Derive Approximation Formulas Research Articles

Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

The goal of this article is to develop local discontinuous Galerkin (LDG) schemes for solving a time fractional equation describing the ultrasonic wave in a homogeneous isotropic porous material. Two novel semi-discrete LDG schemes are designed for the considered model. The semi-discrete LDG schemes are constructed by splitting the original model into a coupled system. The first semi-discrete scheme follows the traditional LDG method by splitting second-order space derivative. The second one splits the original model for both time and space derivatives. The discontinuous Galerkin is used for the spatial discretization. Two kinds of fully discrete LDG schemes are presented by using the Grünwald-Letnikov and L1 approximation formulas for the time fractional derivatives. The $ L^2 $ norm stability and convergence analysis are carried out for both semi-discrete and fully discrete LDG schemes. The stability analysis reveals that the numerical schemes are unconditionally stable in $ L^2 $ norm and convergence with optimal convergence rate. Finally, numerical examples are presented to test the effectiveness of the proposed schemes and the correctness of the theoretical analysis.

The closed-form approximation to price basket options under stochastic interest rate

The paper presents closed-form approximation formulas for pricing basket options. We assume that the underlying asset prices follow geometric Brownian motions and the interest rate follows the one-factor Hull-White model. Under a given forward measure, we obtain the analytical lower and upper bounds of basket options. By finding a simple random variable to replace the sum of the lognormal random variables, we combine the conditioning and the moment matching approaches, and derive approximation formulas to price basket options. Numerical results illustrate that our results fall in the sharp lower and upper bounds of basket options and are consistent with Monte Carlo simulation results.

Efficient Bayesian credible-region certification for quantum-state tomography

Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte~Carlo sampling of the state space followed by sample rejection. This conventional method typically gives negligible yield for very small error regions originating from large datasets. We propose an operational reformulated theory to compute both size and credibility from region-average quantities that in principle convey information about behavior of these two properties as the credible-region changes. We next suggest the accelerated hit-and-run Monte~Carlo sampling, customized to the construction of Bayesian error-regions, to efficiently compute region-average quantities, and provide its complexity estimates for quantum states. Finally by understanding size as the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class or Bures distance), we derive approximation formulas to analytically estimate both distance-induced size and credibility under the pseudo-Bloch parametrization without resorting to any Monte~Carlo computation.

EXPANSION FORMULAS FOR EUROPEAN QUANTO OPTIONS IN A LOCAL VOLATILITY FX-LIBOR MODEL

We develop an expansion approach for the pricing of European quanto options written on LIBOR rates (of a foreign currency). We derive the dynamics of the system of foreign LIBOR rates under the domestic forward measure and then consider the price of the quanto option. In order to take the skew/smile effect observed in fixed income and FX markets into account, we consider local volatility models for both the LIBOR and the FX rate. Because of the structure of the local volatility function, a closed form solution for quanto option prices does not exist. Using expansions around a proxy related to log-normal dynamics, we derive approximation formulas of Black–Scholes type for the price, that have the benefit of giving very rapid numerical procedures. Our expansion formulas have the major advantage that they allow for an accurate estimation of the error, using Malliavin calculus, which is directly related to the maturity of the option, the payoff, and the level and curvature of the local volatility function. These expansions also illustrate the impact of the quanto drift adjustment, while the numerical experiments show an excellent accuracy.

Fractional-Compact Numerical Algorithms for Riesz Spatial Fractional Reaction-Dispersion Equations

Abstract It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives plays a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

Second-order approximations for variable order fractional derivatives: Algorithms and applications

Fractional calculus allows variable-order of fractional operators, which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we derive two second-order approximation formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation. We then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub-diffusion and super-diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable-order function. Finally, we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super-diffusion, and also simulations of the Burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and/or time. Taken together, our results demonstrate that variable-order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long-standing issues of outflow boundary conditions and monotonicity of integer-order PDEs.

Message fragmentation for a chain of disrupted links

We investigate the problem of estimating the transmission time of fragmented messages over multiple disrupted links. We build a system model for the case where a single message is sent over a chain of links and the disruptions in these links are identically and independently distributed. For this case, we derive approximation formulas for the mean transmission time, based on number of links, length of fragments and distributions of disruptions. The formulas are verified against simulation experiments in the cases of uniform and exponential distributions for disruptions.

EXPANSION FORMULAS FOR BIVARIATE PAYOFFS WITH APPLICATION TO BEST-OF OPTIONS ON EQUITY AND INFLATION

A wide class of hybrid products are evaluated with a model where one of the underlying price follows a local volatility diffusion and the other asset value a log-normal process. Because of the generality for the local volatility function, the numerical pricing is usually much time consuming. Using proxy approximations related to log-normal modeling, we derive approximation formulas of Black–Scholes type for the price, that have the advantage of giving very rapid numerical procedures. This derivation is illustrated with the best-of option between equity and inflation where the stock price follows a local volatility model and the inflation rate a Hull–White process. The approximations possibly account for Gaussian HJM (Heath-Jarrow-Morton) models for interest rates. The experiments show an excellent accuracy.

Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition

In this paper, we study the spatial-temporal patterns of the solutions to the diffusive non-local Nicholson's blowflies equations with time delay (maturation time) subject to the no flux boundary condition. We establish the existence of both spatially homogeneous periodic solutions and various spatially inhomogeneous periodic solutions by investigating the Hopf bifurcations at the spatially homogeneous steady state. We also compute the normal form on the centre manifold, by which the bifurcation direction and stability of the bifurcated periodic solutions can be determined. The results show that the bifurcated homogeneous periodic solutions are stable, while the bifurcated inhomogeneous periodic solutions can only be stable on the corresponding centre manifold, implying that generically the model can only allow transient oscillatory patterns. Finally, we present some numerical simulations to demonstrate the theoretic results. For these transient patterns, we derive approximation formulas which are confirmed by numerical simulations.

An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

Quadratic spline quasi-interpolants and collocation methods

Univariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated and compared with finite difference methods on some numerical examples of elliptic boundary value problems.

Computational aspects of fuzzy arithmetics based on Archimedean t-norms

For fuzzy multiplication and division of (L,R)-numbers we derive approximation formulas by including the exact results using (L,R)-numbers. This permits an error control and meets user's ideas on tractable solutions better than the level sets approach does. As potential application the resolution of linear systems is considered.

Relationships between antiviral treatment effects and biphasic viral decay rates in modeling HIV dynamics

Recently, potent combination antiviral therapies consisting of reverse transcriptase inhibitor (RTI) drugs and protease inhibitor (PI) drugs, have been developed which can rapidly suppress HIV below the limit of detection. Two phases of plasma viral decay after initiation of treatment have been observed in clinical studies. Some researchers have suggested that the viral decay rates may reflect the potency (efficacy) of antiviral therapies. In this paper we model the effect of RTI drugs and PI drugs as inhibition rates of cell infection and infectious virus production, respectively, based on the biological mechanisms of these two different types of drugs. Through rigorous mathematical derivation, we show that the two viral decay rates are monotone functions of the treatment effects of these antiviral therapies. We derive approximation formulas for the relationships between viral decay rates and treatment effects. Computer simulations show that the approximation formulas approximate the true values very well. These formulas may be used to study what factors really affect the viral decay rates. The results in this paper provide a theoretical justification for using both viral decay rates for evaluation of the treatment efficacy of antiviral therapies.

Light-traffic approximations for Markov-modulated multi-server queues

A general concept is considered of expanding the expectation of a wide class of functional of marked point processes, which expresses this expectation by a sum of integrals over higher-order factorial moment measures of the underlying point process. The idea of factorial moment expansion is applied in order to derive approximation formulas for stationary characteristics of multi-server queues with Markov-modulated arrival process and with the first-come-first-served queueing discipline. Besides real-valued queueing characteristics like waiting time and total work load, we also give approximations for the Kiefer-Wolfowitz work-load vector. A boundedness condition on the service time distributions is given which ensures that the components of the expected stationary work-load vector are analytic functions of the arrival intensity in a neighborhood of zero. If the service times have phase-type distributions, the factorial moment expansion provides a useful computational technique for approximations of moment...

Inefficiency of democratic decision making in an unstable society

The political power is said to be efficient if good results in decision making are achieved by moderate means. For example, if a large parliament makes the same decisions as a given president, the power efficiency of the president is better than that of the parliament. We study the efficiency of three forms of political power, president, cabinet of ministers, and the parliament, using the model from the author's earlier publications on the mathematical theory of democracy. We define an indicator of power quality, referring to the weight of the coalition whose preference is satisfied by each decision. We prove that a presidential power can be absolutely efficient when the society is divided by preferences into (almost) equal coalitions, which is interpreted as social instability, because a small change in preferences breaks the equilibrium. When the society is stable, i.e. there are predominant coalitions, the cabinet and the parliament provide a better power quality. For a model with a large number of independent individuals we derive approximation formulas of the indicator of the power quality which imply a geometric interpretation of optimal president, cabinet, and parliament as providing the greatest projections of their characteristic vectors on the characteristic vector of the society. We draw an analogy to the geometric interpretation of forces in physics. From our standpoint, democratic (collective) forms of power are efficient in a stable society. In an unstable society, a parliament and a cabinet are less efficient than personal power. Linking social stability to economic development, we conclude that a democratic political system is not the cause but rather a result of economic development which makes the society stable. We comment on the political development in the ex-Soviet Union. In particular, we justify the separation of the Soviet Union into independent states as a measure to make the power more powerful when its potential has been exhausted. However, we doubt that democratic transformations can contribute to solving economic problems and that real democratic systems will be established in the ex-Soviet republics before at least major economic difficulties have been overcome. We also outline some applications of our results to Gallup polls of public opinion and to multicriteria decision making.

The velocity of spatial population expansion

We consider the velocity with which an invading population spreads over space. For a general linear model, originally due to Diekmann and Thieme, it is shown that the asymptotic velocity of population expansion can be calculated if information is available on: (i) the net-reproduction, Ro; i.e. the expected number of offspring produced by one individual throughout its life, and (ii) the (normalized) reproduction-and-dispersal kernel, β(a, χ − ξ); i.e. the density of newborns produced per unit of time at position χ by an individual of age a born at ξ By means of numerical examples we study the effect of the net-reproduction and the shape of the reproduction-and-dispersal kernel on the velocity of population expansion. The reproduction-and-dispersal kernel is difficult to measure in full. This leads us to derive approximation formulas in terms of easily measurable parameters. The relation between the velocity of population expansion calculated from the general model and that from the Fisher/Skellam diffusion model is discussed. As a final step we use the model to analyse some real-life examples, thus showing how it can be put to work.

Application of differential quadrature to transport processes

The methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented. Application of the method is demonstrated by solving a simple one-dimensional, time-dependent (transient) diffusion process involving an irreversible reaction without any flux across the end boundary. In addition, the same technique is used (for the first time to the authors' knowledge) to solve a steady-state problem. For this purpose, a convection-diffusion problem involving an irreversible reaction is considered. The demonstration is carried out in two ways, (1) using the Bellman et al. technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative, and (2) using individual quadratures to approximate the partial derivatives of first, as well as higher orders, as suggested by Mingle. Both approaches give the same results; however, the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature. Since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points, both the programming task and computational effort are alleviated considerably. For these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena.