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In signals of opportunity (SOPs)-based positioning utilizing low Earth orbit (LEO) satellites, ephemeris data derived from two-line element files can introduce increasing error over time. To handle the erroneous measurement, an additional base receiver with a known position is often used to compensate for the effect of ephemeris error when positioning the user terminal (UT). However, this approach is insufficient for the long baseline (the distance between the base receiver and UT) as it fails to adequately correct Doppler shift measurement errors caused by ephemeris inaccuracies, resulting in degraded positioning performance. Moreover, the lack of clock synchronization between the base receiver and UT exacerbates erroneous Doppler shift measurements. To address these challenges, we put forth a robust double-difference Doppler shift-based positioning framework, coined 3DPose, to handle the clock synchronization issue between the base receiver and UT, and positioning degradation due to the long baseline. The proposed 3DPose framework leverages double-difference Doppler shift measurements to eliminate the clock synchronization issue and incorporates a novel ephemeris error correction algorithm to enhance UT positioning accuracy in case of the long baseline. The algorithm specifically characterizes and corrects the Doppler shift measurement errors arising from erroneous ephemeris data, focusing on satellite position errors in the tangential direction. To validate the effectiveness of the proposed framework, we conduct comparative analyses across three different scenarios, contrasting its performance with the existing differential Doppler positioning method. The results demonstrate that the proposed 3DPose framework achieves an average reduction of 90% in 3-dimensional positioning errors compared to the existing differential Doppler approach., Comment: 32 pages, 8 figures, 2 tables

Let $\tau: I=[0, 1]\to [0, 1]$ be a piecewise convex map with countably infinite number of branches. In \cite{GIR}, the existence of absolutely continuous invariant measure (ACIM) $\mu$ for $\tau$ and the exactness of the system $(\tau, \mu)$ has been proven. In this paper, we develop an Ulam method for approximation of $f^*$, the density of ACIM $\mu$. We construct a sequence $\{\tau_n\}_{n=1}^\infty$ of maps $\tau_n: I\to I$ s. t. $\tau_n$ has a finite number of branches and the sequence $\tau_n$ converges to $\tau$ almost uniformly. Using supremum norms and Lasota-Yorke type inequalities, we prove the existence of ACIMs $\mu_n$ for $\tau_n$ with the densities $f_n$. For a fixed $n$, we apply Ulam's method with $k$ subintervals to $\tau_n$ and compute approximations $f_{n,k}$ of $f_n$. We prove that $f_{n,k}\to f^*$ as $n\to \infty, k\to \infty,$ both a.e. and in $L^1$. We provide examples of piecewise convex maps $\tau$ with countably infinite number of branches, their approximations $\tau_n$ with finite number of branches and for increasing values of parameter $k$ show the errors $\|f^*-f_{n,k}\|_1$., Comment: 19 pages, 9 figures

This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in details, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory.

To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such as Galerkin, Least Square, and Collocation for the efficient numerical solution of nonlinear two-point boundary value problems. Some nonlinear cases are examined for observing the maximum absolute errors by the considered methods, demonstrating the accuracy and reliability of the present technique using the modified Legendre and modified Bernoulli polynomials as weight functions. The mathematical formulations and computational algorithms are more straightforward and uncomplicated to understand. Absolute errors and the graphical representation reflect that our method is more accurate and reliable., Comment: 14 Pages, 10 Figures

The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L infinity norm.

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases, Comment: Accepted for publication into the journal:European Journal of Mathematics and Statistics, 2023

The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science and engineering. The initial and first approximations of parabolic nonlinear PDEs associated with initial conditions have been generated by utilizing this method. Then the Galerkin method is applied to estimate the coefficients that remain unknown. Finally, the values of the coefficients generated by the Galerkin method have been inserted into the first approximation. In each example, all numerical computations and corresponding absolute errors are provided in schematic and tabular representations. The rate of convergence attained by the proposed method is depicted in tabular form, Comment: Accepted for Publication into the journal GANIT: Journal of Bangladesh Mathematical Society, 2023

We introduce a glide symmetric optical waveguide exhibiting a stationary inflection point (SIP) in the Bloch wavenumber dispersion relation. An SIP is a third order exceptional point of degeneracy (EPD) where three Bloch eigenmodes coalesce to form a so-called frozen mode with vanishing group velocity and diverging amplitude. We show that the incorporation of chirped distributed Bragg reflectors and distributed coupling between waveguides in the periodic structure facilitates the SIP formation and greatly enhances the characteristics of the frozen mode regime. We confirm the existence of an SIP in two ways: by observing the flatness of the dispersion diagram and also by using a coalescence parameter describing the separation of the three eigenvectors collapsing on each other. We find that in the absence of losses, both the quality factor and the group delay at the SIP grow with the cubic power of the cavity length. The frozen mode regime can be very attractive for light amplification and lasing, in optical delay lines, sensors, and modulators., Comment: 11 pages, 14 figures

We design a three-way silicon optical waveguide with the Bloch dispersion relation supporting a stationary inflection point (SIP). The SIP is a third order exceptional point of degeneracy (EPD) where three Bloch modes coalesce forming the frozen mode with greatly enhanced amplitude. The proposed design consists of a coupled resonators optical waveguide (CROW) coupled to a parallel straight waveguide. At any given frequency, this structure supports three pairs of reciprocal Bloch eigenmodes, propagating and/or evanescent. In addition to full-wave simulations, we also employ a so-called ''hybrid model'' that uses transfer matrices obtained from full-wave simulations of sub-blocks of the unit cell. This allows us to account for radiation losses and enables a design procedure based on minimizing the eigenmodes' coalescence parameter. The proposed finite-length CROW displays almost unitary transfer function at the SIP frequency, implying a nearly perfect conversion of the input light into the frozen mode. The group delay and the effective quality factor at the SIP frequency show an $N^{3}$ scaling, where $N$ is the number of unit cells in the cavity. The frozen mode in the CROW can be utilized in various applications like sensors, lasers and optical delay lines., Comment: 14 pages, 17 figures

The optoacoustic process can solve the longstanding challenge of wireless information transmission from an airborne unit to an underwater node (UWN). The nonlinear optoacoustic signal generated by proper laser parameters can propagate long distances in water. However, forming such a signal requires a high-power laser, and the buildup of a vapor cloud precludes the subsequent acoustic signal generation. Therefore, pursuing the traditional on-off keying (OOK) modulation technique will limit the data rate and power efficiency. In this paper, we analyze different modulation techniques and propose a vapor cloud delayed-differential pulse position modulation (VCD-DPPM) technique to improve the data rate and achieve high power efficiency for a single stationary laser transmitter. The symbol rate of VCD-DPPM is approximately 6.9 times and 1.69 times higher than OOK in our text communication simulation using a laser repetition rate of 10 kHz and 40 Hz, respectively. Furthermore, VCD-DPPM is 137% more power efficient than the OOK technique for both cases. We have generated different acoustic signal levels in laboratory conditions and simulated the bit error rate (BER) for different depths and positions of the UWN, while considering ambient underwater noises. Our results indicate that VCD-DPPM enables efficient data transmission., Comment: This paper is accepted in GLOBECOM 2022 - IEEE Global Communications Conference, Rio de Janeiro, Brazil