Complex Partial Differential Equations Research Articles

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

PurposeThis paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.Design/methodology/approachA bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.FindingsBi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.Originality/valueADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

Homogenization of the dielectric and magnetic permeability tensors for anisotropic and bianisotropic layered media

The paper deals with the problem of calculating the effective properties of a multilayer system consisting of layers of anisotropic or bianisotropic media. It is assumed that the electromagnetic field is described by the Maxwell equations, and the layers are coordinate surfaces in some orthogonal coordinate system. No assumptions are made in advance about the properties of the dielectric and magnetic permeability tensors. The thickness of the layers is considered to be small in comparison with the wavelengths of all waves that can propagate in the system (long-wave approximation) and in comparison with the curvature radii of the layers as well as with the characteristic size of inhomogeneity in the layers. To construct an approximate solution, it is proposed to use a generalization of the matrix homogenization method. For the applicability of the method, it is necessary to rewrite the system of initial equations for a layered medium in a special form. The main advantage of the proposed approach, to find the parameters of an effective medium, is the absence of the need to solve complex partial differential equations. It is enough only to operate with matrices. The matrix homogenization method also makes it possible to obtain effective characteristics of a periodic structure with each period consisting of a set of thin layers. The study is illustrated by examples of finding effective tensors of dielectric and magnetic permeabilities for a system of planar, cylindrical, and spherical layers. In addition, the dependence of the magnetoelectric coefficient on the layer thicknesses for the cases listed above are compared.

A Physics-Based Electrochemical Model for Lithium-Ion Battery State-of-Charge Estimation Solved by an Optimised Projection-Based Method and Moving-Window Filtering

State-of-charge (SOC) is one of the most critical parameters in battery management systems (BMSs). SOC is defined as the percentage of the remaining charge inside a battery to the full charge, and thus ranges from 0% to 100%. This percentage value provides important information to manufacturers about the performance of the battery and can help end-users identify when the battery must be recharged. Inaccurate estimation of the battery SOC may cause over-charge or over-discharge events with significant implications for system safety and reliability. Therefore, it is crucial to develop methods for improving the estimation accuracy of battery SOC. This paper presents an electrochemical model for lithium-ion battery SOC estimation involving the battery’s internal physical and chemical properties such as lithium concentrations. To solve the computationally complex solid-phase diffusion partial differential equations (PDEs) in the model, an efficient method based on projection with optimized basis functions is presented. Then, a novel moving-window filtering (MWF) algorithm is developed to improve the convergence rate of the state filters. The results show that the developed electrochemical model generates 20 times fewer equations compared with finite difference-based methods without losing accuracy. In addition, the proposed projection-based solution method is three times more efficient than the conventional state filtering methods such as Kalman filter.

On a type of matrix splitting preconditioners for a class of block two-by-two linear systems

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting iterative method, called preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a preconditioner. In this study, the CGNR method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned Krylov subspace methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.

A complex elastographic hyperbolic solver (CEHS) to recover frequency dependent complex shear moduli in viscoelastic models utilizing one or more displacement data-sets

In this paper, we present a linear marching scheme to recover frequency-dependent complex shear moduli in viscoelastic models utilizing two sets of single component displacement data. The proposed method is designed to provide stable and accurate estimation of the tissue viscoelastic stiffness parameters by solving a first-order complex partial differential equation. To control the exponential growth of the numerical error resulting from one of the complex coefficients in the inverse equation, a modified upwind discretization is utilized on the first-order derivative terms of the target parameter. The algorithm is fully stablized when: (1) carefully chosen multiple data-sets are combined to eliminate the remaining complex coefficient that contributes to exponential error growth; and (2) a modified Tikhonov regularization is applied to the inversion method. We obtain the stability result in the norm so that the numerical scheme is convergent at fractional 1 / 2 order. Its performance is compared with the performance of the Algebraic Inversion Model previously investigated. We present shear modulus reconstructions from synthetic data, from laboratory phantom data and match frequency-dependent complex moduli from phantom data to several viscoelastic models. Since we have previously presented phase wave speed images from interference patterns, we exhibit those images here for comparison.

Entropy Generation in Thermal Radiative Loading of Structures with Distinct Heaters

Thermal loading by radiant heaters is used in building heating and hot structure design applications. In this research, characteristics of the thermal radiative heating of an enclosure by a distinct heater are investigated from the second law of thermodynamics point of view. The governing equations of conservation of mass, momentum, and energy (fluid and solid) are solved by the finite volume method and the semi-implicit method for pressure linked equations (SIMPLE) algorithm. Radiant heaters are modeled by constant heat flux elements, and the lower wall is held at a constant temperature while the other boundaries are adiabatic. The thermal conductivity and viscosity of the fluid are temperature-dependent, which leads to complex partial differential equations with nonlinear coefficients. The parameter study is done based on the amount of thermal load (presented by heating number) as well as geometrical configuration parameters, such as the aspect ratio of the enclosure and the radiant heater number. The results present the effect of thermal and geometrical parameters on entropy generation and the distribution field. Furthermore, the effect of thermal radiative heating on both of the components of entropy generation (viscous dissipation and heat dissipation) is investigated.

Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino singular integrals

The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincare–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in $$\mathbb {R}^4$$ is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system.

A New Covariance Function and Spatio‐Temporal Prediction (Kriging) for A Stationary Spatio‐Temporal Random Process

Consider a stationary spatio‐temporal random process and let be a sample from the process. Our object here is to predict, given the sample, for all t at the location so. To obtain the predictors, we define a sequence of discrete Fourier transforms using the observed time series. We consider these discrete Fourier transforms as a sample from the complex valued random variable . Assuming that the discrete Fourier transforms satisfy a complex stochastic partial differential equation of the Laplacian type with a scaling function that is a polynomial in the temporal spectral frequency ω, we obtain, in a closed form, expressions for the second‐order spatio‐temporal spectrum and the covariance function. The spectral density function obtained corresponds to a non‐separable random process. The optimal predictor of the discrete Fourier transform is in terms of the covariance functions. The estimation of the parameters of the spatio‐temporal covariance function is considered and is based on the recently introduced frequency variogram method. The methods given here can be extended to situations where the observations are corrupted by independent white noise. The methods are illustrated with a real data set.

Wave motion equation and the dynamic Green’s function for a transverse isotropic multilayered half-space

A new formulation is presented here for harmonic wave motion in a transverse isotropic multilayered half-space. By means of a Fourier-Bessel transform, the complex partial differential equations of wave motion can be uncoupled into a pair of second order ordinary differential equations: one for SV-P vectorial waves (matrix size 2×2) and the other for SH scalar waves (matrix size 1×1). They have the same form as that for isotropic media. Thus, the same solution procedure as that for isotropic media is equally applicable to transverse isotropic media, which considerably simplifies the solution. Furthermore, by introducing a mixed variable formulation of the wave motion solution, the matrix form of Green’s function for various boundary conditions of stratified soil is analytically derived. Numerical examples of Green’s function and the dynamic foundation impedance demonstrate the accuracy and the efficiency of the proposed approach. The computation is unconditionally stable.

A stochastically weighted operator splitting Monte Carlo (SWOSMC) method for the numerical simulation of complex aerosol dynamic processes

PurposeThe purpose of this paper is to study the complex aerosol dynamic processes by using this newly developed stochastically weighted operator splitting Monte Carlo (SWOSMC) method.Design/methodology/approachStochastically weighted particle method and operator splitting method are coupled to formulate the SWOSMC method for the numerical simulation of particle-fluid systems undergoing the complex simultaneous processes.FindingsThis SWOSMC method is first validated by comparing its numerical simulation results of constant rate coagulation and linear rate condensation with the corresponding analytical solutions. Coagulation and nucleation cases are further studied whose results are compared with the sectional method in excellent agreement. This SWOSMC method has also demonstrated its high numerical simulation capability when used to deal with simultaneous aerosol dynamic processes including coagulation, nucleation and condensation.Originality/valueThere always exists conflict and tradeoffs between computational cost and accuracy for Monte Carlo-based methods for the numerical simulation of aerosol dynamics. The operator splitting method has been widely used in solving complex partial differential equations, while the stochastic-weighted particle method has been commonly used in numerical simulation of aerosol dynamics. However, the integration of these two methods has not been well investigated.

Symmetry reductions, exact equations and the conservation laws of the generalized (3+1) dimensional Zakharov-Kuznetsov equation

Because the nonlinear evolution equations can describe the complex phenomena of physical, chemical and biological field, many methods have been proposed for investigating such types of equations, and the Lie symmetry analysis method is one of the powerful tools for studying the nonlinear evolution equations. By using the Lie symmetry analysis method, we can obtain the symmetries, reduced equations, group invariant solutions, conservation laws, etc. In the reduction process, we can reduce the order and dimension of the equations, and a complex partial differential equations (PDE) can be reduced to ordinary differential equations directly, which simplifies the solving process. Meanwhile, the symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. For example, we can obtain a lot of new exact solutions by the known symmetries of the original equation; through the analysis of the special form of solution we can better explain some physical phenomena. In addition, the studying of conservation laws and symmetry groups is also the central topic of physical sciencein both classical mechanics and quantum mechanics. Lie symmetry analysis method is suitable for not only constant coefficient equations, but also variable coefficient equations and PDE systems. By using Lie symmetry analysis method, the symmetries and corresponding symmetry reductions of the (3+1) dimensional generalized Zakharov-Kuzetsov (ZK) equation are obtained. Combining the homogeneous balance principle, the trial function method and exponential function method, the group invariant solutions and some new exact explicit solutions are obtained, including the shock wave solutions, solitary wave solutions, etc. Then, we give the conservation laws of the generalized (3+1) dimensional ZK equation in terms of the Lagrangian and adjoint equation method.

Passivity-Based Control for a Micro Air Vehicle Using Unit Quaternions

In this paper the development and practical implementation of a Passivity-Based Control (PBC) algorithm to stabilize an Unmanned Aerial Vehicle (UAV) described with unit quaternions are presented. First, a mathematical model based on Euler-Lagrange formulation using a logarithmic mapping in the quaternion space is introduced. Then, a new methodology: a quaternion-passivity-based control is derived, which does not compute excessive and complex Partial Differential Equations (PDEs) for synthesizing the control law, making a significant advantage in comparison with other methodologies. Therefore, the control design to a system as the quad-rotor is easily solved by the proposed methodology. Another advantage is the possibility to stabilize quad-rotor full dynamics which may not be possible with classical PBC techniques. Experimental results and numerical simulations to validate our proposed scheme are presented.

Derivation and Tuning of a Compact Differential-Algebraic Equations Model for LiFePO4-Graphite Li-Ion Batteries

This paper presents a procedure for deriving and tuning a compact and solvable differential-algebraic equations (DAE) model for the LiFePO4-graphite battery cell. Electrochemical models of the battery cell are typically represented by complex partial differential equations of state variables and parameters whose relationships can be highly nonlinear. Several software packages are available for numerically solving these equations for simulations. However, their long computing time due to the complexity of the model is a major bottleneck for control and monitoring applications. A reduced order model (ROM) can drastically decrease the simulation time with a minimal loss of the prediction accuracy. A ROM is constructed by simplifying or approximating parameters of the full-order model of the battery cell. For on-line diagnostics and real-time control purposes, a lower-order compact representation of complicated transport and diffusion phenomena of a lithium-ion battery is highly desirable. In this paper, the psedo-2D (P2D) electrochemical model of the battery cell is simplified based on polynomial representations of pore-wall flux and lithium concentrations in a similar manner to previous studies. However, instead of adopting the standard Galerkin method which tends to increase the number of equations, this paper adopts Subramanian et al.’s method which demonstrated a potential for producing a compact description of a lithium-ion battery cell. While a traditional Galerkin method requires selecting particular polynomial representations which satisfy the boundary conditions, Subramanian et al.’s method selects a simple polynomial which can be analytically integrated foregoing setting up separate polynomials for solid and electrolyte potentials. This technique, therefore, has a potential to drastically reduce the size of simplified system. However, since the boundary conditions should be satisfied by additional analytical equations, one requires judicious choice of Galerkin formulations for electrolyte concentration in order to build a solvable DAE system. This paper introduces a systematic way of assessing a Galerkin formulation for the electrolyte concentration based on the theorem of Weierstrass. It is shown that each different Galerkin formulation converts the governing equation and its boundary conditions for the electrolyte concentration into a different set of linear DAE’s. However, not all DAE systems are solvable and some may induce numerical instabilities. It is shown that a full rank of the matrix pencil can ensure that a certain Galerkin formulation produces a solvable DAE system. Usefulness of the proposed method is illustrated by comparing several different Galerkin formulations for the LiFePO4-graphite system. In addition, this paper improves the accuracy of the model by accounting for the hysteresis of open-circuit voltage in LiFePO4 using a differential equation model. When the order of the polynomials are chosen at fours, the P2D model of the cell is simplified into a system of 24 DAE’s. This size is significantly smaller than those of comparable ROMs in the literature. The developed DAE system is solved by using SUNDIAL’s IDA solver where a typical charge/discharge cycle can be simulated under 10 seconds on a regular personal computer. The parameters of the developed DAE model are tuned based on charge/discharge experimental data from a commercial LiFePO4-graphite battery. In this paper, a systematic tuning of the model parameters is investigated by exploiting the fast simulation capability of the developed model. Based on a sensitivity study, six model parameters are chosen for tuning. These tuning parameters, thus selected, include the solid phase diffusivity of the negative electrode, initial hysteresis parameters, reaction rate constants, and the contact resistance. The genetic algorithm is applied to simultaneously find the six model parameters that minimize the sum of squared errors in charge/discharge voltages. The tuned model shows a good agreement with the experimental data at rates upto 4C. Contributions of this paper can be summarized as follows. This paper proposes a new method to ensure a compact DAE system is solvable when it is driven from the original higher-order model of the Lithium-ion battery cell. Moreover, a systematic tuning of the model parameters using the global optimization method is demonstrated by exploiting the computational efficiency of the simplified model. When coupled with the model for describing the hysteresis of the battery cell, the tuned model shows a good agreement with the experimental data from a LiFePO4-graphite battery cell.

The approximation theorem of Baouendi and Treves

This article presents some of the applications of the Baouendi–Treves approximation theorem to complex analysis and partial differential equations. An extension of the theorem for vector fields with only continuous coefficients is also discussed

A fractional order variational model for the robust estimation of optical flow from image sequences

In this paper, a fractional order variational model for estimating the optical flow is presented. In particular, the proposed model generalizes the integer order variational optical flow models. The fractional order derivative describes discontinuous information about texture and edges, and therefore a more suitable in estimating the optical flow. The proposed variational functional is a combination of a global model of Horn and Schunck and the classical model of Nagel and Enkelmann. This formulation yields a dense flow and preserves discontinuities in the flow field and also provides a significant robustness against outliers. The Grünwald–Letnikov derivative is used for solving complex fractional order partial differential equations. The corresponding linear system of equations is solved by an efficient numerical scheme. A detailed stability and convergence analysis is given in order to show the mathematical applicability of the numerical algorithm. Experimental results on various datasets verify the validity of the proposed model.

Modelling of Complexities in PA of Laplace Equation

Dominant trends in parallel and distributed computing (parallel computers, parallel algorithms) aim to connect a number of powerful workstations (WS) based on single personal computers (PC) or symmetrical multiprocessor system (SMP) to solve complex problems in a parallel way. Typical representatives of these trends are network of workstation (NOW) and Grid as a high integrated network of NOW modules. In relation to it there is a bad need of unified modelling in traditionally evolved parallel computing and distributed computing as two separate research disciplines. Likewise current trends in high performance computing (HPC) aim to use NOW modules based on SMP parallel computers as cheaper alternatives to the traditionally used massive parallel multiprocessors or supercomputers and to profit from unifying modelling in parallel and distributed computing too. To exploit the parallel processing capability of such cluster, any application program must be paralleled. A way how to do it (decomposition model) belongs to a most important step in developing process of effective parallel algorithm. This article at first discusses the development of effective (optimised) iterative parallel algorithms (IPA) to solve partial differential equations (PDE). By carefully choosing an illustrative typical problem of PDE (Laplace equation) the proposed research has demonstrated the most important complexities in the process of complete modelling of substantial complexities and their optimisation in order to develop an effective PA. Such approach could be used as illustrative example for a complete modelling and effective solution of even other more complex PDE´s in a parallel way.

Guest Editorial: Shape Analysis Beyond the Eikonal Equation

“Shape Analysis” is an interesting research area which contains many thought-provoking problems and mathematical challenges, especially at the representation level. Shape of an object is considered as the enclosing surface (the boundary) or the enclosed interior. One of the major challenges is to define representations of “shapes” that can be manipulated by image processing methods. This is an important need in practice because more often than not shapes need to be extracted or recovered from raw images; moreover, the process of recovering shapes from raw images highly benefits from prior knowledge of candidate shapes. One of the key means of shape representation is the Euclidean Distance Transform (EDT), which is governed by the Eikonal Equation. The values of the EDT at each interior point give the distance to the nearest boundary point. This distance notion is used in almost any problem that requires integrating shape and image information; e.g., a rich variety of shape guided image-processing or shape extraction tasks. Especially in the 1990s, the idea of embedding shapes as zeros of the signed EDT is highly popularized by the level-set framework and its fast implementation, leading to a booming research area in implicit shape representations. However, as challenging problems are developed, formulations that move beyond the classical Eikonal Equation emerged. This special issue consists of six papers each exploring an interesting direction moving beyond the Eikonal formulation in the field of Shape Analysis. The first two papers, “Analysis and Approximation of some Shape–from–Shading Models for non-Lambertian Surfaces” and “A Unified Variational Volume Registration Method based on Automatically Learned Brain Structures”, deal with two key applications that jointly involve shapes and images. In the first paper, Silvia Tozza and Maurizio Falcone present a unified mathematical formulation of some non-Lambertian models, which lead to more complex stationary nonlinear partial differential equations of Hamilton-Jacobi type which can be regarded as a generalization of the classical Eikonal Equation corresponding to the Lambertian case. In the second paper, Carl Lederman et al. present an active volume technique employing level sets of the distance function. Unlike the common practice, the level sets in their formulation are only utilized on their respective regions and not on the whole domain, making the computations feasible as the number of regions increases. The next four papers propose alternative distance notions or computations. They are briefly discussed as follows. In “Affine Invariant Distance Using Multiscale Analysis”, Luis Alvarez et al. present a distance transform for planar shapes that is invariant under any planar projection of the

Analysis and Approximation of Some Shape-from-Shading Models for Non-Lambertian Surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal-type equation when the light source is on the vertical axis. In this paper, we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary non-linear partial differential equations of Hamilton---Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren---Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren---Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.

A real-time Lie-group differential algebraic equations method to solve the inverse nonlinear vibration problems

Desiring a real-time recovery of an unknown external force in the nonlinear inverse vibration problem, we transform the nonlinear ordinary differential equation (ODE) of motion into a nonlinear parabolic type partial differential equation (PDE), which can raise the robustness against large noise. Then, we come to a heat source identification problem, of which the numerical method of lines is used to discretize the PDE into a system of differential algebraic equations (DAEs). Consequently, we can develop an implicit Lie-group scheme and a Newton algorithm to stably solve the DAEs for finding the unknown force, damping function, or stiffness function, which is well recovered even under a large noise. Superficially, we seem to transform a simple ODE to a more complex PDE; however, the advantages gained in this transformation will be seen when we test some nonlinear inverse vibration problems with large time span and under large noise.

Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems

The superconvergence in the finite element method is a phenomenon in which the fi-nite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. However, since the conforming finite element method (CFEM) requires a strong continuity, it is not easy to construct such finite elements for the complex partial differential equations. Thus, the nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM. The objective of this paper is to establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L2-projection methods by applying the idea presented in Wang. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in NCFEM by L2- projections.