Interpolating Scaling Functions Research Articles

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Numerical Scheme with Convergence Analysis and Error Estimate for Variable Order Weakly Singular Integro-Differential Equation

This paper represents a new application of Legendre wavelet and interpolating scaling function to discuss the approximate solution of variable order integro-differential equation having weakly singular kernel. So far, this technique has been used to solve variable order integro differential equation. In this paper, it is extended to solve variable order integro differential equation with weakly singular kernel. For this purpose, we derive the operational matrices of Legendre wavelets and interpolating scaling function. The resulting operational matrices along with the collocation method transform the original problem into a system of algebraic equation. By solving this system, the approximate solution is obtained. The convergence and error estimate of the presented method have been rigorously investigated. We also discuss the numerical stability of the method. The numerical result of some inclusive examples has been provided through a table and graph for both basis functions that support the robustness and desired precision of the method.

Interpolating Scaling Functions Tau Method for Solving Space–Time Fractional Partial Differential Equations

This paper is devoted to an innovative and efficient technique for solving space–time fractional differential equations (STFPDEs). To this end, we apply the Tau method such that the bases used are interpolating scaling functions (ISFs). The operational metrics for the derivative operator and fractional integration operator are used to introduce the operational matrix for the Caputo fractional derivative. Due to some characteristics of ISFs, such as interpolation, computation costs can be significantly reduced. We investigate the convergence of the technique, and some numerical implementations show that the method is effective for solving such equations.

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

An Efficient Algorithm for Solving Hyperbolic Partial Differential Equations with a Nonlocal Conservation Condition

In this paper, a numerical scheme based on the Galerkin method is extended for solving one-dimensional hyperbolic partial differential equations with a nonlocal conservation condition. To achieve this goal, we apply the interpolating scaling functions. The most important advantages of these bases are orthonormality, interpolation, and having flexible vanishing moments. In other words, to increase the accuracy of the approximation, we can individually or simultaneously increase both the degree of polynomials (multiplicity r ) and the level of refinement (refinement level J ). The convergence analysis is investigated, and numerical examples guarantee it. To show the ability of the proposed method, we compare it with existing methods, and it can be confirmed that our results are better than them.

On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

An interpolating scaling functions method with low-storage five-stage fourth-order explicit Runge-Kutta schemes for 3D ground penetrating radar simulation

Accurate and efficient 3D simulation is extremely important for both grasping the features of radar wave propagation in realistic models and the high-resolution inversion. In this paper, we propose a new numerical algorithm called the low-storage five-stage fourth-order explicit Runge-Kutta interpolation scaling function method (LSERK-ISFM) and apply it into the 3D simulation of ground penetrating radar (GPR). Firstly, we derive the auxiliary differential equation (ADE) of the 3D Maxwell's equations based on the complex-frequency shifted perfectly matched layer (CFS-PML), discretize the spatial derivative of the Maxwell equation by ISFM, and integrate the temporal dimension through the LSERK scheme. Compared with the traditional finite-difference time-domain (FDTD) method, the LSERK-ISFM algorithm can effectively improve the precision of space-time dispersion of the Maxwell's equations. Specifically, by the homogeneous vacuum medium model, we compare the forward modeling results of the different number of grids per wavelength, the comparison results prove that a smaller number of grids per wavelength can be selected in the LSERK-ISFM scheme, which can greatly contribute to reducing the consumption of memory. Then we build a simple 3D model to properly compare the proposed algorithm with the coarse-mesh FDTD and the fine-mesh FDTD, the results show that, with the same mesh generation, the LSERK-ISFM algorithm outperforms the conventional FDTD algorithms in terms of both accuracy and efficiency. Finally, we test the algorithm on a 3D complex model by a series of 3D forward profiles and the corresponding slices, both simulated reflections and diffractions clearly depict the shape and location of the anomalous bodies, which shows great potential of application in the field of 3D exploration and interpretation for GPR community.

Lagrange Operational Matrix Methods to Lane–Emden, Riccati’s and Bessel’s Equations

The current study is presented to develop two approaches and methodologies to find the numerical solution of linear and non-linear initial value problems such as Lane–Emden type equation, Riccati’s equation and Bessel’s equation of order zero based on approximation. The function approximations (scheme-I and scheme-II) are presented to find the numerical solutions of linear and non-linear initial value problems by using Gauss Legendre roots as node points and random node points in the domain [0, 1]. In the scheme-I, the roots of Legendre polynomial are used as node points for Lagrange polynomials and in scheme-II, we have taken random node points in the domain [0, 1] and orthogonalize the resulting Lagrange polynomials using Gram–Schmidt orthogonalization process. Firstly, we have introduced the function approximations by using generating interpolating scaling functions (ISF) and orthonormal Lagrangian basis functions (OLBF) over the space \(L^{2}[0,1]\) then we have constructed the operational matrices of integration and product operational matrices based on newly designed approximations namely ISF and OLBF. These operational matrices convert given linear and non-linear initial value problems into the associated system of algebraic equations. Finally, we have established error bounds (Lemmas 1, 2) of both scheme-I and scheme-II including the function approximations. The efficiency of the proposed schemes has been confirmed with several test examples including numerical stability. So, the schemes are simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort.

Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

This paper presents the formulation of finite elements based on Deslauriers-Dubuc interpolating scaling functions, also known as Interpolets, for their use in wave propagation modeling. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Expressions for stiffness and mass matrices are developed based on connection coefficients, which are inner products of basis functions and their derivatives. An example in 1-D was formulated using Central Difference and Newmark schemes for time differentiation. Encouraging results were obtained even for large time steps. Results obtained in 2-D are compared with the standard Finite Difference Method for validation.

ADAPTIVE GRID SIMULATION OF HYPERBOLIC EQUATIONS

We are interested in an adaptive grid method for hyperbolic equations. A multiresolution analysis, based on a biorthogonal family of interpolating scaling functions and lifted interpolating wavelets, is used to dynamically adapt grid points according to the physical field profile in each time step. Traditional finite-difference schemes with fixed stencils produce high oscillations around sharp discontinuities. In this paper, we hybridize high-resolution schemes, which are suitable for capturing singularities, and apply a finite-difference approach to the scaling functions at non-singular points. We use a total variation diminishing Runge-Kutta method for the time integration. The computational cost is proportional to the number of points present after compression. We provide several numerical examples to verify our approach.

Artificial Neural Network Simulation for Finite Element Analysis

Artificial neural networks are composed of interconnecting artificial neurons. Artificial neural networks may either be used to gain an understanding of biological neural networks, or for solving artificial intelligence problems without necessarily creating a model of a real biological system. Average interpolating scaling function is constructed with symmetric interpolating scaling function, and the two scaling functions are given the relationship between the derivatives, which provides a convenient approach for the interpolation, and greatly improve accuracy.

Mixed finite difference and Galerkin methods for solving Burgers equations using interpolating scaling functions

The current paper proposes a technique for the numerical solution of Burgers equations. The method is based on finite difference formula combined with the Galerkin method, which uses the interpolating scaling functions. Several test problems are given, and the numerical results are reported to show the accuracy and efficiency of the new algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

Wavelet Finite Element Method Analysis of Bending Plate Based on Hermite Interpolation

In this paper, a new kind of finite element method (FEM) is proposed, which use the two-dimensional Hermite interpolation scaling function constructed by tensor product as the basis interpolation function of field function, and then combine with the energy functional with related mechanics and variational principle, the wavelet finite element equations for solving elastic thin plate unit that constructed in this paper are derived. Then the bending problem of thin plate is solved very quickly and availably through the matlab program. The numerical example in this paper indicates the correctness and validity of this method, and has high calculation precision and convergence speed. Moreover, it also provides a reliable method to solve the free vibration problem of thin plate and the pipe crack problems.

A New Ant Colony Algorithm for Finite Element Analysis

By means of measuring creep curves, microstructure observation and FEM analysis of the stress field near the hole; an investigation has been made into the influence of the defects on creep behaviors and microstructure evolution of single crystal nickel-based superalloys. Results show that the creep lifetimes and plasticity of the single crystal nickel based superalloys are obviously decreased by microstructure defects. During high temperature creep, the stress isoline near the holes region displays the feature of the acetabuliform distribution, and possesses the bigger stress value at 45 angle direction relative to the applied stress axis. Average interpolating scaling function is constructed with symmetric interpolating scaling function, and the two scaling functions are given the relationship between the derivatives, which provides a convenient approach for the interpolation, and greatly improve accuracy.

Solution of nonlinear optimal control problems by the interpolating scaling functions

This paper presents a numerical method for solving nonlinear optimal control problems including state and control inequality constraints. The method is based upon interpolating scaling functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation

Two numerical techniques based on the interpolating scaling functions are presented for the solution of the generalized Burgers–Huxley equation. Some properties of the interpolating scaling functions are presented and are utilized to reduce the solution of the generalized Burgers–Huxley equation to the solution of a system of algebraic equations. Also another method which is based on the mixed collocation finite difference schemes is developed to solve this important nonlinear partial differential equation. Illustrative examples are included to demonstrate the validity and applicability of the presented techniques.

B-Spline Wavelet on the Interval Element for the Solution of Hamilton Canonical Equation

Based on the modified Hellinger-Reissner (H-R) variational principle for the elastic material, the formulation of B-spline wavelet on the interval (BSWI) element of the Hamilton canonical equation was derived through the use of the interpolation scaling function of BSWI. To demonstrate the excellent predictive capability of the formulation, numerical studies were conducted on a thick plate and a three-layer plate with various common boundary conditions. The basic steps from which the formulation of the BSWI element of the Hamilton canonical equation was derived can also be extended to deduce the expressions of other wavelet elements in the Hamiltonian system, such as Daubechies orthogonal wavelet, biorthogonal wavelet based on lifting scheme, and so on.

Numerical solution of telegraph equation using interpolating scaling functions

A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses interpolating scaling functions. The method consists of expanding the required approximate solution as the elements of interpolating scaling functions. Using the operational matrix of derivatives, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

Implementation of Exact Exchange with Numerical Atomic Orbitals

A method to calculate Hartree-Fock-type exact exchange has been implemented in the electronic structure code SIESTA based on a localized numerical atomic orbital basis set. In our implementation, the electron repulsion integrals are calculated by solving Poisson's equation using the interpolating scaling function method and then doing numerical integration in real-space. Test calculations for both isolated and periodic systems are performed, and good agreement with results calculated by Gaussian03 or Crystal06 packages is obtained.