Linear Combination Of Eigenfunctions Research Articles

Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.

A fast convergent solution of wave propagation for multilayer inhomogeneous cylindrical dielectric waveguides using a semianalytical method

Abstract This paper presents an accurate and efficient semianalytical method based on the Galerkin procedure for solving electromagnetic wave propagation problems in multilayer inhomogeneous cylindrical dielectric waveguides. The method represents the field in each inhomogeneous layer by a linear combination of eigenfunctions with unknown coefficients, which are expressed using the inner products of a series of basis functions, following the Galerkin procedure. The continuity of the field and its radial derivative is enforced at the interface between adjacent layers. By applying this procedure to all inhomogeneous layers, the Helmholtz equations are transformed into linear algebraic equations with expanded coefficients in matrix form, allowing the complicated wave propagation problem in a multilayer inhomogeneous waveguide to be solved as a matrix eigenvalue problem. The method is validated by providing detailed propagation characteristics for various multilayer inhomogeneous cylinders with different permittivity profiles. The accuracy and efficiency of the proposed method are demonstrated through comparisons with results obtained using other numerical techniques.

An adaptive eigenfunction basis strategy to reduce design dimension in topology optimization

AbstractThe concept of adaptive eigenspace basis (AEB) has recently proved effective for solving medium imaging problems. In this article, we present an AEB strategy for design parameterization in topology optimization (TO) problems. We seek the density design field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the structural domain, and solve for the associated eigenfunction coefficients. Restriction to this truncated eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution, removing the need for auxiliary filtering operations and design‐variable bound constraints. We furthermore develop the basis adaptation scheme inherent in the AEB, which iteratively recomputes the eigenfunction basis to conform to the evolving density field, enabling further dimension reduction and acceleration of the optimization process. The known aptitude of the adapted eigenfunctions to approximate piecewise constant fields is especially useful for TO as relevant design subspaces can be given low‐dimensional representations. We propose criteria for the selection of the basis dimension and demonstrate the use of basis function selection as means for length scale control. We compare performance of the AEB against conventional TO implementations in problems for static linear‐elasticity, showing comparable structural solutions, computational cost benefits, and consistent design dimension reduction.

Functional Principal Component Analysis for Extrapolating Multistream Longitudinal Data

In this article, we present a nonparametric approach to predict the evolution of multistream longitudinal data. Our approach first decomposes each stream into a linear combination of eigenfunctions and their corresponding functional principal component (FPC) scores. A Gaussian process prior for the FPC scores is then induced based on a functional semimetric that introduces a similarity measure across streams. Finally, an empirical Bayesian updating strategy is derived to update the established prior using real-time stream data. Empirical evidence shows that the proposed framework outperforms state-of-the-art approaches and can effectively account for heterogeneity as well as achieve high predictive accuracy.

On the controllability of a Boussinesq system for two-way propagation of dispersive waves

In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. The main focus is on the boundary controllability property, which corresponds to the question of whether the solutions can be driven to a given state at a given final time by means of controls acting at one endpoint of the interval. We first show that the system is not spectrally controllable. This means that no finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero. Although the system is not spectrally controllable, it can be shown that it is approximately controllable, i.e., any state can be steered arbitrarily close to another state.

On the spectral resolution of products of Laplacian eigenfunctions

We study products of eigenfunctions of the Laplacian $-\Delta \phi\_{\lambda} = \lambda \phi\_{\lambda}$ on compact manifolds. If $\phi\_{\mu}, \phi\_{\lambda}$ are two eigenfunctions and $\mu \leq \lambda$, then one would perhaps expect their product $\phi\_{\mu}\phi\_{\lambda}$ to be mostly a linear combination of eigenfunctions with eigenvalue close to $\lambda$. This can faily quite dramatically: on $\mathbb{T}^2$, we see that $$ 2\sin{(n x)} \sin{((n+1) x)} = \cos{(x)} - \cos{( (2n+1) x)} $$ has half of its $L^2-$mass at eigenvalue 1. Conversely, the product $$ \sin{(n x)} \sin{(m y)} $$ lives at eigenvalue $$ \max{\left{m^2,n^2\right}} \leq m^2 + n^2 \leq 2\max{\left{m^2,n^2\right}} $$ and the heuristic is valid. We show that the main reason is that in the first example 'the waves point in the same direction': if the heuristic fails and multiplication carries $L^2-$mass to lower frequencies, then $\phi\_{\mu}$ and $\phi\_{\lambda}$ are strongly correlated at scale $\sim \lambda^{-1/2}$ (the shorter wavelength) $$ \left| \int\_{M}{ p(t,x,y)( \phi\_{\lambda}(y) - \phi\_{\lambda}(x))( \phi\_{\mu}(y) - \phi\_{\mu}(x)) dy} \right|{L^2\_x} \gtrsim | \phi{\mu}\phi\_{\lambda}|\_{L^2}, $$ where $p(t,x,y)$ is the classical heat kernel and $t \sim \lambda^{-1}$. This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.

On the spectral resolution of products of Laplacian eigenfunctions

We study products of eigenfunctions of the Laplacian -\Delta \phi_{\lambda} = \lambda \phi_{\lambda} on compact manifolds. If \phi_{\mu}, \phi_{\lambda} are two eigenfunctions and \mu \leq \lambda , then one would perhaps expect their product \phi_{\mu}\phi_{\lambda} to be mostly a linear combination of eigenfunctions with eigenvalue close to \lambda . This can faily quite dramatically: on \mathbb{T}^2 , we see that 2\sin{(n x)} \sin{((n+1) x)} = \cos{(x)} - \cos{( (2n+1) x)} has half of its L^2- mass at eigenvalue 1. Conversely, the product \sin{(n x)} \sin{(m y)} lives at eigenvalue \max{\left\{m^2,n^2\right\}} \leq m^2 + n^2 \leq 2\max{\left\{m^2,n^2\right\}} and the heuristic is valid. We show that the main reason is that in the first example 'the waves point in the same direction': if the heuristic fails and multiplication carries L^2- mass to lower frequencies, then \phi_{\mu} and \phi_{\lambda} are strongly correlated at scale \sim \lambda^{-1/2} (the shorter wavelength) \left\| \int_{M}{ p(t,x,y)( \phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy} \right\|_{L^2_x} \gtrsim \| \phi_{\mu}\phi_{\lambda}\|_{L^2}, where p(t,x,y) is the classical heat kernel and t \sim \lambda^{-1} . This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.

Two-Wavelet Multipliers on the Dual of the Laguerre Hypergroup and Applications

In this paper, we are interested in the Laguerre hypergroup $$\mathbb {K} = [0,\infty )\times {\mathbb {R}}$$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup $$\widehat{\mathbb {K}}$$ which can be topologically identified with the so-called Heisenberg fan, the subset of $${\mathbb {R}}^{2}$$ $$\begin{aligned} \cup _{j\in {\mathbb {N}}}\left\{ (\lambda ,\mu )\in {\mathbb {R}}^{2}:\mu =|\lambda |(2j+\alpha +1), \; \lambda \ne 0\right\} \cup \left\{ (0,\mu )\in {\mathbb {R}}^{2}:\mu \ge 0\right\} , \end{aligned}$$ by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on $$L^{p}_{\alpha }(\mathbb {K})$$ , $$1 \le p \le \infty $$ . Afterwards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving $$\varepsilon $$ -concentration of the generalized two-wavelet multiplier operators. Moreover, we study functions whose time–frequency content is concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

Parameter identification through mode isolation for reaction–diffusion systems on arbitrary geometries

We present a computational framework for isolating spatial patterns arising in the steady states of reaction–diffusion systems. Such systems have been used to model many natural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction–diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally, we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.

Phase space localized functions in the Dunkl setting

Time–frequency analysis plays a central role in signal analysis, because signals that have highly concentrated time–frequency content are used in many applications. The uncertainty principle in Fourier analysis sets a limit to the possible simultaneous concentration of a function and its Dunkl transform. To localize signals in the time–frequency plane we use time–frequency localization operators in the Dunkl setting to measure their time–frequency content on some subset of finite measure. Then, using eigenfunctions of these operators, which are maximally time–frequency concentrated, we prove a characterization of functions that are time–frequency concentrated in the region of interest, and we obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions.

Fractional optimal control problem of an axis-symmetric diffusion-wave propagation

This paper presents the formulation of an axis-symmetric fractional optimal control problem (FOCP). Dynamic characteristics of the system are defined in terms of the left and right Riemann–Liouville fractional derivatives (RLFDs). The performance index of a FOCP is described with a state and a control function. Furthermore, dynamic constraints of the system are given by a fractional diffusion-wave equation. It is preferred to use the method of separation of variables for finding the analytical solution of the problem. In this way, the closed form solution of the problem is obtained by a linear combination of eigenfunctions and eigencoordinates. For numerical evaluation, the Grünwald–Letnikov approximation is applied to the problem. Consequently, some simulation results show that analytical and numerical solutions overlap for α=1. This numerical approach is applicable and effective for such a kind of FOCP. In addition, the changing of some variables related to the problem formulation is analyzed.

Application of spectral decomposition of Green's function to linear inverse problem

An inverse analysis method using the spectral decomposition of Green's function is proposed for linear inverse problems of identifying inner sources from data of surface responses. It is assumed that Green's function of the corresponding physical problem is given. Applying the spectral decomposition, Green's function is discretized as a sum of eigen-values and eigen-functions. From the comparison of the measurement accuracy with the eigen-values, it is shown that responses which can be actually measured are given as a linear combination of eigen-functions corresponding to larger eigen-values. Such responses are found by determining coefficients of the eigen-functions from the measured data, and then sources which are predictable are determined just by calculating their coefficients for the eigen-functions. Without any ambiguity, the proposed method can determine the predictable inner sources from the data which are measured with the limited accuracy. A numerical simulation of solving a simple example problem is carried out to demonstrate the usefulness of the proposed inverse analysis method, and the results are discussed.

A robust timing synchronization scheme in multiple antenna systems with Doppler frequency shifts

We propose a timing synchronization scheme for a dual antenna system in Rayleigh-fading environments. Instead of assuming the channel gain to be constant during the training duration, we consider the time-variant nature of the multiplicative distortions and formulate them with linear combinations of eigenfunctions. Then, we derive a formula to find the maximum-likelihood estimation of the channel timing and simulate the performance with both ideal and nonideal channel state information. The results show that this approach outperforms the conventional ones especially when the Doppler spread is severe.

Heat transfer in turbulent tube flow

The differential equation for the temperature distribution in a laminar flow through a smooth cylindrical tube can be solved on certain assumptions by separating the variables; the result is a temperature distribution given by a linear combination of eigenfunctions. For a fully developed turbulent flow the same differential equation applies in principle with the following differences:a) the velocity distribution is only known from measurement; consequently only empirical formulae are available for the calculations;b) the thermal diffusivitya is increased by an amountAq due to turbulent mixing, and the value of this thermal eddy diffusivity depends upon the velocity distribution. The cross-section of the tube is divided into three concentric parts, according to the magnitude of the frictional eddy diffusivityAm: the turbulent core, where the molecular momentum diffusion is negligible in relation to the eddy diffusivity of momentum; the laminar boundary layer, where the molecular momentum diffusion prevails; and the transition layer, where eddy diffusivity and molecular diffusion of momentum are of the same order. Separately for each of these regions formulae have been drawn up for the velocity and momentum eddy diffusivityAm. Given the hypothesis thatAq/Am=A is a constant,Aq follows fromAm. In the present article, for each of these three regions a general solution is given for the differential equation valid in that area, by separating the variables. The constants in these solutions have been chosen in such a way as to make the temperatures and heat currents continuous at the boundaries between the three regions. For a constant wall temperature the first two eigenfunctions and eigenvalues are calculated for severalRe andPr values. Furthermore, the heat transfer and temperature distribution for a homogeneous entrance temperature are given. The calculated eigenfunctions can also be used for calculating the heat transfer and heat distribution for a non-homogeneous entrance temperature, provided the latter is symmetrical with respect to the axis of the tube.