Precorrected-FFT Accelerated Singular Boundary Method for High-Frequency Acoustic Radiation and Scattering

This paper presents a precorrected-FFT (pFFT) accelerated singular boundary method (SBM) for acoustic radiation and scattering in the high-frequency regime. The SBM is a boundary-type collocation method, which is truly free of mesh and integration and easy to program. However, due to the expensive CPU time and memory requirement in solving a fully-populated interpolation matrix equation, this method is usually limited to low-frequency acoustic problems. A new pFFT scheme is introduced to overcome this drawback. Since the models with lots of collocation points can be calculated by the new pFFT accelerated SBM (pFFT-SBM), high-frequency acoustic problems can be simulated. The results of numerical examples show that the new pFFT-SBM possesses an obvious advantage for high-frequency acoustic problems.

An approximation of one-dimensional nonlinear Kortweg de Vries equation of order nine

This research presents the approximate solution of nonlinear Korteweg-de Vries equation of order nine by a hybrid staggered one-dimensional Haar wavelet collocation method. In literature, the underlying equation is derived by generalizing the bilinear form of the standard nonlinear KdV equation. The highest order derivative is approximated by Haar series, whereas the lower order derivatives are attained by integration formula introduced by Chen and Hsiao in 1997. The findings are shown in the form of tables and a figure, demonstrating the proposed technique’s convergence, robustness, and ease of application in a small number of collocation points.

A COMPUTATIONAL ALGORITHM FOR THE NUMERICAL SOLUTION OF NONLINEAR FRACTIONAL INTEGRAL EQUATIONS

In this paper, we develop a numerical method for the solution of nonlinear fractional integral equations (NFIEs) based on Haar wavelet collocation technique (HWCT). Under certain conditions, we also prove the uniqueness and existence as well as Hyers–Ulam (HU) stability of the solution. With the help of the mentioned technique, the considered problem is transformed to a system of algebraic equations which is then solved for the required results by using Broyden algorithm. To check the validation and convergence of the proposed technique, some examples are given. For different number of collocation points (CPs), maximum absolute and mean square root errors are computed. The results show that for solving these equations, the HWCT is effective. The convergence rate is also measured for different CPs, which is nearly equal to [Formula: see text].

Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021).

Deep Learning of Two-Phase Flow in Porous Media via Theory-Guided Neural Networks

Summary A theory-guided neural network (TgNN) is proposed as a prediction model for oil/water phase flow in this paper. The model is driven by not only labeled data, but also scientific theories, including governing equations, boundary and initial conditions, and expert knowledge. Two independent neural networks (NNs) are built in the TgNN for oil/water phase flow problems, with one approximating pressure and the other approximating saturation. The two networks are connected by loss functions, which include a data mismatch term, as well as theory-guided terms. The desired parameters in NNs are trained by a certain optimization algorithm to decrease the value of the loss function. The training process uses a two-stage strategy as follows: (1) after one of the two NNs obtains a satisfactory result, parameters in the network with better performance are fixed in calculating the nonlinear terms and (2) the other NN continues to be trained until satisfactory performance is also obtained. The proposed TgNN offers an effective way to solve the coupled nonlinear two-phase flow problem. Numerical results demonstrate that the proposed TgNN achieves better accuracy than the traditional deep neural network (DNN). This is because the governing equation can constrain spatial and temporal derivatives, and other physical constraints (i.e., boundary and initial conditions, expert knowledge) can make the outputs more scientifically consistent. The effect of sparse data (including labeled data and collocation points) is tested, and the results show that more labeled data and collocation points lead to improved long-term prediction performance. However, the TgNN can also be successfully trained in the absence of labeled data by merely adhering to the above-mentioned scientific theories. In addition, several more complicated scenarios are tested, including the existence of data noise, changes in well condition, transfer learning, and the existence of different levels of dynamic capillary pressure. Compared with the traditional DNN, TgNN possesses superior stability with the guidance of theories for the considered complex situations.

ON APPROXIMATION OF PERIODIC ATEB-FUNCTIONS

Two versions of approximation formulae for periodic Ateb-sine and Ateb-cosine in the first quarter of their common period are proposed. The first version is a Pade type approximation derived when constructing analytical solution of corresponding integral equation by iteration method with transforming the power series into a closed sum by Shanks’ formula. Two iteration approximations are considered. The first one is more concise but of worse approximation accuracy which deteriorates with increasing the argument value. To improve the approximation accuracy a hybrid approximation is proposed when the values of the Ateb-functions in the beginning (for the cosine) and in the end (for the sine) of the quarter period are computed by a separate formula obtained a priory by the asymptotic method. The comparison analysis of the approximate and exact values of the special functions indicates the error of the approximation proposed to be less than one per cent. The second variant of approximation is by replacing the periodic Ateb-functions by trigonometric functions of specific argument. The arguments are chosen so that the values of the special functions are exact at specific points of the quarter period. Five such collocation points are introduced in the paper. To implement this version of approximation a separate table of the values of the periodic Ateb-functions at the collocation points is compiled. The computational examples presented in the paper show the approximate values of the special functions obtained by the second version of approximation to have a good accuracy.

A CHEBYSHEV PROJECTION METHOD FOR SOLVING SHALLOW-WATER EQUATIONS

The aims of this paper is to propose a numerical approach to simulate water flows in a 2D shallow medium. We consider the 2D Shallow water equations following the velocity-denivelation formulation. We solve these equations by a projection technique using a $\mathbb{P}_{N,M}$-type Chebyshev spectral approach which uses the Chebyshev-Gauss-Lobatto collocation points.

Generalized Bessel QuasilinearizationTechnique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations.

Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

In this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

The Quasi-Optimal Radial Basis Function Collocation Method: A Technical Note

The traditional radial basis function parameter controls the flatness of these functions and influences the precision and stability of approximation solution. The coupled radial basis function, which is based on the infinitely smooth radial basis functions and the conical spline, achieves an accurate and stable numerical solution, while the shape parameter values are almost independent. In this paper, we give a quasi-optimal conical spline which can improve the numerical results. Besides, we consider the collocation points in the Chebyshev-type which improves solution accuracy of the method with no additional computational cost.