numerical solutions for systems of fractional differential equations by the decomposition method in this paper we use adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi term fractional differential equation by reducing it to a system of fractional equations each of order at most unity we begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method moreover we show how the method can be applied to a general linear multi term equation and solve several applied problems

a paradox for the smooth ambiguity model of preference two ellsberg style thought experiments are described that reflect on the smooth ambiguity decision model developed by klibanoff marinacci and mukerji 2005 the first experiment poses difficulties for the model s axiomatic foundations and as a result also for its interpretation particularly for the claim that the model achieves a separation between ambiguity and the attitude toward ambiguity given the problematic nature of its foundations the behavioral content of the model and how it differs from multiple priors for example are not clear the second thought experiment casts some light on these questions

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

static and stability analyses of arbitrary straight sided quadrilateral thin plates by dqm abstract a differential quadrature dq methodology is employed for the static and stability analysis of irregular quadrilateral straight sided thin plates a four noded super element is used to map the irregular physical domain into a square computational domain second order transformation schemes with relative ease and low computational effort are employed to transform the fourth order governing equations of thin plates between the domains within the domain the displacements are the only degrees of freedom whereas along the boundaries the displacements as well as the second order derivatives of the displacements with respect to the associated normal coordinate variables in the computational domain are the two sets of degrees of freedom the implementation procedures for different boundary conditions including free edge boundaries are formulated to demonstrate the accuracy convergency and stability of the methodology detailed studies of skewed and trapezoidal plates for different geometries under different boundary and loading conditions are made good agreement is achieved between the results of the present methodology and those of other dq methodologies or other comparable numerical algorithms

dynamics of the fractional oscillator abstract the integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator the solution is obtained in terms of mittag leffler functions using laplace transforms the expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained numerical application and the phase plane representation of the dynamics are discussed

a smooth model of decision making under ambiguity we propose and axiomatize a model of preferences over acts such that the decision maker prefers act f to act g if and only if e e u f e e u g where e is the expectation operator u is a vn m utility function is an increasing transformation and is a subjective probability over the set of probability measures that the decision maker thinks are relevant given his subjective information a key feature of our model is that it achieves a separation between ambiguity identified as a characteristic of the decision maker s subjective information and ambiguity attitude a characteristic of the decision maker s tastes we show that attitudes towards risk are characterized by the shape of u as usual while attitudes towards ambiguity are characterized by the shape of we also derive x 1 e x as the special case of constant ambiguity aversion ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures this characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by epstein zhang 2001 and ghirardato marinacci 2002 one advantage of this model is that the welldeveloped machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes the model is also distinct from many in the literature on ambiguity in that allows smooth rather than kinked indifference curves this leads to different behavior and improved tractability while still sharing the main features e g ellsberg s paradox etc the maxmin eu model e g gilboa and schmeidler 1989 with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion two illustrative applications to portfolio choice are offered

a numerical scheme for the solution of multi order fractional differential equations we present an algorithm for the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 0 combined with suitable initial conditions the fractional derivatives are described in the caputo sense the algorithm is based on adomian s decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components several numerical examples are presented

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

vibrations of mindlin rectangular plates with elastically restrained edges using static timoshenko beam functions with the rayleigh ritz method abstract a set of static timoshenko beam functions is developed as the admissible functions for the free vibration analysis of mindlin rectangular plates with uniform elastic translational and or rotational edge constraints by the use of the rayleigh ritz method this set of beam functions is made up of two parts the transverse deflection functions and rotation functions which are the static solutions of a timoshenko beam under the transverse loads of a fourier sinusoidal series distributed along the length of the beam therefore the effect of shear deformation on the admissible functions is taken into account in fact the beam is considered to be a unit width strip taken from the rectangular plate in a direction parallel to the edge of the plate and has the end supports corresponding to the edge conditions of the particular plate under consideration it can be seen that the method is sound theoretically each of the beam functions is a polynomial function of not more than third order plus a sine or cosine function for all cases and the change of the boundary supports only results in a corresponding change of the coefficients of the low order polynomial therefore a unified program with little computational efforts can be easily developed for the vibration analysis of mindlin rectangular plates with arbitrary boundary conditions and thickness span ratios the accuracy and efficiency of the proposed method are demonstrated by comparison and convergency studies finally sets of reasonably accurate natural frequencies are presented which can serve as a supplement to the database of vibration characteristics of moderately thick plates

numerical methods for multi term fractional arbitrary orders differential equations our main concern here is to give a numerical scheme to solve a nonlinear multi term fractional arbitrary orders differential equation

fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two component mixture fractional calculus is a powerful tool that has been applied successfully for the analysis of the complex systems one interesting example of a complex mixture is given by the multicomponent pharmaceutical samples having constant matrix content the main aim of this study is to develop a new approach based on the combined use of the fractional wavelet transform fwt and the continuous wavelet transform cwt in order to quantify atorvastatin ato and amlodipine aml in their mixtures without requiring a chemical pretreatment in the first step the absorption spectra of the compounds and their samples were processed by the fwt method in the next step the cwt approach was applied to the fractional wavelet spectra obtained in the above step the aim of the application of fwt is data reduction corresponding to the spectra of compounds and their commercial samples in the following step the cwt was used for the quantitative resolution of the composite signals of the analyzed compounds after method validation the proposed signal processing methods based on the combined use of the fwt and the cwt were successfully applied to the resolution of the composite spectra for the quantitation of atorvastatin ato and amlodipine aml in tablets

fractional variational principles with delay the fractional variational principles within riemann liouville fractional derivatives in the presence of delay are analyzed the corresponding euler lagrange equations are obtained and one example is analyzed in detail

on the smooth ambiguity model a reply epstein 2009 describes three ellsberg style thought experiments and argues that they pose difficulties for the smooth ambiguity model of decision making under uncertainty developed by klibanoff marinacci and mukerji 2005 we revisit these thought exeperiments and find to the contrary that they either point to strengths of the smooth ambiguity model compared to other models such as the maximum expected utility model gilboa and schmeidler 1989 or in the case of one thought experiment raise criticisms that apply equally to a broad range of current ambiguity models

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

an algorithm for solving multi term diffusion wave equations of fractional order in this paper an algorithm based on a new modified homotopy perturbation method mhpm is presented to obtain approximate solutions of multi term diffusion wave equations of fractional order to illustrate the method some examples are provided the results show the simplicity and the efficiency of the algorithm

fractional differential transform method combined with the adomian polynomials abstract a modification of the fractional differential transform method fdtm for solving nonlinear fractional differential equations fdes is presented in this technique the nonlinear term is replaced by its adomian polynomial of index k then the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index thus nonlinear fdes can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the adomian polynomials numerical examples with different types of nonlinearities are solved and good results are obtained

a semi analytical dqem for free vibration analysis of thick plates with two opposite edges simply supported this paper is introducing a semi analytical differential quadrature element method dqem based on first order shear deformation theory for free vibration analysis of thick plates by decomposing the plate into a series of sub domains or elements any discontinuity in geometry material properties and elastic foundations in one direction can be handled conveniently classical as well as non classical boundary conditions can be imposed one benefits of this semi analytical dqem is reducing the storage requirements and the computational efforts needed to assemble the global equations the method would also increase the computational efficiency in comparison with the finite element method fem or differential quadrature method dqm analysis of such problems the accuracy versatility stability and convergence of the algorithm are certified

rectangular thick plates on winkler foundation differential quadrature element solution abstract this paper deals with the static analysis of homogenous isotropic rectangular plates on winkler foundation on the basis of first order shear deformation theory an improved differential quadrature dq method called the differential quadrature element method dqem has been developed for this analysis the plates considered are subjected to a patch load or a concentrated line load which are not solvable by the global dq method the convergence and comparison studies are carried out to establish the reliability of the dqem results then the numerical results for different boundary conditions i e ssss cccc s s s s and sfsf are presented showing the parametric effects of dimensions of loading area line relative thickness ratio and elastic foundation modulus on the deflection bending and twisting moments and shear forces at selected locations most of these data are new and due to the high accuracy of the dq solution they can be useful for benchmarking future work

ellsberg revisited an experimental study an extension to ellsberg s experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated the sample is decomposed into three main groups subjective expected utility subjects who reduce compound objective lotteries and are ambiguity neutral and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior

ambiguity learning and asset returns we develop a consumption based asset pricing model in which the representative agent is ambiguous about the hidden state in consumption growth he learns about the hidden state under ambiguity by observing past consumption data his preferences are represented by the smooth ambiguity model axiomatized by klibanofi et al 2005 2006 unlike the standard bayesian theory this utility model implies that the posterior of the hidden state and the conditional distribution of the consumption process given a state cannot be reduced to a predictive distribution by calibrating the ambiguity aversion parameter the subjective discount factor and the risk aversion parameter with the latter two values between zero and one our model can match the flrst moments of the equity premium and riskfree rate found in the data in addition our model can generate a variety of dynamic asset pricing phenomena including the procyclical variation of price dividend ratios the countercyclical variation of equity premia and equity volatility and the mean reversion and long horizon predictability of excess returns

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

two alternating direction implicit difference schemes with the extrapolation method for the two dimensional distributed order differential equations the grunwald formula is used to solve the two dimensional distributed order differential equations two alternating direction implicit adi difference schemes are derived it is proved that the schemes are unconditionally stable and convergent with the convergence orders o h 1 2 h 2 2 2 and o h 1 4 h 2 4 4 in h 1 norm respectively where h 1 h 2 and are the step sizes in time space in x and y direction and distributed order the extrapolation method is applied to improve the approximate accuracy to the orders o 2 ln 2 h 1 2 h 2 2 2 and o 2 ln 2 h 1 4 h 2 4 4 respectively several numerical examples are given to confirm the theoretical results

ambiguity and second order belief anscombe and aumann 1963 wrote a classic characterization of subjective expected utility theory this paper employs the same domain for preference and a closely related but weaker set of axioms to characterize preferences that use second order beliefs beliefs over probability measures such preferences are of interest because they accommodate ellsberg type behavior

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method