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## Partial Differential Equations (partial + differential_equation)
Kinds of Partial Differential Equations Terms modified by Partial Differential Equations
## Selected Abstracts## A Cartesian-grid collocation technique with integrated radial basis functions for mixed boundary value problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2010Phong B. H. LeAbstract In this paper, high-order systems are reformulated as first-order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D-integrated radial basis function networks (1D-IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23:1192,1210). The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well-known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009Yi ZhouAbstract In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263,1317; Nonlinear Anal. 1997; 28:1299,1322; Chin. Ann. Math. 2004; 25B:37,56). We give a new, very simple proof of this result and also give a sharp point-wise decay estimate of the solution. Then, we consider the mixed initial-boundary-value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59,78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point-wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Interaction of elementary waves for scalar conservation laws on a bounded domain MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Hongxia LiuAbstract This paper is concerned with the interaction of elementary waves on a bounded domain for scalar conservation laws. The structure and large time asymptotic behaviours of weak entropy solution in the sense of Bardos et al. (Comm. Partial Differential Equations 1979; 4: 1017) are clarified to the initial,boundary problem for scalar conservation laws ut+,(u)x=0 on (0,1) × (0,,), with the initial data u(x,0)=u0(x):=um and the boundary data u(0,t)=u -,u(1,t)=u+, where u±,um are constants, which are not equivalent, and ,,C2 satisfies ,,,>0, ,(0)=f,(0)=0. We also give some global estimates on derivatives of the weak entropy solution. These estimates play important roles in studying the rate of convergence for various approximation methods to scalar conservation laws. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Second-order Galerkin-Lagrange method for the Navier-Stokes equations (retracted article), NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2005Mohamed BensaadaAbstract It has come to the attention of the editors and publisher that an article published in Numerical Methods and Partial Differential Equations, "Second-order Galerkin-Lagrange method for the Navier-Stokes equations," by Mohamed Bensaada, Driss Esselaoui, and Pierre Saramito, Numer Methods Partial Differential Eq 21(6) (2005), 1099,1121 included large portions that were copied from the following paper without proper citation: "Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations," Endre Suli, Numerische Mathematik, Vol. 53, No. 4, pp. 459,486 (July, 1988). We have retracted the paper and apologize to Dr. Suli Numer Methods Partial Differential Eq (2007)23(1)211. [source] ## Hamiltonian particle-mesh simulations for a non-hydrostatic vertical slice model ATMOSPHERIC SCIENCE LETTERS, Issue 4 2009Seoleun ShinAbstract A Lagrangian particle method is developed for the simulation of atmospheric flows in a non-hydrostatic vertical slice model. The proposed particle method is an extension of the Hamiltonian particle mesh (HPM) [Frank J, Gottwald G, Reich S. 2002. The Hamiltonian particle-mesh method. In Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 26, Griebel M, Schweitzer M (eds). Springer-Verlag: Berlin Heidelberg; 131,142] and provides preservation of mass, momentum, and energy. We tested the method for the gravity wave test in Skamarock W, Klemp J. 1994. Efficiency and accuracy of the Klemp-Wilhelmson time-splitting technique. Monthly Weather Review 122: 2623,2630 and the bubble experiments in Robert A. 1993. Bubble convection experiments with a semi-implicit formulation of the Euler equations. Journal of the Atmospheric Sciences 50: 1865,1873. The accuracy of the solutions from the HPM simulation is comparable to those reported in these references. A particularly appealing aspect of the method is in its non-diffusive transport of potential temperature. The solutions are maintained smooth largely due to a ,regularization' of pressure, which is controlled carefully to preserve the total energy and the time-reversibility of the model. In case of the bubble experiments, one also needs to regularize the buoyancy contributions. The simulations demonstrate that particle methods are potentially applicable to non-hydrostatic atmospheric flow regimes and that they lead to a highly accurate transport of materially conserved quantities such as potential temperature under adiabatic flow regimes. Copyright © 2009 Royal Meteorological Society [source] ## Partial differential equations of chemotaxis and angiogenesis MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2001B. D. SleemanThe topic of this paper is concerned with an investigation of the qualitative properties of solutions to the following problem. Let ,,Rn be a bounded domain with boundary ,,. We seek solutions P,,,Rm+1 of the system (1) subject to the ,no-flux' boundary condition (2) where n denotes the inward pointing normal to ,,. To close the system we prescribe the initial conditions (3) In this system D is a constant diffusion coefficient, P is a population density and , is a vector of nutrients or chemicals whose dynamics influences the movement of P. Notice here that the substances , do not diffuse. If they do then the second equation of (1) is modified to (4) where d is a positive semi-definite diagonal matrix. This more general system includes the so-called Keller,Segel model of Biology ([1] Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology 1970; 26: 339,415). To motivate our study of system (1),(3) we begin by outlining two themes. One basic to developmental biology and the other from angiogenesis. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005Mehdi DehghanAbstract Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] ## He's homotopy perturbation method for two-dimensional heat conduction equation: Comparison with finite element method HEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 4 2010M. JalaalAbstract Heat conduction appears in almost all natural and industrial processes. In the current study, a two-dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy-based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20292 [source] ## Axisymmetric interaction of a rigid disc with a transversely isotropic half-space INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 12 2010Amir Aabbas KatebiAbstract A theoretical formulation is presented for the determination of the interaction of a vertically loaded disc embedded in a transversely isotropic half-space. By means of a complete representation using a displacement potential, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order partial differential equation. With the aid of Hankel transforms, a relaxed treatment of the mixed-boundary value problem is formulated as dual integral equations, which, in turn, are reduced to a Fredholm equation of the second kind. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary-layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy and compared with existing solutions. Further numerical examples are also presented to elucidate the influence of the degree of the material anisotropy on the response. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodes INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2008Hang MaAbstract To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Boundary solution of Poisson's equation using radial basis function collocated on Gaussian quadrature nodes INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2001M. ElansariAbstract In the solution of Poisson's equation using either the dual reciprocity boundary element method or the method of fundamental solution, radial basis functions (RBFs) are used to approximate the right-hand side of the governing partial differential equation to eliminate the domain integration. This paper shows that if the RBF interpolation is collocated on the Gaussian quadrature nodes, we seem to observe superconvergence behaviour. This behaviour is demonstrated using a series of numerical examples. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Torsion of orthotropic bars with L -shaped or cruciform cross-section INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001Y. Z. ChenAbstract For an orthotropic torsion bar with L -shaped or cruciform cross-section, the studied torsion problem can be reduced to a boundary value problem of elliptic partial differential equation. The studied region is separated into several rectangular sub-regions, and the series solution is suggested to solve the problem for the individual sub-region. By using the continuation condition for the functions on the neighbouring sub-regions, the investigated solution is obtainable. Finally, numerical results for the torsion rigidities of bars are given to demonstrate the influence of the degree of orthotropy. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain-gradient theory of elasticity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010G. F. KarlisAbstract An advanced boundary element method (BEM) for solving two- (2D) and three-dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form-II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6-7 2009F. NobileAbstract We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen,Ločve expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Local discretization error bounds using interval boundary element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009B. F. ZalewskiAbstract In this paper, a method to account for the point-wise discretization error in the solution for boundary element method is developed. Interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented. The developed method does not assume any special properties besides the Laplace equation being a linear elliptic partial differential equation whose Green's function for an isotropic media is known. Numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A space,time discontinuous Galerkin method for the solution of the wave equation in the time domain INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2009Steffen PetersenAbstract In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space,time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave-like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space,time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A dynamic approach for evaluating parameters in a numerical method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005A. A. OberaiAbstract A new methodology for evaluating unknown parameters in a numerical method for solving a partial differential equation is developed. The main result is the identification of a functional form for the parameters which is derived by requiring the numerical method to yield ,optimal' solutions over a set of finite-dimensional function spaces. The functional depends upon the numerical solution, the forcing function, the set of function spaces, and the definition of the optimal solution. It does not require exact or approximate analytical solutions of the continuous problem, and is derived from an extension of the variational Germano identity. This methodology is applied to the one-dimensional, linear advection,diffusion problem to yield a non-linear dynamic diffusivity method. It is found that this method yields results that are commensurate to the SUPG method. The same methodology is then used to evaluate the Smagorinsky eddy viscosity for the large eddy simulation of the decay of homogeneous isotropic turbulence in three dimensions. In this case the resulting method is found to be more accurate than the constant-coefficient and the traditional dynamic versions of the Smagorinsky model. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Non-reflecting artificial boundaries for transient scalar wave propagation in a two-dimensional infinite homogeneous layer INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2003Chongbin ZhaoAbstract This paper presents an exact non-reflecting boundary condition for dealing with transient scalar wave propagation problems in a two-dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x ,, and another extends toward x,- ,, the non-reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time-dependent partial differential equation with only one spatial variable, can be further changed into a linear first-order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non-reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non-reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi-infinite media. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Adaptive moving mesh methods for simulating one-dimensional groundwater problems with sharp moving fronts INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2002Weizhang HuangAbstract Accurate modelling of groundwater flow and transport with sharp moving fronts often involves high computational cost, when a fixed/uniform mesh is used. In this paper, we investigate the modelling of groundwater problems using a particular adaptive mesh method called the moving mesh partial differential equation approach. With this approach, the mesh is dynamically relocated through a partial differential equation to capture the evolving sharp fronts with a relatively small number of grid points. The mesh movement and physical system modelling are realized by solving the mesh movement and physical partial differential equations alternately. The method is applied to the modelling of a range of groundwater problems, including advection dominated chemical transport and reaction, non-linear infiltration in soil, and the coupling of density dependent flow and transport. Numerical results demonstrate that sharp moving fronts can be accurately and efficiently captured by the moving mesh approach. Also addressed are important implementation strategies, e.g. the construction of the monitor function based on the interpolation error, control of mesh concentration, and two-layer mesh movement. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## A control volume capacitance method for solidification modelling with mass transport INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002K. DaveyAbstract Capacitance methods are popular methods used for solidification modelling. Unfortunately, they suffer from a major drawback in that energy is not correctly transported through elements and so provides a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where mass transport and solidification are combined. The approach adopted is founded on theory that describes energy transfer through a control volume (CV) moving relative to the transporting mass. An equivalent governing partial differential equation is established, which is designed to be transformable into a finite element system that is commonly used to model transient heat-conduction problems. This approach circumvents the need to use the methods of Bubnov,Galerkin and Petrov,Galerkin and thus eliminates many of the stability problems associated with these approaches. An integration scheme is described that accurately caters for enthalpy fluxes generated by mass transport. Shrinkage effects are neglected in this paper as all the problems considered involve magnitudes of velocity that make this assumption reasonable. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Instationary aeroelastic computation of yacht sails INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001Heinrich SchoopAbstract Effective schemes exist to calculate aerodynamic forces for thin bodies and structural dynamics of flexible membranes. The fluid dynamic of thin wings in a irrotational flow leads to the lifting surface theory. Neglecting the inertia of the membrane the structural dynamics are solved by the non-linear (FEM). But the interaction of flexible membranes and an irrotational flow causes problems due to the different nature of the mathematical equations. On the one hand, there is a partial differential equation for the structural dynamics and on the other hand, there is a singular integral equation for the aerodynamics. The numerical discretization scheme has to fit these different types of equation. Our work introduces a new interaction scheme to couple the singular integral equation of the lifting surface theory with the non-linear FEM of the membrane static. The fundamental examinations, showed by Schoop et al. (International Journal for Numerical Methods in Engineering 1998; 41: 217,219), are applied to realistic sail geometries and the aerodynamics is extended to instationary flow conditions. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Further constructive results on interconnection and damping assignment control of mechanical systems: the Acrobot example INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 14 2006Arun D. MahindrakarAbstract Interconnection and damping assignment passivity-based control is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Hamiltonian structure with a desired energy function,that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. A class of underactuation degree one systems for which the partial differential equations can be explicitly solved,making the procedure truly constructive,was recently reported by the authors. In this brief note, largely motivated by the interesting Acrobot example, we pursue this investigation for two degrees-of-freedom systems where a constant inertia matrix can be assigned. We concentrate then our attention on potential energy shaping and give conditions under which an explicit solution of the associated partial differential equation can be obtained. Using these results we show that it is possible to swing-up the Acrobot from some configuration positions in the lower half plane, provided some conditions on the robot parameters are satisfied. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Generalized Diffusion Tensor Imaging (GDTI): A Method for Characterizing and Imaging Diffusion Anisotropy Caused by Non-Gaussian Diffusion ISRAEL JOURNAL OF CHEMISTRY, Issue 1-2 2003Chunlei LiuFor non-Gaussian distributed random displacement, which is common in restricted diffusion, a second-order diffusion tensor is incapable of fully characterizing the diffusion process. The insufficiency of a second-order tensor is evident in the limited capability of diffusion tensor imaging (DTI) in resolving multiple fiber orientations within one voxel of human white matter. A generalized diffusion tensor imaging (GDTI) method was recently proposed to solve this problem by generalizing Fick's law to a higher-order partial differential equation (PDE). The relationship between the higher-order tensor coefficients of the PDE and the higher-order cumulants of the random displacement can be derived. The statistical property of the diffusion process was fully characterized via the higher-order tensor coefficients by reconstructing the probability density function (PDF) of the molecular random displacement. Those higher-order tensor coefficients can be measured using conventional diffusion-weighted imaging or spectroscopy techniques. Simulations demonstrated that this method was capable of quantitatively characterizing non-Gaussian diffusion and accurately resolving multiple fiber orientations. It can be shown that this method is consistent with the q-space approach. The second-order approximation of GDTI was shown to be DTI. [source] ## Dynamic optimization of the methylmethacrylate cell-cast process for plastic sheet production AICHE JOURNAL, Issue 6 2009Martín Rivera-ToledoAbstract Traditionally, the methylmethacrylate (MMA) polymerization reaction process for plastic sheet production has been carried out using warming baths. However, it has been observed that the manufactured polymer tends to feature poor homogeneity characteristics measured in terms of properties like molecular weight distribution. Nonhomogeneous polymer properties should be avoided because they give rise to a product with undesired wide quality characteristics. To improve homogeneity properties force-circulated warm air reactors have been proposed, such reactors are normally operated under isothermal air temperature conditions. However, we demonstrate that dynamic optimal warming temperature profiles lead to a polymer sheet with better homogeneity characteristics, especially when compared against simple isothermal operating policies. In this work, the dynamic optimization of a heating and polymerization reaction process for plastic sheet production in a force-circulated warm air reactor is addressed. The optimization formulation is based on the dynamic representation of the two-directional heating and reaction process taking place within the system, and includes kinetic equations for the bulk free radical polymerization reactions of MMA. The mathematical model is cast as a time dependent partial differential equation (PDE) system, the optimal heating profile calculation turns out to be a dynamic optimization problem embedded in a distributed parameter system. A simultaneous optimization approach is selected to solve the dynamic optimization problem. Trough full discretization of all decision variables, a nonlinear programming (NLP) model is obtained and solved by using the IPOPT optimization solver. The results are presented about the dynamic optimization for two plastic sheets of different thickness and compared them against simple operating policies. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source] ## Robust detection and accommodation of incipient component and actuator faults in nonlinear distributed processes AICHE JOURNAL, Issue 10 2008Antonios ArmaouAbstract A class of nonlinear distributed processes with component and actuator faults is presented. An adaptive detection observer with a time varying threshold is proposed that provides additional robustness with respect to false declarations of faults and minimizes the fault detection time. Additionally, an adaptive diagnostic observer is proposed that is subsequently utilized in an automated control reconfiguration scheme that accommodates the component and actuator faults. An integrated optimal actuator location and fault accommodation scheme is provided in which the actuator locations are chosen in order to provide additional robustness with respect to actuator and component faults. Simulation studies of the Kuramoto-Sivashinsky nonlinear partial differential equation are included to demonstrate the proposed fault detection and accommodation scheme. © 2008 American Institute of Chemical Engineers AIChE J, 2008 [source] ## Air drying of milk droplet under constant and time-dependent conditions AICHE JOURNAL, Issue 6 2005Xiao Dong ChenAbstract Spray drying is the prime process for many years for manufacturing food powders. Dairy powders are one of the main products consumed worldwide. There has been a stream of studies published previously on both modeling the drying characteristics of a single milk droplet and the dryer wide simulations incorporating computational fluid dynamics (CFD). In CFD simulations, large numbers of particles of different sizes need be tracked to represent the size distribution; it is desirable to have an accurate yet simple model for drying of a single droplet, which does not require partial differential equation. Here for the first time, two such models are validated. One model is of the characteristic drying rate curve approach and the other (new) model is of the reaction engineering approach. The model predictions are compared against a very wide range of experimental results including isothermal and time-varying temperature conditions. © 2005 American Institute of Chemical Engineers AIChE J, 2005 [source] ## LOCAL WELL-POSEDNESS OF MUSIELA'S SPDE WITH LÉVY NOISE MATHEMATICAL FINANCE, Issue 3 2010Carlo MarinelliWe determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight. [source] ## STOCHASTIC HYPERBOLIC DYNAMICS FOR INFINITE-DIMENSIONAL FORWARD RATES AND OPTION PRICING MATHEMATICAL FINANCE, Issue 1 2005Shin Ichi AiharaWe model the term-structure modeling of interest rates by considering the forward rate as the solution of a stochastic hyperbolic partial differential equation. First, we study the arbitrage-free model of the term structure and explore the completeness of the market. We then derive results for the pricing of general contingent claims. Finally we obtain an explicit formula for a forward rate cap in the Gaussian framework from the general results. [source] ## Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2009Hassan A. ZedanAbstract In this paper, the method of deriving the Bäcklund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrödinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Zero-coupon bond prices in the Vasicek and CIR models: Their computation as group-invariant solutions, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2008W. SinkalaAbstract We compute prices of zero-coupon bonds in the Vasicek and Cox,Ingersoll,Ross interest rate models as group-invariant solutions. Firstly, we determine the symmetries of the valuation partial differential equation that are compatible with the terminal condition and then seek the desired solution among the invariant solutions arising from these symmetries. We also point to other possible studies on these models using the symmetries admitted by the valuation partial differential equations. Copyright © 2007 John Wiley & Sons, Ltd. [source] |