Initial Value Problems For Differential Equations Research Articles

A Predictor-Corrector Method for Nonlinear Fractional Differential Equation Initial Value Problems on Graded Grids

A Predictor-Corrector Method for Nonlinear Fractional Differential Equation Initial Value Problems on Graded Grids

On Euler methods for Caputo fractional differential equations

Numerical methods for fractional differential equations have specific properties with respect to the ones for ordinary differential equations. The paper discusses Euler methods for Caputo differential equation initial value problem. The common properties of the methods are stated and demonstrated by several numerical experiments. Python codes are available to researchers for numerical simulations.

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.

Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

We consider a one‐dimensional aggregation equation for a non‐negative density associated with a quartic potential ( , ). We show that for the case of symmetric initial data [ ], the solution of the aggregation equation can be expressed in terms of an explicit function of , , and , where the functions and are determined by an ordinary differential equation initial value problem, the numerical solution of which is relatively straightforward. The function , which can be interpreted as an upper bound for the radius of the support of , is finite for all , even if the support of is unbounded, while is a potential related to the time integral of the second spatial moment of . We develop various bounds on and and a number of results concerning convexity and monotonicity that require no knowledge of other than its symmetry. We show that many, but not all, of these results persist if the assumption of symmetry is relaxed. Our general results are tested against numerical solutions for two examples of symmetric . We also exhibit some counterintuitive behaviour of the model, by finding conditions under which as , even if we start with and take , which ensures ultimate collapse to a single Dirac component at the origin.

Tomographic Inverse Problem with Estimating Missing Projections

Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed.

Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova & Rakhlin,2013; Farmer et al., 2013; Donier et al., 2015; Tóth (2016).Mathematically, the Hamilton–Jacobi–Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; Toth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.

Filtered Leapfrog Time Integration with Enhanced Stability Properties

The Leapfrog method for the solution of Ordinary Differential Equation initial value problems has been historically popular for several reasons. The method has second order accuracy, requires only one function evaluation per time step, and is non-dissipative. Despite the mentioned attractive properties, the method has some unfavorable stability properties. The absolute stability region of the method is only an interval located on the imaginary axis, rather than a region in the complex plane. The method is only weakly stable and thus exhibits computational instability in long time integrations over intervals of finite length. In this work, the use of filters is examined for the purposes of both controlling the weak instability and also enlarging the size of the absolute stability region of the method.

Solving hybrid fuzzy differential equations by Chebyshev wavelet

In this paper, a new approach for solving hybrid fuzzy differential equation initial value problems under generalized differentiability is proposed. The method is based on Chebyshev wavelets approximations. The accuracy and efficiency of the proposed method are demonstrated by applying it to some different numerical experiments.

On Poisson's state-dependent delay

In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. Initial data in the singular set produce multiple solutions.

An iterated, multipoint differential transform method for numerically evolving partial differential equation initial-value problems

Traditional numerical techniques for solving time-dependent partial differential equation (PDE) initial-value problems (IVPs) store a truncated representation of the function values and a certain number of their time derivatives at each time step. Although redundant in the dx → 0 limit, what if spatial derivatives were also stored? This paper presents an iterated, multipoint differential transform method (IMDTM) for numerically evolving PDE IVPs. Using this scheme, we demonstrate that stored spatial derivatives can be propagated in an efficient and self-consistent manner and can effectively contribute to the evolution procedure in a way that can confer several advantages, including aiding in solution verification. Lastly, in order to efficiently implement the IMDTM scheme, a generalized finite-difference stencil formula is derived that can take advantage of multiple higher-order spatial derivatives when computing even-higher-order derivatives. As demonstrated here, the performance of these techniques compares favorably to other explicit evolution schemes in terms of speed, memory footprint and accuracy.

The Nyström method for hybrid fuzzy differential equation IVPs

In this paper, the Nyström method is developed to approximate the solutions for hybrid fuzzy differential equation initial value problems (IVPs) using the Seikkala derivative. A proof of convergence of this method is also discussed in detail. The accuracy and efficiency of the proposed method are demonstrated by applying it to two different numerical experiments.

Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem

In this paper, we study hybrid fuzzy differential equation initial value problems (IVPs). We consider the problem of finding their numerical solutions by using a recent characterization theorem of Bede for fuzzy differential equations. We prove a corollary to Bede’s characterization theorem and give a characterization theorem for hybrid fuzzy differential equation IVPs. Then we prove that any suitable numerical method for ODEs can be applied piecewise to numerically solve hybrid fuzzy differential equation IVPs. Numerical examples are provided which connect the new results with previous findings.

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

On the perturbation of the observability equation in linear control systems

The aim of this paper is to investigate stability and sensitivity of the observability variable in linear control systems, (LCS) for short. We first present two results of Holder continuity in the abstract framework of the ordinary differential equation initial-value problem x?(t) = f(t,x(t)),x(t 0) = x 0. Afterwards, we apply our results to automatic systems, providing henceforth the sharpest bounds for the parametric input-output relation in LCS.

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h is large ( h ⩾ 0.1 ) . Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.

Mixed collocation methods for y″= f( x, y)

The collocation methods introduced here are based on linear combinations of trigonometric functions and powers. The motivation is to provide better approximations for oscillatory solutions of initial-value problems for differential equations of the special form y″= f( x, y). The resulting methods, for two or more collocation points, are implicit Runge–Kutta–Nyström methods with coefficients which depend on both the fitted angular frequency and the steplength. Algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined. Particular mixed collocation methods, and other methods for the same class of problems, are compared by applying them to a variety of test problems.

The starting procedure in variable-stepsize variable-order PECE codes

An existing starting procedure used in codes for solving ordinary differential equation initial-value problems by variable-stepsize variable-order PECE formulas is studied. The actual error propagation and the estimate formed in the code are analyzed and compared. Based on the behavior of a bound on error propagation, we present a modified starting procedure which improves efficiency significantly for stringent error tolerances.

Automated selection of mathematical software

Current approaches to recommending mathematical software are qualitative and categorical. These approaches are unsatisfactory when the problem to be solved has features that can “trade-off” in the recommendation process. A quantitative system is proposed that permits tradeoffs and can be built and modified incrementally. This quantitative approach extends other knowledge-engineering techniques in its knowledge representation and aggregation facilities. The system is demonstrated on the domain of ordinary differential equation initial value problems. The results are significantly superior to an existing qualitative (decision tree) system.

Parallel multischeme computation with the switch of computation history

A parallel multischeme computation in the solutions of differential equation initial-value problems has been studied. The mathematical switch of computation history is used successfully in the identification of the best approximation among all available ones at a computing step. A solution correction factor is also developed to achieve an extra four digits in solution accuracy. Based on our results, if the truncation error of computation history as we defined it can be properly utilized, then a computation engaging high-order schemes or using fine grids may be unnecessary.