Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Torsion problem for elastic cylinder with inserts and holes

An integral equation method to solve the classical torsion problem for an elastic cylinder with inserts and holes is treated. The bounded region outside the inserts and the holes will be termed a matrix. As is well-known the solution depends on finding plane harmonic functions in the matrix and inserts such that (a) on the outer boundary of the matrix and the boundaries of the holes the harmonic function in the matrix takes the values $\text{1}\text{2}\text{(x}{}^2\text{+y}{}^2\text{)+c}{}_{\mathrm{j}}$, and (b) on the interfaces of the matrix and the inserts relations exist between the harmonic functions and between their normal derivatives. Here (x, y) are the coordinates of the point on the boundary and c_j, are unknown constants. The usual methods are cumbersome and lengthy. In this paper a straightforward method is presented which is easily programmable. The numerical solution is obtained by evaluating a few integrals either analytically or numerically and solving a system of linear simultaneous equations. An example of a cylinder with an eccentric insert is given which substantiates the theory developed in this paper and is found to agree with known results. However, the method is general and may be applied to a variety of problems.     

Implicit finite difference techniques for the advection–diffusion equation using spreadsheets

This study proposes one-dimensional advection–diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implementation, solutions are implicitly obtained for the BTCS, Upwind and Crank–Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. Thus, it is not required a solution of simultaneous equations for each time step using matrix algebra. Two examples which, have the numerical and analytical solutions in literature, are solved in order to test the ADEISS performance. Both examples are solved for three schemes. It has been determined that the Crank–Nicolson scheme is in good agreement with the analytical solution; however the results of the BTCS and the Upwind schemes are lower than the analytical solution. The Upwind scheme suffers from considerably numerical diffusion whereas the BTCS scheme does not produce numerical diffusion. Thus, it provided better results than Upwind scheme which are closer to analytical results depending on the selected parameters. The ADEISS implementation is a computationally convenient procedure for the three well-known methods in the literature: The BTCS, Upwind and Crank–Nicolson.     

Application of the Galerkin and Least-Squares Finite Element Methods in the solution of 3D Poisson and Helmholtz equations

This paper presents the numerical solution, by the Galerkin and Least Squares Finite Element Methods, of the three-dimensional Poisson and Helmholtz equations, representing heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used to compare with the numerical solutions. The analysis of results was made from the L₂ norm (average error throughout the domain) and L_∞ norm (maximum error in the entire domain). The results of the two applications (Poisson and Helmholtz equations) are presented and discussed for testing of the efficiency of the methods.     

Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption

The numerical errors associated with explicit upstream finite difference solutions of two-dimensional advection—Dispersion equation with linear sorption are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation. The numerical truncation errors are defined using Peclet and Courant numbers in the X and Y direction, a sink/source dimensionless number and new Peclet and Courant numbers in the XY plane. The effects of these truncation errors on the explicit solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution in uniform flow field. The results show that these errors are not negligible and correcting the finite difference scheme for them results in a more accurate solution.     

A numerical treatment to the solution of certain first kind Fredholm integral equations

This paper presents results obtained by an implementation of the kernel basis functions method to the solution of a special Fredholm integral equation of the first kind. The approximate solution is expressed in the linear form u_k(t)=∑w_jK(s_j, t), j=1, 2,..., k, where the unknown parameters w_j and s_j are determined by solving two linear overdetermined systems and a polynomial equation of the k-th order. Test examples are used to show that the numerical solution is comparable to the exact one.     

An algorithm for determining the size of symmetry groups

To determine the symmetry group of pointor Lie-symmetries of a differential equation is of great theoretical and practical importance, in particular for determining closed form solutions. There does not seem to exist an algorithm that finds this group in general. However, it is always possible to determine thesize of the symmetry group. In this article an algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and the number of unspecified functions and its arguments if it is infinite. To this end the so calleddetermining system is transformed into aninvolutive system by means of a critical-pair/completion algorithm similar like it is applied for computing Gröbner bases in polynomial ideal theory. The foundation for obtaining this form is the theory of Riquier and Janet for partial differential equations. The algorithmInvolution System has been implemented in several computer algebra systems as part of the packageSPDE. Various results that have been obtained by applying it are presented as well. If symmetry analysis is considered as part of the more general process of obtaining the best possible information on the solutions of a differential equation, the algorithm described in this article removes the heuristics which is usually involved in making the transition from analytical to numerical methods.     

Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem

The paper considers the determination of heat sources in unsteady 2-D heat conduction problem. The determination of the strength of a heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the θ-method with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem the Tikhonov regularization method based on SVD decomposition was used. In order to determine the optimum value of the regularization parameter the L-curve criterion was used. For testing purposes of the proposed algorithm the 2-D inverse boundary-initial-value problems in square region Ω with the known analytical solutions are considered. The numerical results show that the proposed method is easy to implement and pretty accurate. Moreover the accuracy of the results does not depend on the value of the θ parameter and is greater in the case of the identification of the temperature field than in the case of the identification of the heat sources function.     

Symbolic-numerical analysis of equilibrium solutions in a restricted four-body problem

Algorithms for searching equilibrium solutions of the circular restricted four-body problem formulated on the basis of triangular Lagrange solutions of the three-body problem are discussed. For small values of one of the two system parameters, equilibrium solutions are found in the form of power series. For large values of this parameter, an algorithm for numerical solution is presented. The proposed algorithms are implemented in the computer algebra system Mathematica.     

An efficient analytical approach for solving fourth order boundary value problems

Based on the homotopy analysis method (HAM), an efficient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. We apply the approach to a linear problem which involves a parameter c and cannot be solved by other analytical methods for large values of c, and obtain convergent series solutions which agree very well with the exact solution, no matter how large the value of c is. Consequently, we give an affirmative answer to the open problem proposed by Momani and Noor in 2007 [S. Momani, M.A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput. 191 (2007) 218-224]. We also apply the approach to a nonlinear problem, and obtain convergent series solutions which agree very well with the numerical solution given by the Runge-Kutta-Fehlberg 4-5 technique.     

On stable and explicit numerical methods for the advection–diffusion equation

In this paper two stable and explicit numerical methods to integrate the one-dimensional (1D) advection–diffusion equation are presented. These schemes are stable by design and follow the main general concept behind the semi-Lagrangian method by constructing a virtual grid where the explicit method becomes stable. It is shown that the new schemes compare well with analytic solutions and are often more accurate than implicit schemes. In particular, the diffusion-only case is explored in some detail. The error produced by the stable and explicit method is a function of the ratio between the standard deviation σ₀ of the initial Gaussian state and the characteristic virtual grid distance ΔS. Larger values of this ratio lead to very accurate results when compared to implicit methods, while lower values lead to less accuracy. It is shown that the σ₀/ΔS ratio is also significant in the advection–diffusion problem: it determines the maximum error generated by new methods, obtained with a certain combination of the advection and diffusion values. In addition, the error becomes smaller when the problem becomes more advective or more diffusive.     

A Functional Equation For The Computation Of Parameter Symmetries Of Path Functions

The paper considers a particular general class of parametrised path function used in computer graphics, geometric modeling and approximation theory. General methods are developed for the identification of the conditions under which parameter transformations preserve the path geometry. The determination of these 'parameter symmetries' is shown to be equivalent to the identification of the solution space of a functional equation. Case studies, each with distinctive features, are presented to illustrate the concepts developed and the generality of the approach. Full solutions are obtained and seen to provide both known and new results.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

Analytical solution methods for the fuzzy weighted average

For the fuzzy weighted average (FWA), despite various discrete solution algorithms and their improvements, attempts at analytical solutions are very rare. This paper provides an analytical solution method for the FWA based on the conclusions of the Karnik–Mendel (KM) algorithm. Compared with the two current popular kinds of α-cut based computational methods for the FWA (mathematical programming transformations and direct iterate computations), our method is precise, and, has a concise structure, efficient computation process, and sound theoretical proofs. We propose two algorithms for computing the analytical solution of the FWA. Two numerical examples illustrate our proposed approach.     

Distributions of the potential and electric field of an electrode elliptic array used in tumor electrotherapy: Analytical and numerical solutions

To estimate the potential and electric field generated by any electrode array is very useful in effective tumor destruction. At present, an electrode array that takes into account the ellipsoidal geometry of the solid tumors has not been proposed. We present both analytical and numerical solutions for the potential and electric field in a solid tumor established by an electrode array with elliptic shape which may be used in vitro, in vivo and in clinical studies for cancer treatment with electrotherapy. These analytical and numerical solutions are obtained using multipole expansion and the finite difference method. Distributions of potential and electric field magnitudes are computed in function of the eccentricity of an elliptical array and compared with those obtained with a circular array of electrode. Maximum difference and Root Means Square Error are used to compare the distributions of the potential and electric field in leading-order and first-order correction and between the analytical and numerical solutions. The results show a good agreement between these distributions in both orders and the analytical and numerical solutions. It was concluded that the mathematical approach presented in this study is a tool for a rapid design of electrode elliptical arrays in order to induce the maximum destruction of the tumor. Moreover, it is shown that, for all values of eccentricity, there is a good correspondence between the distributions of the potential and the electric field for leading-order and first-order correction and for both the analytical and numerical solutions.     

2D and 3D transient heat conduction analysis by BEM via particular integrals

Particular integral formulations are presented for 2D and 3D transient potential flow (heat conduction) analysis. The results of the analysis are compared with an alternative formulation developed using the volume integral conversion approach. Although the mathematical foundation of the two methods are different both formulations are shown to produce almost identical results.For the particular integral formulation, the steady-state heat conduction equation is used as the complementary solution and two global shape functions (GSFs) are considered to approximate the transient term of the heat conduction equation.The numerical results for three example problems are given and compared with their analytical solutions.     

The global approximate particular solution meshless method for two-dimensional linear elasticity problems

The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function (RBF) is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.