Trigonometric Basis Functions Research Articles

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Enhancing curve and surface applications with trigonometric polynomial basis functions.

This article primarily focuses on the utilization and importance of parametric curves in surface design. It delves into the construction and applications of parametric curves, exploring the implementation of trigonometric polynomial basis functions that possess two shape parameters. Initially, these basis functions are employed in constructing both rational and non-rational curves. Later, they are employed to define the surfaces generated by these curves. The discussion includes rational surfaces, tensor product surfaces, and various specialized surfaces. The aim is to provide a comprehensive understanding of the role and potential of parametric curves in surface design.

Numerical analysis with a class of trigonometric functions for nonlinear time fractional Wu-Zhang system

We study the implementation of a numerical method to solve the time fractional (2+1)-dimensional Wu–Zhang system focusing on properties of trigonometric basis functions (TBFs). The approximation of the system describing dispersive long waves by appropriate discretization of the system and Grünwald difference operators is discussed and a linear system of the resulting discrete systems is given. This process includes a presentation of a general framework to approximate the solution by a mesh-free scheme which is stable and converges. Finally, various numerical experiments are present to illustrate the exceptional effect of fractional-order on the WZ system.

Time discretization for modeling migration of groundwater contaminant in the presence of micro-organisms via a semi-analytic method

The parabolic systems for migration of groundwater contaminants are studied through the semi-analytic method (SAM). For systems, a modification of the classic trigonometric basis function (TBF) is proposed that successfully obtains the solution on the available boundary data and improves accuracy. The method is based on a theory that removes instability and keeps the same accuracy. However, especially when using SAM schemes with acceptable stability properties, one is still faced with the considerable task of linearizing the systems, consequently, the linearization of the systems which approximate the nonlinear term with the first order derivative is introduced. The final approximation is given by the summation of the primary approximation, radial basis functions (RBFs), and the related TBFs which are determined by the homogeneous boundary conditions. Then the approximation is substituted back to the governing equations where the unknown coefficients can be determined. The efficiency of the algorithm is highlighted by numerical simulations relating to a model on test cases with analytical solutions and without an analytical solution.

An accurate RBF–based meshless technique for the inverse multi-term time-fractional integro-differential equation

Using the Grünwald difference operator one reduces the inverse boundary value problem of the multi-term time-fractional integro-differential equation (TFIDE) in two dimensions to a system L[u]=F. The backward substitution method (BSM) which is based on the trigonometric basis function (TBF) is developed to a new version and examined for the system. The solution of ill-conditioned TFIDE is expressed in terms of the TBFs on the corrupted boundary data with a small parameter in the noisy data. Finally, the TFIDE model in the complex computational domain is assessed through the methodology with reasonable selective parameters, in addition, high-accuracy solutions and acceptable estimation for the errors are illustrated.

Summation-by-Parts Operators for General Function Spaces

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

Difference mode decomposition for adaptive signal decomposition

Adaptive extraction of concerned components (CC) from mixed frequency components remains to be a challenging topic in various research domains. Most existing adaptive mode decomposition algorithms for extracting CC, such as wavelet transform, wavelet packet transform, singular value decomposition, empirical mode decomposition, empirical wavelet transform, and variational mode decomposition, are intrinsically adaptive band-pass filter banks and they face the following two tough problems. The first problem is that they cannot separate CC from interferential components in same frequency bands. The second problem is that it is not fully effective in automatically selecting CC distributed in different frequency bands by using some designed criteria. In this paper, a new decomposition approach called difference mode decomposition (DMD) is proposed to adaptively decompose a mixed signal into CC, reference components, and noise, and enrich the domain of adaptive mode decomposition. The proposed DMD relies on convex optimization and Fourier transform, and its decomposition is mathematically justified and composes physical interpretations. Analyses of simulated and real-world bearing and gear vibration signals are used to verify the effectiveness and superiority of the proposed DMD over existing adaptive mode decomposition algorithms. It is demonstrated that the proposed DMD can effectively extract CC such as repetitive transients caused by bearing and gear faults. Moreover, since the proposed DMD is based on Fourier transform expanded by trigonometric basis functions, the proposed DMD can be easily extended to other domains by expanding basis functions besides the frequency domain.

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

Fourier Analysis: A new computing approach [Lecture Notes

A Fourier expansion is a special case of signal decomposition that decomposes a signal into oscillatory components. In this method, the signal is represented as a linear combination of trigonometric or exponential basis functions. The expansion coefficients (or weights) are then computed by correlating the signal with the corresponding basis functions <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> . The process of computing the coefficients is known as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fourier analysis</i> . In real applications, we are interested in using a few terms of a Fourier expansion, or it may be impossible to use all of the terms to approximate the signal. Therefore, a truncated Fourier expansion is used instead <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> . However, when a truncated Fourier expansion is used to approximate a signal with a jump discontinuity, an overshoot/undershoot at the discontinuity occurs, which is known as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gibbs phenomenon</i> . The correct size of the overshoot and the undershoot of a truncated Fourier expansion near the point of discontinuity was computed by Gibbs; the size of the overshoot/undershoot is approximately 9% of the magnitude of the jump <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> .

Space-time backward substitution method for nonlinear transient heat conduction problems in functionally graded materials

In this paper, an efficient space-time backward substitution method (STBSM) is presented for solving nonlinear transient heat conduction problems in 2D and 3D functionally graded materials (FGMs). To improve the computation efficiency of the traditional backward substitution method (BSM) for transient problems, the space-time radial basis function (STRBF) and space-time trigonometric basis function (STTBF) are introduced for interpolation in the proposed method. The time dimension is regarded as a pseudo-space dimension to generate the space-time Euler metric r T in STRBF, which can be applied to transform the d -dimensional transient governing equation into ( d + 1 ) -dimensional steady-state governing equation. Benefiting from merits of the traditional BSM, an accurate numerical solution is available with only a few of computational points. Several examples are solved to verify the accuracy and efficiency of the proposed method and its merits and flaws are discussed in the end. • A space-time method is attempted for 2D and 3D heat conduction problems in FGMs. • Space-time domain and space-time radial basis function are introduced into backward substitution method. • By introducing pseudo-space-time dimension to transform the origin transient problem into steady-state problem. • The proposed method has excellent computational efficiency.

On some quasi-periodic approximations

Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.

Construction of Cubic Trigonometric Curves with an Application of Curve Modelling

This paper introduces new trigonometric basis functions (TBF) in polynomial and rational form with two shape parameters (SPs). Some classical characteristics, such as the partition of unity, positivity, symmetry, CHP, local control and invariance under affine transformation properties are proven mathematically and graphically. In addition, different continuity conditions at uniform knots (UK) are proven. Some open and closed curves from TBS and trigonometric rational B-spline (TRBS) are generated to test the applicability of the suggested technique, and the influence of the shape parameter is also noted. Furthermore, various objects, such as designing an alphabet, star, butterfly, leaf and 3D cube.

Rational cubic trigonometric exponential Bézier curve

The cubic trigonometric basis functions are constructed by including two exponential functions with five shape parameters and a tension parameter. Also, the properties of the basis functions are discussed. We propose to define the rational cubic trigonometric exponential Bezier curve with weight and the continuity condition of the curve. Visual effects of parameters and tension parameter along with weight are presented. We compare the proposed method of constructing the desired curve with the existing method. The conic sections of the curve are also discussed.

Dynamic fracture analysis using a high-accuracy manifold element modelling scheme

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.

Shape preserving rational cubic trigonometric fractal interpolation functions

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form pi(θ)/qi(θ), where pi(θ) and qi(θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C1-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

A Computational Approach for One and Two Dimensional Fisher’s Equation Using Quadrature Technique

In this paper, a refined form of the differential quadrature method is proposed to compute the numerical solution of one and two-dimensional convection-diffusion Fisher’s equation. The cubic trigonometric B-spline basis functions are applied in the differential quadrature method in a modified form to obtain the weighting coefficients. The application of the method reduces nonlinear Fisher’s partial differential equation into a system of ordinary differential equations which can be solved by applying the Runge-Kutta method. Six numerical test problems of Fisher’s equation are analyzed numerically to establish the efficiency of the proposed method. The stability of the method is also discussed using the matrix method.

A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α(t) and β(t) are constructed in a generalized quasi cubic trigonometric space span{1,sin2t,(1−sint)2α(t),(1−cost)2β(t)}, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions αi(t) and βi(t) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.

Epsilon kappa -Curves: controlled local curvature extrema

The kappa -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or epsilon kappa -curves.kappa -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using epsilon kappa -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

Approximate Solution of Bratu Differential Equations Using Trigonometric Basic Functions

In this paper, I have proposed a method for finding an approximate function for Bratu differential equations (BDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, I define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the approximate function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get an approximate solution function with discrete derivatives of the solution function, we have determined the approximate solution function which satisfies in the Bratu differential equations (BDEs). In the end, the algorithm of the method is elaborated with several examples. In one example, I have presented an absolute error comparison of some approximate methods.

A novel meshless space-time backward substitution method and its application to nonhomogeneous advection-diffusion problems

For time-dependent problems, a novel space-time backward substitution method is proposed in this paper. By employing the space-time radial basis function and the space-time trigonometric basis function in the original backward substitution method, the space-time backward substitution method is obtained. Compared with the traditional backward substitution method, this novel numerical method avoids the time-difference measurement in the temporal approximation, which may result in accumulative errors, and at the same time maintains the existing advantages. Furthermore, the proposed method can be easily used to solve more general time-dependent problems. To evaluate the accuracy and efficiency, several numerical examples are tested and compared.