Two-dimensional Elliptic Partial Differential Equations Research Articles

Adaptive multilayer neural network for solving elliptic partial differential equations with different boundary conditions

In recent years, elliptic partial differential equations (PDEs) have been challenging in numerical mathematics, engineering, and physics. In this paper, we proposed a novel approach called Multilayer Neural Network for Partial Differential Equations (MLNPDE) based on deep learning algorithms by exploring the elliptic PDE family under certain conditions such as initial conditions and boundary conditions to govern approximation solutions of PDEs. In the proposed model, we include a multilayer neural network using a densely connected network. Moreover, the approximation solution underlying PDEs can be expressed in two terms: the first term satisfies the boundary conditions, and the second term is a function of the unknown parameters that were estimated by our proposed model. Furthermore, we derive a normal form of the boundary conditions on the first term of the approximation solution for two-dimensional elliptic partial differential equation problems with an arbitrary domain. The numerical results show that our method achieved tremendous state-of-the-art performance in accuracy and efficiency compared to the classical methods.

A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

In this paper, we have extended the operational matrix method for approximating the solution of the fractional-order two-dimensional elliptic partial differential equations (FPDEs) under nonlocal boundary conditions. We use a general Legendre polynomials basis and construct some new operational matrices of fractional order operations. These matrices are used to convert a sample nonlocal heat conduction phenomenon of fractional order to a structure of easily solvable algebraic equations. The solution of the algebraic structure is then used to approximate a solution of the heat conduction phenomena. The proposed method is applied to some test problems. The obtained results are compared with the available data in the literature and are found in good agreement.Dedicated to my father Mr. Sher Mumtaz, (1955-2021), who gave me the basic knowledege of mathematics.

Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains

In this paper, a novel collocation method is presented for the efficient and accurate evaluation of the two-dimensional elliptic partial differential equation. In the new method, the physical domain is discretized into a series of overlapping small (local) subdomains, and in each of the subdomain, a localized Chebyshev collocation method is applied in which the unknown functions at every node can be computed by using a linear combination of unknowns at its near-by nodes. The Chebyshev polynomials employed here can provide the spectral accuracy of new approach. The concept of the local subdomain is introduced to derive a sparse system, which ensures the feasibility for large-scale simulation. This paper aims at proposing a new method to solve general partial differential equations accurately and efficiently. Several numerical examples including Poisson equation, Helmholtz-type equation and transient heat conduction equation are provided to demonstrate the validity and applicability of the proposed method. Numerical experiments indicate that the localized Chebyshev collocation method is very promising for the efficient and accurate solution of large-scale problems.

Analytical estimation of solution parameters for elliptic equations with variable coefficients in fluid dynamics

In this paper we consider multidimensional and two-dimensional elliptic partial differential equations of the second order with variable coefficients. They are often used in the oil and gas industry to calculate the inflow volume of liquids and gases from natural reservoirs to exciting structures. The analytical solution of that equations with variable values of the permeability coefficient is possible in rare cases, when the change in the formation permeability is expressed in the form of simple analytical dependencies over the area of fluid filtration. In this paper, we give estimated formulas for fluid inflow to structures under conditions of reservoir heterogeneity, which retain an analytical relationship with the permeability coefficient. The value of the proposed method lies in the ability to establish important qualitative regularities of an underground flow formation, based on simple formulas. The obtained expressions of the above theorems are consistent with the calculation formulas of underground fluid dynamics, and allows one to perform calculations of oil production in heterogeneous formations.

Wavelet neural networks based solutions for elliptic partial differential equations with improved butterfly optimization algorithm training

In this study, a machine learning approach based on the unsupervised version of wavelet neural networks (WNNs) is used to solve two-dimensional elliptic partial differential equations (PDEs). The design of the WNNs must be judiciously addressed, particularly, the adopted training algorithm, since it greatly influences the generalization performance and the convergence rate of WNNs. Although the gradient information of the commonly used gradient descent training algorithm in WNNs may direct the search to optimal weight solutions that minimize the error function, the learning process is slow due to the complex calculation of the partial derivatives. To date, on account of the derivative free characteristic and adaptability to respond to the complex dynamic changes of the interdependencies, numerous studies explored the potential benefit of integrating a meta-heuristic algorithm as the training algorithm of WNNs, where encouraging results are achieved. In this paper, an improved butterfly optimization algorithm (IBOA) is proposed and subsequently integrated into the training process of the WNNs. To evaluate the performance of the proposed IBOA training method, the obtained results are compared to the results of the momentum backpropagation (MBP), the particle swarm optimization (PSO) and the standard butterfly optimization algorithm (BOA) training methods. Statistical analyses of the results based on a sufficient number of independent runs validate the effectiveness of the proposed method in terms of accuracy, robustness and convergence.

Accuracy analysis of different higher-order Laplacian models of incompressible SPH method

PurposeThis paper aims to examine the accuracy of several higher-order incompressible smoothed particle hydrodynamics (ISPH) Laplacian models and compared with the classic model (Shao and Lo, 2003).Design/methodology/approachThe numerical errors in solving two-dimensional elliptic partial differential equations using the Laplacian models are investigated for regular and highly irregular node distributions over a unit square computational domain.FindingsThe numerical results show that one of the Laplacian models, which is newly developed by one of the authors (Shobeyri, 2019) can get the smallest errors for various used node distributions.Originality/valueThe newly proposed model is formulated by the hybrid of the standard ISPH Laplacian model combined with Taylor expansion and moving least squares method. The superiority of the proposed model is significant when multi-resolution irregular node distributions commonly seen in adaptive refinement strategies used to save computational cost are applied.

Tension spline method for the solution of elliptic equations

In this paper, two classes of methods are developed for the solution of two-dimensional elliptic partial differential equations. We have used tension spline function approximation in both x and y spatial directions and a new scheme of order has been obtained. The convergence analysis of the methods has been carried out. Numerical examples are given to illustrate the applicability and accurate nature of our approach.

The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy–Navier equations

In this paper, we extended the method of approximate particular solutions (MAPS) using trigonometric basis functions to solve two-dimensional elliptic partial differential equations (PDEs) with variable-coefficients and the Cauchy–Navier equations. The new approach is based on the closed-form particular solutions for second-order differential operators with constant coefficients. For the Cauchy–Navier equations, a reformulation of the equations is required so that the particular solutions for the new differential operators are available. Five numerical examples are provided to demonstrate the effectiveness of the proposed method.

The method of particular solutions using trigonometric basis functions

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method.

Three-dimensional semi-idealized model for estuarine turbidity maxima in tidally dominated estuaries

We develop a three-dimensional idealized model that is specifically aimed at gaining insight in the physical mechanisms resulting in the formation of estuarine turbidity maxima in tidally dominated estuaries. First, the three-dimensional equations for water motion and suspended sediment concentration together with the so-called morphodynamic equilibrium condition, are scaled. Next, surface elevation, velocity and sediment concentration are expanded in a small parameter ϵ=A¯M2/H, where A¯M2 is the mean amplitude of the M2 tide and H is the mean water depth at the seaward side. This results in a system of equations at each order in this small parameter. This ordering allows solving for the vertical structure of the velocity and suspended sediment concentration, independently of the horizontal dimension. After obtaining these vertical structures, the horizontal dependencies of the physical variables follow from solving a two-dimensional elliptic partial differential equation for the surface elevation. The availability of fine sediments in the estuary follows from a two-dimensional elliptic partial differential equation which results from requiring the system to be in morphodynamic equilibrium, and prescribing the total amount of easily erodible sediments available in the estuary. These elliptic equations for the surface elevation and sediment availability are solved numerically using the finite element method with cubic polynomials as basis functions. As a first application, the model is applied to the Ems estuary using a simplified geometry and bathymetric profiles characteristic for the years 1980 and 2005. The availability of fine sediments and location of maximum concentration are investigated for different lateral depth profiles. In the first experiment, a uniform lateral depth is considered. In this case, both the sediment availability and suspended sediment concentration are, as expected, uniform in the lateral direction. In 1980, the sediment is mainly trapped near the entrance, while in 2005, the sediment is mostly trapped in the freshwater zone. In the next experiment, the lateral bathymetry is varied parabolically while keeping the mean depth unchanged. In this case, the fine sediment is mainly found at the shallow sides, but the maximum sediment concentration is found in the deeper channel where the bed shear stress is much larger than on the shoals. As a final experiment, a more realistic (but smoothed) geometry and bathymetry for the Ems estuary are considered, showing the possibilities of applying the newly developed model to complex geometries and bathymetries.

High Accuracy Spline Explicit Group (SEG) Approximation for Two Dimensional Elliptic Boundary Value Problems.

Over the last few decades, cubic splines have been widely used to approximate differential equations due to their ability to produce highly accurate solutions. In this paper, the numerical solution of a two-dimensional elliptic partial differential equation is treated by a specific cubic spline approximation in the x-direction and finite difference in the y-direction. A four point explicit group (EG) iterative scheme with an acceleration tool is then applied to the obtained system. The formulation and implementation of the method for solving physical problems are presented in detail. The complexity of computational is also discussed and the comparative results are tabulated to illustrate the efficiency of the proposed method.

THE EIGENVALUE PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH TWO-POINT NONLOCAL CONDITIONS

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the two-dimensional problem by the separation of variables technique.

A New Fourth-Order Compact Off-Step Discretization for the System of 2D Nonlinear Elliptic Partial Differential Equations

AbstractThis paper discusses a new fourth-order compact off-step discretization for the solution of a system of two-dimensional nonlinear elliptic partial differential equations subject to Dirichlet boundary conditions. New methods to obtain the fourth-order accurate numerical solution of the first order normal derivatives of the solution are also derived. In all cases, we use only nine grid points to compute the solution. The proposed methods are directly applicable to singular problems and problems in polar coordinates, which is a main attraction. The convergence analysis of the derived method is discussed in detail. Several physical problems are solved to demonstrate the usefulness of the proposed methods.

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm–Liouville problem

article i nfo abstract A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm- Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for

Numerical solution of elliptic partial differential equations on parallel systems

The numerical solution of two-dimensional, linear and non-linear elliptic partial differential equations (PDEs) using two parallel algorithms namely Lawrie Sameh and domain decomposition has been computed on three parallel architectures. The PDEs considered here, describe the diffusion of a pollutant released from a constant source, variable source and a point source. In addition, an example with diffusion and chemical removal modelled by a non-linear PDE and three-dimensional diffusion equation is given. The parallel algorithms have been implemented on three-different systems (i) SUNFIRE (ii) IBM and (iii) PARAM. The scalability analysis has been done to analyze the performance of both the algorithms on all three parallel systems.

Convergence of the compact finite difference method for second-order elliptic equations

In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two-dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus we prove the compact finite difference scheme converges with the fourth order of accuracy. To solve the discretized block tri-diagonal matrix equations of the two-dimensional elliptic PDE’s, we develop an efficient iterative method: Full Multigrid Method. In our numerical experiments, we compare the compact finite difference method with the finite element method and Crank–Nicolson finite difference method. The results show that the compact finite difference scheme is a highly efficient and accurate method.

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations.

Algebraic Analysis of the Hierarchical Basis Preconditioner

The use of the hierarchical basis in finite element discretizations of two-dimensional elliptic partial differential equations produces matrices with condition numbers of order $O((\log h^{ - 1} )^2 )$. Standard proofs of this result are functional analytic in style. In this paper, it is shown that for uniform grids the result can be obtained using a purely linear algebraic argument.

Parallel Implementation of the $hp$-Version of the Finite Element Method on a Shared-Memory Architecture

The costs incurred by an implementation of the $hp$-version of the finite element for solving two-dimensional elliptic partial differential equations on a shared-memory parallel computer are studied. For a collection of benchmark problems, the costs in CPU time of various individual subtasks performed by the finite element solver are systematically examined, including construction of local stiffness matrices, elimination of unknowns associated with element interiors, and global solution on element interfaces by a preconditioned conjugate gradient method. General observations are that the costs of the “naturally” parallel computations associated with local elements are significantly higher than any global computations, so that the latter do not represent a significant bottleneck to parallel efficiency. However, memory conflicts place some limitations on the sizes or number of local problems that can be handled efficiently in parallel.

A fast adaptive grid scheme for elliptic partial differential equations

We describe the Recursive Subdivision (RS) method-an efficient and effective adaptive grid scheme for two-dimensional elliptic partial differential equations (PDEs). The RS method generates a new grid by recursively subdividing a rectangular domain. We use a heuristic approach which attempts to equidistribute a given density function over the domain. The resulting grid is used to generate an adaptive grid domain mapping (AGDM), which may be applied to transform the PDE problem to another coordinate system. The PDE is then solved in the transformed coordinate system using a uniform grid. We believe parallelism is most easily exploited when computations are carried out on uniform grids; the AGDM framework allows the power of adaptation to be applied while still preserving this uniformity. Our method generates good adaptive grid domain mappings at a small cost compared to the cost of the entire computation. We describe the RS algorithm in detail, briefly describe the AGDM framework, and illustrate the effectiveness of our scheme on several realistic test problems.