(2002-5686) Milstein method for solving fuzzy differential equation

On new solutions of fuzzy differential equations

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

With fuzzy set theory, the uncertainty nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs). The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by two type of Bernstein neural networks. We first transform the FDE into four ordinary differential equation (ODEs) with Hukuhara differentiability. Then we construct neural models with the structure of ODEs. With modified backpropagation method for fuzzy variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

Theory of fuzzy differential equations is the important new developments to model various science and engineering problems of uncertain nature because this theory represents a natural way to model dynamical systems under uncertainty. Since, it is too difficult to obtain the exact solution of fuzzy differential equations so one may need reliable and efficient numerical techniques for the solution of fuzzy differential equations. In this chapter we have presented various numerical techniques viz. Euler and improved Euler type methods and Homotopy Perturbation Method (HPM) to solve fuzzy differential equations. Also application problems such as fuzzy continuum reaction diffusion model to analyse the dynamical behaviour of the fire with fuzzy initial condition is investigated. To analyse the fire propagation, the complex fuzzy arithmetic and computation are used to solve hyperbolic reaction diffusion equation. This analysis finds the rate of burning number of trees in bounds where wave variable/ time are defined in terms of fuzzy. Obtained results are compared with the existing solution to show the efficiency of the applied methods.

In this paper, we investigate linear partial differential equations with fuzzy source function, and with fuzzy initial and boundary conditions. Usually, researchers consider solutions of fuzzy differential equations in the form of fuzzy-valued functions. On the contrary, in this study, we are looking for a solution in the form of fuzzy set (bunch) of real functions. To demonstrate the proposed approach we use Dirichlet problem for the heat equation. We assume the source function, and the initial and boundary conditions to be in a special form, which we name as triangular fuzzy function. We show that the uncertainties of the solution due to these parameters are triangular fuzzy functions too. The solution for the example, which we discuss in the paper, is expressed by an analytical formula. If we use numerical methods, we can find the solution in the suggested sense for each problem from the examined class.

In this paper, the stability theory for fuzzy differential equations in the quotient space of fuzzy numbers was essentially investigated with Lyapunov-like functions. Some sufficient criteria for the stability, uniformly stability and expo- nentially stability of the trivial solution of the fuzzy differential equations were obtained by using the differential inequalities and the comparison principle for Lyapunov-like functions.

We consider fuzzy differential equations with data L-R type fuzzy numbers. First we construct special classes of fuzzy numbers that will be closed under algebraic and analytic operations on fuzzy numbers. These spaces allow us to solve fuzzy differential equations. We present a general example of a problem which has fuzzy parameters and also fuzzy initial data.

In this paper, the solutions of fuzzy differential equations (FDEs) are estimated by using two types of Bernstein neural networks. Here, the uncertainties are in the form of Z numbers. Firstly, we transform the FDE to four ordinary differential equations (ODEs) at par with Hukuhara differentiability. After that we develop neural models having the structure of ODEs. By using modified backpropagation technique for Z number variables, the training of neural networks are carried out. The results of the simulation illustrate that these innovative models, Bernstein neural networks, are efficient to approximate the solutions of FDEs which are on the basis of Z-numbers.

Coal fired power plants contribute significantly to greenhouse gas emission, notably CO2. A novel method to effectively reduce the amount of CO2 emission is to cofire a high fraction of biomass and coal at oxygen enriched environments. However, this technique has not been demonstrated on a large-scale yet. The objective of this PhD research is to perform a detailed modeling study of combustion of a single biomass particle and to establish reduced models which can be used for describing the main characteristics of pyrolyzing and combusting single biomass particles to be employed in design codes of industrial furnaces. To accomplish the goals of this PhD thesis, the research has been carried out in two main stages. First, the sub-processes involved in the biomass combustion are identified and described with a one-dimensional mathematical model based on conservation of mass, energy and momentum. The model encompasses the kinetics of biomass pyrolysis, homogeneous reactions and heterogeneous char oxidation and gasification reactions, coupled with transient transport equations. Subsequently, this model is implemented in an in-house code and a comprehensive numerical study on pyrolysis and combustion of single biomass particles is conducted. The accuracy of the model is examined by comparing its predictions with several experimental data obtained from the literature on pyrolysis and combustion of various types of single biomass particles. The computer code based on the detailed model allows one to observe time and space evolution of several parameters including biomass and char densities, gaseous species mass fractions, porosity, internal pressure, mass flux of volatiles within the pores of the solid matrix, and temperature. The model is used for simulation of combustion of particles of three common shapes; i.e. slab, cylinder and sphere. The results of the detailed modeling study reveal that the combustion of a single biomass particle at the conditions of industrial furnaces (small particles and high heating conditions) consists of three main sub-processes: preheating, pyrolysis, and char oxidation. Therefore, any simplified/reduced particle model should account for these three processes. In the next stage of the project, simplified models are developed to predict the main characteristics of pyrolyzing and combusting single biomass particles. Initially, the preheating stage is modeled using a time and space integral method, which allows one to convert the partial differential form of the heat transfer equation into an algebraic equation. This treatment is then applied to model the pyrolysis process. Two possible regimes are identified: thermally thin and thermally thick particles. A model is established for both regimes, which consists of a set of algebraic equations. This treatment highly simplifies the pyrolysis model so that it can be used in practical applications, which may involve thousands or even millions of particles. The validation of the simplified preheating and pyrolysis models is carried out using various experiments and the results of the detailed model based on partial differential equations (PDEs). The char particle oxidation and gasification processes as the last stages of the particle combustion process, are modeled using the shrinking core approximation. The accuracy of this model is assessed using the experiments reported in past studies. The model is used to study the dynamics of biomass char combustion at oxy-fuel conditions. The effects of the main process parameters on the maximum particle temperature and burnout time are examined. It is found that oxy-fuel combustion with an oxygen mass fraction of 0.3 and higher may lead to a considerable reduction in particle temperature and burnout time compared to the conventional operating case with air as the gasifying agent. In the last stage of this PhD research, the simplified models of biomass particle pyrolysis and char combustion are combined to establish a reduced model for combustion of a single biomass particle. The accuracy of this simplified combustion model is examined using the measured data available in the literature as well as the results of the detailed model. As a conclusion, the simplified models developed in this thesis for pyrolysis and combustion of single biomass particles are efficient enough to capture the main process parameters, and computationally cheaper than the PDE -based models so that they can be used in the design codes of biomass furnaces.

In this paper, we use Chebyshev polynomials to seek the numerical solution of a class of multi-variable order fractional differential equation (MVODEs) that the fractional derivative is described in the Caputo-Prabhakar sense. Using operational matrices, the original equations are transferred to a system of algebraic equations. By solving the system of equations, the numerical solutions are acquired that this system may be solved numerically using an iterative algorithm. The effectiveness and convergence analysis of the numerical scheme is illustrated through four numerical examples.

The approximate dynamics of many physical phenomena, including turbulence, can be represented by dissipative systems of ordinary differential equations. One often turns to numerical integration to solve them. There is an incompatibility, however, between the answers it can produce (i.e., specific solution trajectories) and the questions one might wish to ask (e.g., what behavior would be typical in the laboratory?) To determine its outcome, numerical integration requires more detailed initial conditions than a laboratory could normally provide. In place of initial conditions, experiments stipulate how tests should be carried out: only under statistically stationary conditions, for example, or only during asymptotic approach to a final state. Stipulations such as these, rather than initial conditions, are what determine outcomes in the laboratory. This theoretical study examines whether the points of view can be reconciled: What is the relationship between one's statistical stipulations for how an experiment should be carried out--stationarity or asymptotic approach--and the expected results? How might those results be determined without invoking initial conditions explicitly? To answer these questions, stationarity and asymptotic approach conditions are analyzed in detail. Each condition is treated as a statistical constraint on the system--a restriction on the probability density of states that might be occupied when measurements take place. For stationarity, this reasoning leads to a singular, invariant probability density which is already familiar from dynamical systems theory. For asymptotic approach, it leads to a new, more regular probability density field. A conjecture regarding what appears to be a limit relationship between the two densities is presented. By making use of the new probability densities, one can derive output statistics directly, avoiding the need to create or manipulate initial data, and thereby avoiding the conceptual incompatibility mentioned above. This approach also provides a clean way to derive reduced-order models, complete with local and global error estimates, as well as a way to compare existing reduced-order models objectively. The new approach is explored in the context of five separate test problems: a trivial one-dimensional linear system, a damped unforced linear oscillator in two dimensions, the isothermal Rayleigh-Plesset equation, Lorenz's equations, and the Stokes limit of Burgers' equation in one space dimension. In each case, various output statistics are deduced without recourse to initial conditions. Further, reduced-order models are constructed for asymptotic approach of the damped unforced linear oscillator, the isothermal Rayleigh-Plesset system, and Lorenz's equations, and for stationarity of Lorenz's equations.

In this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

Due to transmission losses and lack of initial flow, flood routing in ephemeral streams is not possible with common methods and it is necessary the flood routing models have been developed for these streams. Therefore in this study a computer model for natural river cross section has been developed in that after linearization of partial differential equations of unsteady none uniform flow, they are solved by stagger method. This model can consider tributary flow and infiltration into river bed simultaneity. For estimation of transmission losses Muscat, Davis-Wilson, and Ingham methods have used and liked with unsteady flow equations in prepared model. Evaluation of model accuracy viewpoint programming, ability to simulate uniform flow and satisfying the continuity equation performed using 60 Garasoo River cross sections in a reach with about 18 km length. Lane’s hydrograph and Hughes Wash river properties were used to investigation model accuracy to estimate flow behavior and transmission losses. The result showed that prepared model can simulate uniform flow and satisfies continuity equation with height accuracy. Additionally when Muscat relation is used developed model can predict start and peak flood times correctly. Also transmission losses and volume of output hydrograph have predicted with maximum error less than 20 percent. While application of Davis – Wilson and Ingham relations showed unsatisfied result compare in situ measurement data. Keywords: Transmission losses, Ephemeral stream, Flood Routing, Saint-Venant equations