Approximate Analytical Technique Research Articles

Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation

Abstract In this article, we have implemented reconstruction of variational iteration method as a new approximate analytical technique for solving fractional Black-Scholes option pricing equations. Indeed, we essentially use the well-known Mittag-Leffler function to obtain explicit solutions for some examples of financial mathematics equations.

Comparison of Optimal Homotopy Asymptotic and Adomian Decomposition Methods for a Thin Film Flow of a Third Grade Fluid on a Moving Belt

We have investigated a thin film flow of a third grade fluid on a moving belt using a powerful and relatively new approximate analytical technique known as optimal homotopy asymptotic method (OHAM). The variation of velocity profile for different parameters is compared with the numerical values obtained byRunge-Kutta Fehlberg fourth-fifth ordermethod and with Adomian Decomposition Method (ADM). An interesting result of the analysis is that the three terms OHAM solution is more accurate than five terms of the ADM solution and this thus confirms the feasibility of the proposed method.

Approximate Transient Analysis of Queuing Networks by Decomposition based on Time-Inhomogeneous Markov Arrival Processes

We address the transient analysis of networks of queues with exponential service times. Such networks can easily have such a huge state space that their exact transient analysis is unfeasible. In this paper we propose an approximate transient analysis technique based on decomposing the queues of the network using a compact and approximate representation of the departure process of each queue. Namely, we apply time-inhomogeneous Markov arrival processes (IMAP) to describe the stream of clients leaving the queues. By doing so, the overall approximate model of the network is a time-inhomogeneous continuous time Markov chain (ICTMC) with significantly less number of states than there are in the original Markov chain. The proposed construction of the output IMAP of a queue is based on its transient state probabilities. We illustrate the approach first on a single M/M/1 queue and analyze the goodness of fitting of the departure process by numerical examples. Then we extend the approach to networks of queues and evaluate the precision of the resulting technique on several simple numerical examples by comparing the exact and the approximate transient probabilities of the queues.

On Critical Buckling Loads of Columns under End Load Dependent on Direction.

Most of the phenomena of various fields of applied sciences are nonlinear problems. Recently, various types of analytical approximate solution techniques were introduced and successfully applied to the nonlinear differential equations. One of the aforementioned techniques is the Homotopy analysis method (HAM). In this study, we applied HAM to find critical buckling load of a column under end load dependent on direction. We obtained the critical buckling loads and compared them with the exact analytic solutions in the literature.

Applying a new method to solve diffusivity equation for gas reservoirs

Diffusivity equation is the general partial differential equation used to describe the flow of any fluid flowing in a radial direction in porous media. The main objective of this study is to present a new numerical scheme based on orthogonal collocation (OC) method to solve the diffusivity equation for heterogeneous and homogeneous gas reservoirs. OC is an approximate analytical technique which categorizes in the weighted residuals methods. The advantage and priority of the OC method over exact analytical solution (i.e., Laplace transform) is in the cases that the heterogeneity and variation of reservoir properties such as porosity and permeability with position or pressure could not be neglected; in these cases, the exact analytical solution is very tedious and may be impossible. The diffusivity equation has been solved by both OC and exact analytical solutions. To demonstrate the reliability of the proposed method, the results of OC method have been compared with those achieved using exact analytical solution. Average absolute deviation percent (AAD%) has been used for determining the suitable number of collocation points to give acceptable error and best matching between approximate and analytical results. Sensitivity analysis indicates that increasing the numbers of collocation points result in significant improvement of OC method accuracy and its capability on dynamic pseudo pressure prediction. The minimum AAD% of 0.113 from the exact analytical predictions has been obtained with 25 collocation points. The results indicate that the proposed approximate method with these numbers of collocation points can predict the reservoir pseudo pressure trend with an acceptable accuracy in heterogeneous and homogeneous gas reservoirs.

Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation

SUMMARYNonlinear viscous dampers are supplemental devices widely used for enhancing the performance of structural systems exposed to seismic hazard. A rigorous evaluation of the effect of these damping devices on the seismic performance of a structural system should be based on a probabilistic approach and take into account the evolutionary characteristics of the earthquake input and of the corresponding system response.In this paper, an approximate analytical technique is proposed for studying the nonstationary stochastic response characteristics of hysteretic single degree of freedom systems equipped with viscous dampers subjected to a fully nonstationary random process representing the seismic input. In this regard, a stochastic averaging/linearization technique is utilized to cast the original nonlinear stochastic differential equation of motion into a simple first‐order nonlinear ordinary differential equation for the nonstationary system response variance. In comparison with standard linearization schemes, the herein proposed technique has the significant advantage that it allows to handle realistic seismic excitations with time‐varying frequency content. Further, it allows deriving a formula for determining the nonlinear system response evolutionary power spectrum. By this way, ‘moving resonance’ effects, related to both the evolutionary seismic excitation and the nonlinear system behavior, can be observed and quantified.Several applications involving various system and input properties are included. Furthermore, various response parameters of interest for the seismic performance assessment are considered as well. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

Scattering of near normal incidence SH waves by sinusoidal and rough surfaces in 3-D: comparison to the scalar wave approximation.

The challenge of accurately simulating how incident scalar waves interact with rough boundaries has made it an important area of research within many scientific disciplines. Conventional methods, which in the majority of cases focus only on scattering in two dimensions, often suffer from long simulation times or reduced accuracy, neglecting phenomena such as multiple scattering and surface self-shadowing. A simulation based on the scalar wave distributed point source method (DPSM) is presented as an alternative which is computationally more efficient than fully meshed numerical methods while obtaining greater accuracy than approximate analytical techniques. Comparison is made to simulated results obtained using the finite element method for a sinusoidally periodic surface where scattering only occurs in two dimensions, showing very good agreement (<0.2 dB). In addition to two-dimensional scattering, comparison to experimental results is also carried out for scattering in three dimensions when the surface has a Gaussian roughness distribution. Results indicate that for two-dimensional scattering and for rough surfaces with a correlation length equal to the incident wavelength (λ) and a root mean square height less than 0.2λ, the scalar wave approximation predicts reflected pulse shape change and envelope amplitudes generally to within 1 dB. Comparison between transducers within a three-element array also illustrate the sensitivity pulse amplitude can have to sensor position above a rough surface, differing by as much as 17 dB with a positional change of just 1.25λ.

Analytical study of nonlinear oscillatory systems using the Hamiltonian approach technique

In this article, we investigate and apply Hamiltonian approach method as one of the analytical approximate techniques, for studying the strongly nonlinear dynamical systems such as the motion of a rigid rod rocking back on the circular surface without slipping and the free vibrations of an autonomous conservative oscillator with inertia and static-type fifth-order nonlinearities. To illustrate the applicability and accuracy of the method, the approximate solution results are compared with exact and numerical solutions.

Stochastic Schrödinger Equations for Markovian and non-Markovian Cases

Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.

Nonstationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism

AbstractA novel approximate analytical technique is developed for determining the nonstationary response probability density function (PDF) of randomly excited nonlinear multidegree-of-freedom (MDOF) systems. Specifically, the concept of the Wiener path integral (WPI) is used in conjunction with a variational formulation to derive an approximate closed-form solution for the system response PDF. Notably, determining the nonstationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by existing alternative numerical path integral solution schemes, which rely on a discrete version of the Chapman-Kolmogorov (C-K) equation. In this manner, the analytical WPI-based technique developed by the authors is extended and generalized herein to account for hysteretic nonlinearities and MDOF systems. This enhancement of the technique affords circumventing approximations associated with the stochastic averaging treatment of the previously developed technique. Hop...

Solutions Exact To Fredholm Fuzzy Integral Equations With Optimal Homotopy Asymptotic Method

A analytic approximate technique for addressing nonlinear problems, namely the Optimal Homotopy Asymptotic Method (OHAM), is proposed. This approach does not depend upon any small/large parameters. This method provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. The series solution is developed and the recurrence relations are given explicitly. The results reveal that the proposed method is effective and easy to use. This method is illustrated by solving an examples.

Survival Probability Determination of Nonlinear Oscillators Subject to Evolutionary Stochastic Excitation

A novel approximate analytical technique for determining the survival probability and first-passage probability density function (PDF) of nonlinear/hysteretic oscillators subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging/linearization treatment of the problem, approximate closed form expressions are derived for the oscillator nonstationary marginal, transition, and joint-response amplitude PDFs and, ultimately, for the time-dependent oscillator survival probability. The developed technique exhibits considerable versatility, as it can handle readily cases of oscillators exhibiting complex hysteretic behaviors as well as cases of evolutionary stochastic excitations with time-varying frequency contents. Further, it exhibits notable simplicity since, in essence, it requires only the solution of a first-order nonlinear ordinary differential equation (ODE) for the oscillator nonstationary response variance. Thus, the computational cost involved is kept at a minimum level. The classical hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in a numerical examples section, in which comparisons with pertinent Monte Carlo simulations data demonstrate the reliability of the proposed technique.

Analytic approximate solution for Falkner-Skan equation.

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques

In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Two efficient and easy mathematical techniques called He's Variational Approach and Laplace Iteration Method are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

Solution of the Differential-Difference Equations by Optimal Homotopy Asymptotic Method

We applied a new analytic approximate technique, optimal homotopy asymptotic method (OHAM), for treatment of differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Volterra equation in different form. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

Optimal Homotopy Asymptotic Approach to Self-Excited Vibrations

This paper is concerned with analytical treatment of nonlinear oscillation of a self-excited system. An analytic approximate technique, namely OHAM is employed for this purpose. Our procedure provides us with a convenient way to optimally control the convergence of solutions, such that the accuracy is always guaranteed. An excellent agreement of the approximate solutions with the numerical ones has been demonstrated. Three examples are given and the results reveal that the procedure is very effective and accurate, demonstrating the general validity and the great potential of the OHAM for solving strongly nonlinear problems.

Tackling continuous state-space explosion in a Markovian process algebra

Fluid or mean-field methods are approximate analytical techniques which have proven effective in tackling the infamous state-space explosion problem which typically arises when modelling large-scale concurrent systems based on interleaving semantics. These methods are particularly suitable in situations which present large populations of simple interacting objects characterised by small local state spaces, since they require the analysis of a problem which is insensitive to the population sizes but is dependent only on the size of the local state spaces. This paper studies the case when the replicated objects are best described as composites which consist of smaller simple objects. A congenial formal modelling framework for situations of this kind may be given by stochastic process algebra. Using PEPA as a representative case, we find that fluid models with replicated copies of composite processes do not scale well with increasing population sizes, thus rendering intractable the analysis of the underlying system of ordinary differential equations (ODEs). We call this problem continuous state-space explosion, by analogy with its counterpart phenomenon in discrete state spaces. The main contribution of this paper is a result of equivalence that simplifies, in an exact way, the potentially massive ODE system arising in those circumstances to one whose size is independent from all the multiplicities in the model. As a byproduct, we find that these simplified ODEs turn out to characterise the fluid behaviour of a family of PEPA models whose elements cannot be related to each other through any known equivalence relation. A substantial numerical assessment investigates the relationship between the different underlying Markov chains and their unique fluid limit, demonstrating its generally good accuracy for all practical purposes.

A New Analytic Approximation Technique for Options in a Regime-Switching Market

Both deterministic and stochastic volatility models have been used to price and hedge European options. Observation of real market data suggests that volatility, while stochastic, is well modelled to alternate between just two states. We consider a set of coupled pricing partial differential equations (PDE) similar to existing regime-switching equations, with the inclusion of a market price of volatility risk term which arises from the typical contract independence assumption. The resulting PDE system has not yet been solved in closed form. Instead, we use the Cauchy-Kowalevski Theorem to obtain an approximate solution in terms of known Black-Scholes formulae. The error in this approximate solution can be quantified in the backward error sense by solving a related root-finding problem. The results obtained from our method are compared to those obtained by implementing a Crank-Nicolson numerical scheme.

Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.