The approximation method presented takes advantage of the fact that, while solutions to differentialequations may be in- volved and complicated the Wigner distribution of the solu- tion may be relatively simple. In addition, the method takes advantage that in the time-frequency plane monocomponent forcing terms can be effectively approximated. Extension to multicomponent forcing terms are now being investigated. Also, we point out that of particular importance are partialdifferentialequations such as wave equations with driving forces. We have recently presented a method for directly writ- ing the equation for the Wigner distribution corresponding to the solution of a linear partialdifferential equation . Our aim is to also develop approximation methods for partial dif- ferential equations along the same lines as we have developed here for ordinarydifferentialequations.
The residual power series method (RPSM) is based on the Taylor series expansion and the concept of a residual error function. It is efficient and conve- nient to use since it does not require discretization or linearization. The RPSM was first developed for solving first-order fuzzy differentialequations. Later, it has been successfully applied to find numerical solutions for other equations, including ordinary and partialdifferentialequations, nonlinear systems of sin- gular initial value problems, pantograph delay differential equation, fractional differentialequations, fuzzy fractional differential models [1, 2, 3, 4, 5, 12, 13, 16, 17, 18, 20, 21, 22].
In general the solution is not easy to obtain since this is usually a system of coupled diﬀerential equations. There is a vast literature regarding the solution of ordinary diﬀerential equations by diﬀerent means and in particular by techniques utilizing generalized analytic functions, see for instance 1–14. These include applications to the three-dimensional Stokes problem, solutions of planar elliptic vector fields with degeneracies, the Dirichlet problem, multidimensional stationary Schr ¨odinger equation, among others 3, 5, 6, 12, 13. In particular, the technique that we present is of interest for people working on vector fields with singularities. For instance, in order to gain insight into the behaviour of analytic vector fields, correct visualization of vector fields in the vicinity of their singular set is required, in the case of visualization of two-dimensional complex analytic vector fields with essential singularities the usual methods only provide partial results see 15–17, whilst the technique which we promote provides accurate and correct solutions 18, 19. These questions arise naturally in discrete and continuous dynamical systems see 20–22.
The minimum energy problem and the associated optimal control problem have been investigated for more than half a century. The system constraints can be ordinarydifferentialequations, partialdifferentialequations, or functional dif- ferential equations. This study introduces a numerical method for finding the minimum energy to satisfy the general criterion that can be adjusted to minim- ize various requirements through the selection of appropriate parameters. One system constraint is the class of equations of the first kind, which originates from an aeroelasticity problem where the mathematical model consists of eight integro-differentialequations . In the model, the most determinate equation How to cite this paper: Chiang, S. (2019)
Because of the increasing demands and complexity in modeling, analysis, and compu- tation, signiﬁcant eﬀorts have been made searching for better mathematical models in recent years. It has been well recognized that many of the systems encountered in the new era cannot be represented by the traditional ordinary diﬀerential equation and/or stochastic diﬀerential equation models alone. The states of such systems have two com- ponents, namely, state = (continuous state, discrete event state). The discrete dynamics may be used to depict a random environment or other stochastic factors that cannot be represented in the traditional diﬀerential equation models. Dynamic systems mentioned above are often referred to as hybrid systems. One of the representatives in the class of hy- brid system is a switching diﬀusion process. A switching diﬀusion process can be thought of as a number of diﬀusion processes coupled by a random switching process. At a ﬁrst glance, these processes are seemingly similar to the well-known diﬀusion processes. A closer scrutiny shows that switching diﬀusions have very diﬀerent behavior compared to traditional diﬀusion processes. Within the class of switching diﬀusion processes, when the discrete event process or the switching process depends on the continuous state, the problem becomes much more diﬃcult; see [, ]. Because of their importance, switch- ing diﬀusions have drawn much attention in recent years. Many results such as smooth dependence of the initial data, recurrence, positive recurrence, ergodicity, stability, and numerical methods for solution of stochastic diﬀerential equations with switching, etc., have been obtained. Nevertheless, certain important concepts are yet fully investigated. The Feynman-Kac formula is one of such representatives.
The main purpose of this research is to utilize the algorithm of Dormand Prince method in solving a differentialequations. Then, the solution obtained is compared with the other numerical methods in term of accuracy. However, this research is limited to the scope of linear ordinarydifferentialequations of first and second order. Initial value problem is emphasized in this research [5, 6].
Impulsive diﬀerential equations, that is, diﬀerential equations involving impulse eﬀect, appear as a natural description of observed evolution phenomena of several real world problems. There are many good monographs on the impulsive diﬀerential equations 1– 6. It is known that many biological phenomena, involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems, do exhibit impulse eﬀects. Let us describe the Kruger-Thiemer model 7 for drug distribution to show how impulses occur naturally. It is assumed that the drug, which is administered orally, is first dissolved into the gastrointestinal tract. The drug is then absorbed into the so-called apparent volume of distribution and finally eliminated from the system by the kidneys. Let xt and yt denote the amounts of drug at time t in the gastrointestinal tract and apparent volume of distribution, respectively, and let k 1 and k 2
In this paper, we have been able to formulate the general linear second order ODE as an optimization problem, and we have also been able to solve the formulated optimization problem using the Differential Evolution algo- rithm. Numerical examples also show that the method gives better approximate solutions. Other evolutionary techniques can be exploited as well.
In this very simple setting, the Bayesian picture is very closely related to the more traditional computational mathematics approach of forming a least-squares objective function (analogous to the log-likelihood), adding a penalty function (analogous to the log of the prior), and optimizing to find a single best parameter value (analogous to computing a point that maximizes the posterior). However, working in terms of the complete posterior density, rather than just presenting an optimal parameter and possibly computing local sensitivity around that value, has benefits when there is more than one region of likely values. Further, by sampling parameter values from the posterior, we can display a set of ‘likely’ trajectories from the model.
Previously, the increasing complexity of systems and the corresponding increases in their computational complexity were matched by faster processor speeds that kept the simulation runtime within reasonable bounds. However, since the year 2005 processor clock speeds have largely leveled off (see Figure 1), and the increase in computing power for commercial chips has been achieved by adding processor cores that can execute in parallel rather than by increasing clock speed . Exploiting this parallel processing power provided by multi-core ar- chitectures to improve the run time of a simulation requires algorithmic changes to the simulation; one has to develop parallel versions of the simulation algorithms to speed up the computation. However, developing these parallel simulation algorithms will require careful consideration of the physical CPU architecture to derive the best parallel performance.
Abstract— Modified Weibull distribution is an appreciable improvement over the Weibull distribution. This paper explores the application of differentiation to obtain the ordinarydifferentialequations (ODE) of the probability functions of the modified Weibull Distribution. The parameters and support that characterized the distribution inevitably determine the behavior, existence, uniqueness and solution of the ODEs. The method is recommended to be applied to other probability distributions and probability functions not considered in this paper. Computer codes and programs can be used for the implementation.
It is suitable to determine the structural behaviour of the classical problem of an arbitrary curved beam element. Normally this problem is formulated in a compact energy equation form, but here the research is approached in an extended system of differentialequations.
properties for the ﬂux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections in [–]. Also, L ∞ -error estimates for general opti- mal control problems using mixed ﬁnite element methods are considered in [, ]. In [, ], a posteriori error estimates of mixed ﬁnite element methods for general convex opti- mal control problems are addressed. However, there does not seem to exist much work on theoretical analysis for mixed ﬁnite element approximation of optimal control problems governed by integro-diﬀerential equations in the literature.