Development of the Kansa method for solving seepage problems using a new algorithm for the shape parameter optimization

In this research, the Kansa or Multiquadric method (MQ) has been developed for solving the seepage problems in 2D and 3D arbitrary domains. This research is the first application of this method for seepage analysis in both confined and unconfined porous media. The domain decomposition approach has been employed for applying MQ method easily in inhomogeneous and irregular complex geometries and decreasing the computational costs. For determining the optimum shape parameter that affects strongly the accuracy of MQ and other RFB methods, a new scheme that decreases drastically the computational time is introduced. The efficiency of the proposed algorithm has been examined under various radial basis functions, variations of number of interpolating points and points distribution, through a numerical example with analytical solution. Eventually, three examples including different boundary conditions are presented. Comparing results of the examples with other numerical methods indicates that the present approach has high capability and accuracy in solving seepage problems.     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

Particular solutions of products of Helmholtz-type equations using the Matern function

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

Radial basis function partition of unity method for modelling water flow in porous media

Water flow in variably-saturated porous media is modelled by using the highly nonlinear parabolic Richards’ equation. The nonlinearity is due to the hydraulic conductivity and moisture content variables. The latter were estimated by using experimental models, including Gardner, Burdine, Mualem and van Genuchten models. The aim of this work is to develop a new technique based on the radial basis function partition of unity method (RBFPUM) and Gardner model in order to solve Richards’ equation in one and two dimensions. We have used Gardner model to handle the nonlinearity issue and the RBFPUM is used to approximate the solution of the linearized Richards’ equation. Our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case. Then we pick up the optimal configuration (partitions number-shape parameter) that yields the best solution in terms of error and conditioning number. By following this procedure, an optimal solution is ensured for our given problem. As numerical tests, we consider the vertical infiltration of water in soils in order to validate our proposed method.     

Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Since neural networks have universal approximation capabilities, therefore it is possible to postulate them as solutions for given differential equations that define unsupervised errors. In this paper, we present a wide survey and classification of different Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network techniques, which are used for solving differential equations of various kinds. Our main purpose is to provide a synthesis of the published research works in this area and stimulate further research interest and effort in the identified topics. Here, we describe the crux of various research articles published by numerous researchers, mostly within the last 10 years to get a better knowledge about the present scenario.     

The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas

This paper investigates the solitary wave solutions of the two-dimensional regularized long-wave equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas. The main idea behind the numerical solution is to use a combination of boundary knot method and the analog equation method. The boundary knot method is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution, the boundary knot method uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to method of fundamental solution, the radial basis function is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method. According to the analog equation method, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Furthermore, in order to show the efficiency and accuracy of the proposed method, the present work is compared with finite difference scheme. The new method is analyzed for the local truncation error and the conservation properties. The results of several numerical experiments are given for both the single and double-soliton waves.     

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.     

Some procedures for function approximation based on the use of sample data and their application in heuristic methods for solving practical problems

Let f(x) be a member of a set of functions over a probability space. Samples of f(x) are 2-tuples (xⁱ,f(xⁱ) where xⁱ is a sample of the random variable X and f(xⁱ) is a sample of f(x) at x = xⁱ. Some procedures and analysis are presented for the approximation of such functions by systems of orthonormal functions. The approximations are based on the data samples. The analysis includes the case of error in the measurement of f(xⁱ). The properties of the expected square error in the approximation are examined for a number of different estimators for the coefficients in the expansion and these well-behaved and easily analyzed estimators are compared to those obtained using the method of least squares. The effectiveness of different sets of basis functions, those involved in the Karhunen-Loeve expansion and others, can be compared and an approach is suggested to adaptive basis selection in order to select that basis which is most efficient in approximating the particular function under examination. The connection between results and applications are discussed in the introduction and conclusion.     

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

Numerical method for the simultaneous determination of unknown parameters in a parabolic equation

Numerical procedures for the solution of an inverse problem of simultaneously determining unknown parameters in a linear parabolic equation are considered. The approach proposed is to approximate unknown functions using Chebyshev polynomials, which are determined consecutively from the solutions of the minimization problems based on overspecified data. Finally, the results of a numerical experiment are displayed.     

Optimal design of radial basis function neural networks for fuzzy-rule extraction in high dimensional data

The design of an optimal radial basis function neural network (RBFNF) is not a straightforward procedure. In this paper we take advantage of the functional equivalence between RBFN and fuzzy inference systems to propose a novel efficient approach to RBFN design for fuzzy rule extraction. The method is based on advanced fuzzy clustering techniques. Solutions to practical problems are proposed. By combining these different solutions, a general methodology is derived. The efficiency of our method is demonstrated on challenging synthetic and real world data sets.     

High-order scheme for determination of a control parameter in an inverse problem from the over-specified data

The problem of finding the solution of partial differential equations with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, thermo-elasticity, chemical diffusion and control theory. In this paper we present a high order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximate the spatial derivative with a fourth order compact scheme and reduce the problem to a system of ordinary differential equations (ODEs). Then we apply a fourth order boundary value method for the solution of resulting system of ODEs. So the proposed method has fourth order accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. Several numerical examples and also some comparisons with other methods in the literature will be investigated to confirm the efficiency of the new procedure.     

Development of a diagnosis technique for failures of V-belts by a cross-spectrum method and a discriminant function approach

This paper describes the development of a failure diagnosis technique for V-belts through vibration monitoring. The V-belt vibration is monitored at a driven bearing body attached to a power transmission device. Seven basic causes of belt failure and their combinations are considered. Power spectra of the vibration data are calculated through noise reduction by a cross-spectrum method. Six parameters characterizing the vibration data are extracted, and 16 typical combinations of the basic causes and a normal belt state are diagnosed successfully by a Bayes' discriminant function approach. Two types of incorrect diagnosis are examined: Type I leaves a failed belt not repaired, and type II causes over-maintenance. A risk ratio for the Bayes' discriminant function is determined to minimize the two types of incorrect diagnosis. Moreover, the risk ratio is determined to minimize type I error.     

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.     

An interactive method for the selection of design criteria and the formulation of optimization problems in computer aided optimal design

This paper presents an interactive method for the selection of design criteria and the formulation of optimization problems within a computer aided optimization process of engineering systems. The key component of the proposed method is the formulation of an inverse optimization problem for the purpose of determining the design preferences of the engineer. These preferences are identified based on an interactive modification of a preliminary optimization result that is the solution of an initial problem statement. A formulation of the inverse optimization problem is presented, which is based on a weighted-sum multi-objective approach and leads to an explicit optimization problem that is computationally inexpensive to solve. Numerical studies on structural shape optimization problems show that the proposed method is able to identify the optimization criteria and the formulation of the optimization problem which drive the interactive user modifications.     

Higher order dispersive and dissipative hybrid method for the numerical solution of oscillatory problems

In this paper, an improved explicit two-step hybrid method with fifth algebraic order is derived. The new method possesses dispersion of order 10 and dissipation of order seven, which is first of its kind in the literature. Numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order.     

The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.