General Partial Differential Equations Research Articles

Modulating Functions Based Algorithm for the Estimation of the Coefficients and Differentiation Order for a Space-Fractional Advection-Dispersion Equation

In this paper, a new method, based on the so-called modulating functions, is proposed to estimate average velocity, dispersion coefficient, and differentiation order in a space-fractional advection-dispersion equation, where the average velocity and the dispersion coefficient are space-varying. First, the average velocity and the dispersion coefficient are estimated by applying the modulating functions method, where the problem is transformed into a linear system of algebraic equations. Then, the modulating functions method combined with a Newton's iteration algorithm is applied to estimate the coefficients and the differentiation order simultaneously. The local convergence of the proposed method is proved. Numerical results are presented with noisy measurements to show the effectiveness and robustness of the proposed method. It is worth mentioning that this method can be extended to general fractional partial differential equations.

Analytical Expansions for Parabolic Equations

We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.

A Penalization Method for Calculating the Flow Beneath Traveling Water Waves of Large Amplitude

We suggest an algorithm for the calculation of large-amplitude traveling water waves in irrotational flows and in flows with constant and nonconstant vorticity, respectively. The algorithm is based on a quite general partial differential equation constrained optimization formulation, maximizing the wave amplitude subject to the PDE constraint that the Euler equations hold. Numerical examples illustrate the performance of the algorithm for irrotational flows and in flows with constant vorticity, where the results are compared with the literature. As a proof of principle we also consider a test case with a nonconstant vorticity.

Short-term Power Demand Forecasting using the Differential Polynomial Neural Network

AbstractPower demand forecasting is important for economically efficient operation and effective control of power systems and enables to plan the load of generating unit. The purpose of the short-term electricity demand forecasting is to forecast in advance the system load, represented by the sum of all consumers load at the same time. A precise load forecasting is required to avoid high generation cost and the spinning reserve capacity. Under-prediction of the demands leads to an insufficient reserve capacity preparation and can threaten the system stability, on the other hand, over-prediction leads to an unnecessarily large reserve that leads to a high cost preparations. Differential polynomial neural network is a new neural network type, which forms and resolves an unknown general partial differential equation of an approximation of a searched function, described by data observations. It generates convergent sum series of relative polynomial derivative terms which can substitute for the ordinary differ...

On integration of a multidimensional version of n-wave type equation

We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the n-wave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)-dimensional n-wave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multi-solitary wave solutions in details. Among examples of explicit particular solutions, we represent a lump-lattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the ∂̄-problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.

A unified Petrov–Galerkin spectral method for fractional PDEs

Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov–Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form 0Dt2τu+∑j=1dcj[ajDxj2μju]+γu=f, where 2τ, μj∈(0,1), in a (1+d)-dimensional space–time domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1-D) and the corresponding convergence study of the scheme (in multi-D). The unified PG spectral method applies to the entire family of linear hyperbolic-, parabolic- and elliptic-like equations. We develop the PG method based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi poly-fractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from poly-fractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Besides the high-order spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in time-integration schemes for solving time-dependent FPDEs as well, rather than employing algebraically accurate traditional methods, especially when 2τ=1. Finally, we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices, which reduces the computational cost drastically. We demonstrate that this framework can reduce to hyperbolic FPDEs such as time- and space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost. Several numerical tests confirm the efficiency and spectral convergence of the unified PG spectral method for the aforementioned families of FPDEs. Moreover, we demonstrate the computational efficiency of the new approach in higher-dimensions e.g., (1+3), (1+5) and (1+9)-dimensional problems.

Applying a new method to solve diffusivity equation for gas reservoirs

Diffusivity equation is the general partial differential equation used to describe the flow of any fluid flowing in a radial direction in porous media. The main objective of this study is to present a new numerical scheme based on orthogonal collocation (OC) method to solve the diffusivity equation for heterogeneous and homogeneous gas reservoirs. OC is an approximate analytical technique which categorizes in the weighted residuals methods. The advantage and priority of the OC method over exact analytical solution (i.e., Laplace transform) is in the cases that the heterogeneity and variation of reservoir properties such as porosity and permeability with position or pressure could not be neglected; in these cases, the exact analytical solution is very tedious and may be impossible. The diffusivity equation has been solved by both OC and exact analytical solutions. To demonstrate the reliability of the proposed method, the results of OC method have been compared with those achieved using exact analytical solution. Average absolute deviation percent (AAD%) has been used for determining the suitable number of collocation points to give acceptable error and best matching between approximate and analytical results. Sensitivity analysis indicates that increasing the numbers of collocation points result in significant improvement of OC method accuracy and its capability on dynamic pseudo pressure prediction. The minimum AAD% of 0.113 from the exact analytical predictions has been obtained with 25 collocation points. The results indicate that the proposed approximate method with these numbers of collocation points can predict the reservoir pseudo pressure trend with an acceptable accuracy in heterogeneous and homogeneous gas reservoirs.

A multiscale discontinuous Galerkin method for convection–diffusion–reaction problems

We provide a general framework of multiscale discontinuous Galerkin methods developed in Buffa et al. (2006), Hughes et al. (2006) for general second-order partial differential equations. We establish stability of the method and prove the error estimates.

On the existence of conservation law multiplier for partial differential equations

In this paper, we use the property of nonlinear self-adjointness with differential substitution to study the existence of conservation law multiplier for partial differential equations (PDEs). Firstly, we give a sufficient and necessary condition for the existence of the multipliers involving only independent and dependent variables, which is the nonlinear self-adjointness of the studying PDEs. Secondly, a necessary condition for the existence of the multipliers involving derivatives is given for the general evolution PDEs, which is the nonlinear self-adjointness with differential substitution. Finally, applications of multiplier and nonlinear self-adjointness with differential substitution methods to nonlinear telegraph equations and a class of Korteweg-de Vries (KdV) type equations are performed and different types of conservation laws are constructed.

Constructing ordinary sum differential equations using polynomial networks

Data relations can define general sum partial differential equations of a composite function additive derivative model. Time-series data observations can analogously describe an ordinary sum differential equation with time derivatives, which is possible to be solved using partial derivative term substitutions of time-dependent series. Differential polynomial neural network is a new type of neural network, which constructs and substitutes for an unknown general partial differential equation from data observations, developed by the author. It generates sum series of convergent partial polynomial derivative terms, which can describe an unknown complex function time-series. This type of non-linear regression decomposes a system model, described by the general differential equation, into many partial low order derivative specifications of selected relative sum terms. Common soft-computing techniques in general can apply input variables of only absolute interval values of a specific data range. The character of relative data allows processing a wider range of test interval values than defined by a training set. The characteristics of the composite sum differential equation solutions can facilitate a much greater variety of model forms than is allowed using standard soft computing methods. Recurrent neural network proved to form simple solid time-series models, most of which it is possible to describe using ordinary differential equations, so the comparisons were done.

Electrostatics simulations on the cloud with NCLab

This survey paper describes NCLab (Network Computing Laboratory), a novel cloud computing service that is pioneering free web browser access to finite element simulations. Presently, these include general second-order linear partial differential equations, linear elasticity problems, and electrostatics problems (all in two spatial dimensions). The modeling process consists of geometry definition in the web browser, mesh generation on a remote server, specification of material parameters and boundary conditions in the web browser, computation on a remote server, and visualization in the web browser. The module supports curvilinear elements and higher-order finite element discretizations. In this paper we present the modeling process using a sample electrostatics problem. Numerical results are provided.

"ALADIN" WEATHER MODEL LOCAL REVISIONS USING THE DIFFERENTIAL POLYNOMIAL NEURAL NETWORK

The 48-hour Aladin forecast model can predict significant meteoro- logical quantities in a middle scale area. Neural networks could try to replace some statistical techniques designed to adapt a global meteorological numerical forecast model for local conditions, described with real data surface observations. They succeed commonly a cut above problem solutions with a predefined testing data set, which provides bearing inputs for a trained model. Time-series predictions of the very complex and dynamic weather system are sophisticated and not any time faithful using simple neural network models entered only some few variables of their own next-time step estimations. Predicted values of a global meteorologi- cal forecast might instead enter a neural network locally trained model, for refine it. Differential polynomial neural network is a new neural network type developed by the author; it constructs and substitutes for an unknown general sum partial differential equation of a system description, with a total sum of fractional poly- nomial derivative terms. This type of non-linear regression is based on trained generalized data relations, decomposed into many partial derivative specifications. The characteristics of composite differential equation solutions of this indirect type of a function description can facilitate a much greater variety of model forms than is allowed using standard soft-computing methods. This adjective derivative model type is supposed to be able to solve much more complex problems than is usual using standard neural network techniques.

A Review on the Modified Finite Point Method

The objective of this paper is to make a review on recent advancements of the modified finite point method, named MFPM hereafter. This MFPM method is developed for solving general partial differential equations. Benchmark examples of employing this method to solve Laplace, Poisson, convection-diffusion, Helmholtz, mild-slope, and extended mild-slope equations are verified and then illustrated in fluid flow problems. Application of MFPM to numerical generation of orthogonal grids, which is governed by Laplace equation, is also demonstrated.

Analytical Expansions for Parabolic Equations

We consider the Cauchy problem associated with a general parabolic partial differential equation in d dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.

Coupling discontinuous Galerkin discretizations using mortar finite elements for advection–diffusion–reaction problems

We investigate DG–DG domain decomposition coupling using mortar finite elements to approximate the solution to general second-order partial differential equations. We weakly impose an inflow boundary condition on the inflow part of the interface and the Dirichlet boundary condition on the elliptic part of the interface via Lagrange multipliers. We obtain the matching condition by imposing the continuity of the total flux through the interface and the continuity of the solution on the elliptic parts of the interface. The diffusion coefficient is allowed to be degenerate and the mortar interface couples efficiently the multiphysics problems. The (discrete) problem is solvable in each subdomain in terms of Lagrange multipliers and the resulting algorithm is easily parallelizable. The subdomain grids need not match and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale h and the coarse mortar scale H are then established. A non-overlapping parallelizable domain decomposition algorithm (Arbogast et al., 2007) reduces the coupled system to an interface mortar problem. The properties of the interface operator are discussed.

A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the “variational string equation”, a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian partial differential equation (PDE) satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the Korteweg–de Vries (KdV) case, we prove the existence of a quasitriviality transformation at any order and for arbitrary initial data.

New solvers for higher dimensional poisson equations by reduced B‐splines

We use higher dimensional B‐splines as basis functions to find the approximations for the Dirichlet problem of the Poisson equation in dimension two and three. We utilize the boundary data to remove unnecessary bases. Our method is applicable to more general linear partial differential equations. We provide new basis functions which do not require as many B‐splines. The number of new bases coincides with that of the necessary knots. The reducing process uses the boundary conditions to redefine a basis without extra artificial assumptions on knots which are outside the domain. Therefore, more accuracy would be expected from our method. The approximation solutions satisfy the Poisson equation at each mesh point and are solved explicitly using tensor product of matrices. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 393–405, 2014

Weakly equilibrated basis functions for elasticity problems

Abstract In this paper weakly equilibrated basis functions (EqBFs) are introduced for the development of a boundary point method. This study is the extension of the one in (Int. J. Numer. Methods Engng. 81 (2010) 971–1018) using exponential basis functions (EBFs) which are available just for partial differential equations (PDEs) with constant coefficients. Here the EqBFs are evaluated numerically to solve more general PDEs with non-constant coefficients. The EqBFs are found through weighted residual integrals defined over a fictitious domain embedding the main domain. A series of Chebyshev polynomials are used for the construction of the basis functions. By properly choosing the weight functions as the product of two unidirectional functions, here with Gaussian distribution, the main 2D integrals are written as the product of the simpler 1D ones. The results of the integrals can be stored for further use; however in some particular cases the EqBFs may be stored as a set of library functions. The results may also be found useful for those who are interested in residual-free functions in other numerical methods. For the verification, we discuss on the validity of the solution through an essential and comprehensive test procedure followed by several numerical examples.

Accurate and efficient evaluation of failure probability for partial different equations with random input data

Several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis. When the dimension of the uncertainties becomes high, well established direct numerical methods can not be employed because of the “curse-of-dimensionality”. Many surrogate models have been proposed with the aim of reducing computational effort. However, most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness. In addition, for a stochastic system modeled by partial differential equations (PDEs) with random input, only a limited number of the underlying PDEs (order of a few tens) are affordable to solve in practice due to the considerable computational effort, therefore preventing the application of many numerical methods especially for high dimensional random inputs. In this work we develop hybrid and goal-oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic PDE. The curse-of-dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal-oriented adaptation. Moreover, an accurate evaluation of the failure probability for PDE system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error. At the end of this paper we suitably extend our proposed method to deal with more general PDE models. Finally we perform several numerical experiments to illustrate its computational accuracy and efficiency.

A kernel-free boundary integral method for implicitly defined surfaces

The kernel-free boundary integral (KFBI) method is a structured grid method for general elliptic partial differential equations. Unlike the standard boundary integral method, it avoids direct evaluation of volume and boundary integrals, which needs to know analytical expressions for the integral kernels. To evaluate a boundary or volume integral, the KFBI method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary. Selection of control points of the boundary plays a key role in the KFBI method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points. This work proposes a new point selection method, based on an overlapping surface decomposition of the boundary, which is implicitly defined by a level set function. The points selected are intersection points of the boundary with the grid lines of an underlying Cartesian grid. By the method, the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions, which allows efficient numerical differentiation and boundary reconstruction/representation. An additional equilibrating process of boundary data further guarantees stable numerical differentiation. Numerical results demonstrating the method with examples in both two and three space dimensions are presented.