderately faster than previous…