Search Results (68)

The Search and Rescue at Sea Manual defines several uncertainties related to the initial position and the time elapsed between an accident and the onset of SAR operations. The present article seeks an approach to address this problem through the assimilation of drifting buoy data and their use in correcting the system parameters via an ill-posed inverse problem. The results demonstrate that, in the search for objects at sea, the uncertainty of their initial position must be explicitly considered. Quantitatively, the proposed methodology reduced the uncertainty of the initial search area by approximately 55–60% compared with the traditional approach that assumes a single deterministic initial point. This outcome underscores the potential of data assimilation techniques to enhance the probabilistic accuracy of maritime search and rescue planning. Full article

In this work, we combine the Natural Transform and generalized-Laplace Transform into a new transform called, the Natural Generalized-Laplace Transform, (NGLT) and some of its properties are provided. Moreover, the existence condition, convolution theorem, periodic theorem, and non-constant coefficient partial derivatives are proved with some details. The (NGLT) is applied to gain the solutions of linear telegraph and partial integro-differential equations. Also, we obtained the solution of the singular one-dimensional Boussinesq equation by employing the Natural Generalized-Laplace Transform Decomposition Method, (NGLTDM). Full article

Kirchhoff’s current law was originally derived for systems such as telegraphs that switch in 0.1 s. It is used widely today to design circuits in computers that switch in ~0.1 nanoseconds, one billion times faster. Current behaves differently in one second and one-tenth of a nanosecond. A derivation of a current law from the fundamental equations of electrodynamics—the Maxwell equations—is needed. Here is a derivation in one line: $\mathbf{d}\mathbf{i}\mathbf{v} \mathbf{c}\mathbf{u}\mathbf{r}\mathbf{l}\mathbf{B}/{\mathsf{\mu }}_0=0=\mathbf{d}\mathbf{i}\mathbf{v}\left(\mathbf{J}+({\mathsf{\epsilon }}_{\mathit{r}}-1){\mathsf{\epsilon }}_0\mathbf{\partial }\mathbf{E}/\mathbf{\partial }\mathbf{t}+{\mathsf{\epsilon }}_0\mathbf{\partial }\mathbf{E}/\mathbf{\partial }\mathbf{t}\right)=\mathbf{d}\mathbf{i}\mathbf{v}{\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$. Maxwell’s ‘true’ current is defined as ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$. The universal displacement current found everywhere is ${\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$. The conduction current $\mathbf{J}$ is carried by any charge with mass, no matter how small, brief, or transient, driven by any source, e.g., diffusion. The second term $({\mathsf{\epsilon }}_r-1){\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$ is the usual approximation to the polarization currents of ideal dielectrics. The dielectric constant ${\mathsf{\epsilon }}_r$ is a dimensionless real number. Real dielectrics can be very complicated. They require a complete theory of polarization to replace the $({\mathsf{\epsilon }}_r-1){\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$ term. The Maxwell current law $\mathrm{d}\mathrm{i}\mathrm{v}{\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}=0$ defines the solenoidal field of total current that has zero divergence, typically characterized in two dimensions by streamlines that end where they begin, flowing in loops that form circuits. Note that the conduction current $\mathbf{J}$ is not solenoidal. Conduction current $\mathbf{J}$ accumulates significantly in many chemical and biological applications. Total current ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ does not accumulate in any time interval or in any circumstance where the Maxwell equations are valid. ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ does not accumulate during the transitions of electrons from orbital to orbital within a chemical reaction, for example. ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ should be included in chemical reaction kinetics. The classical Kirchhoff current law $\mathrm{d}\mathrm{i}\mathrm{v} \mathbf{J}=0$ is an approximation used to analyze idealized topological circuits found in textbooks. The classical Kirchhoff current law is shown here by mathematics to be valid only when $\mathbf{J}\gg {\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t},$ typically in the steady state. The Kirchhoff current law is often extended to much shorter times to help topological circuits approximate some of the displacement currents not found in the classical Kirchhoff current law. The original circuit is modified. Circuit elements—invented or redefined—are added to the topological circuit for that purpose. Full article

In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency. Our observations indicate that the method generates an approximate solution in series form, which converges rapidly to the analytic solution in each instance. The convergence of these series solutions is assessed both geometrically and numerically. Our results demonstrate that LTHAM is a reliable, powerful, and straightforward approach to solving fractional telegraph equations, and it can be effectively extended to solve similar types of equations. Full article

The Pennes bioheat equation is the most widely used model for describing heat transfer in living tissue during thermal exposure. It is derived from the classical Fourier law of heat conduction and assumes energy exchange between blood vessels and surrounding tissues. The literature presents various numerical methods for solving the bioheat equation, with exact solutions developed for different boundary conditions and geometries. However, analytical models based on this framework are rarely reported. This study aims to develop an analytical three-dimensional model using MATHEMATICA software, with subsequent mathematical validation performed through COMSOL simulations, to characterize heat transfer in biological tissues induced by laser irradiation under various therapeutic conditions. The objective is to refine the conventional bioheat equation by introducing three key improvements: (a) incorporating a non-Fourier framework for the Pennes equation, thereby accounting for the relaxation time in thermal response; (b) integrating Dirac functions and the telegraph equation into the bioheat model to simulate localized point heating of diseased tissue; and (c) deriving a closed-form analytical solution for the Pennes equation in both its classical (Fourier-based) and improved (non-Fourier-based) formulations. This paper investigates the nuanced relationship between the relaxation time parameter in the telegraph equation and the thermal relaxation time employed in the bioheat transfer equation. Considering all these aspects, the optimal thermal relaxation time determined for these simulations was 1.16 s, while the investigated thermal exposure time ranged from 0.01 s to 120 s. This study introduces a generalized version of the model, providing a more realistic representation of heat exchange between biological tissue and blood flow by accounting for non-uniform temperature distribution. It is important to note that a reasonable agreement was observed between the two modeling approaches: analytical (MATHEMATICA) and numerical (COMSOL) simulations. As a result, this research paves the way for advancements in laser-based medical treatments and thermal therapies, ultimately contributing to more optimized therapeutic outcomes. Full article

This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate $R_c\left(M\right)$, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision. Full article

The electrodynamics of current provide much of our technology, from telegraphs to the wired infrastructure powering the circuits of our electronic technology. Current flow is analyzed by its own rules that involve the Maxwell Ampere law and magnetism. Electrostatics does not involve magnetism, and so current flow and electrodynamics cannot be derived from electrostatics. Practical considerations also prevent current flow from being analyzed one charge at a time. There are too many charges, and far too many interactions to allow computation. Current flow is essential in biology. Currents are carried by electrons in mitochondria in an electron transport chain. Currents are carried by ions in nerve and muscle cells. Currents everywhere follow the rules of current flow: Kirchhoff’s current law and its generalizations. The importance of electron and proton flows in generating ATP was discovered long ago but they were not analyzed as electrical currents. The flow of protons and transport of electrons form circuits that must be analyzed by Kirchhoff’s law. A chemiosmotic theory that ignores the laws of current flow is incorrect physics. Circuit analysis is easily applied to short systems like mitochondria that have just one internal electrical potential in the form of the Hodgkin Huxley Katz (HHK) equation. The HHK equation combined with classical descriptions of chemical reactions forms a computable model of cytochrome c oxidase, part of the electron transport chain. The proton motive force is included as just one of the components of the total electrochemical potential. Circuit analysis includes its role just as it includes the role of any other ionic current. Current laws are now needed to analyze the flow of electrons and protons, as they generate ATP in mitochondria and chloroplasts. Chemiosmotic theory must be replaced by an electro-osmotic theory of ATP production that conforms to the Maxwell Ampere equation of electrodynamics while including proton movement and the proton motive force. Full article

Local equilibrium approximation (LEA) is a central assumption in many applications of non-equilibrium thermodynamics involving the transport of energy, mass, and momentum. However, assessing the validity of the LEA remains challenging due to the limited development of tools for characterizing non-equilibrium states compared to equilibrium states. To address this, we have developed a theory based on kinetic theory, which provides a nonlinear extension of the telegrapher’s equation commonly discussed in non-equilibrium frameworks that extend beyond LEA. A key result of this theory is a steady-state diffusion equation that accounts for the constraint imposed by available thermal energy on the diffusion flux. The theory is suitable for analysis of steady-state composition profiles and can be used to quantify the deviation from the local equilibrium. To validate the theory and test LEA, we performed molecular dynamics simulations on a two-component system where the two components had identical physical properties. The results show that deviation from the local equilibrium can be systematically quantified, and for the diffusion process we have studied here, we have confirmed that LEA remains accurate even under extreme concentration gradients in gas mixtures. Full article

I study a lattice with periodic boundary conditions using a non-local master equation that evolves over time. I investigate different system regimes using classical theories like Fisher information, Shannon entropy, complexity, and the Cramér–Rao bound. To simulate spatial continuity, I employ a large number of sites in the ring and compare the results with continuous spatial systems like the Telegrapher’s equations. The Fisher information revealed a power-law decay of $t^{-\nu }$, with $\nu =2$ for short times and $\nu =1$ for long times, across all jump models. Similar power-law trends were also observed for complexity and the Fisher information related to Shannon entropy over time. Furthermore, I analyze toy models with only two ring sites to understand the behavior of the Fisher information and Shannon entropy. As expected, a ring with a small number of sites quickly converges to a uniform distribution for long times. I also examine the Shannon entropy for short and long times. Full article

This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms. Full article

The combination of fractional derivatives (due to their global behavior) and the challenges related to hyperbolic PDEs pose formidable obstacles in solving fractional hyperbolic equations. Due to the importance and applications of the fractional telegraph equation, solving it and presenting accurate solutions via a novel and effective method can be useful. This work introduces and implements a method based on the spectral element method (SEM) that relies on interpolating scaling functions (ISFs). Through the use of an orthonormal projection, the method maps the equation to scaling spaces raised from multi-resolution analysis (MRA). To achieve this, the Caputo fractional derivative (CFD) is represented by ISFs as a square matrix. Remarkable efficiency, ease of implementation, and precision are the distinguishing features of the presented method. An analysis is provided to demonstrate the convergence of the scheme, and illustrative examples validate our method. Full article

Electromagnetic wave dissipation is experienced in radiative absorbing-emitting processes and signal transmissions via media. The absorbed wave initiates thermal processes in the conducting medium. Conversely, thermal processes generate electromagnetic waves in the vacuum–material interface region. The two processes do not take place symmetrically, i.e., the incoming and thermalizing electromagnetic spectrum does not occur in the reverse process. The conservation of energy remains in effect, and the loop process “electromagnetic wave–thermal propagation–electromagnetic wave” is dissipative. In the Lagrangian formalism, we provide a unified description of these two interconnected processes. We point out how it involves the origin of the asymmetry. Full article

We consider two different time fractional telegrapher’s equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates due to the resetting mechanism. We also obtain the fractional telegraph process as a subordinated telegraph process by introducing operational time such that the physical time is considered as a Lévy stable process whose characteristic function is the Lévy stable distribution. We also analyzed the survival probability for the first-passage time problem and found the optimal resetting rate for which the corresponding mean first-passage time is minimal. Full article

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics. Full article

Since the discovery of electricity, electric cables have become ubiquitous in human constructions, from machines to buildings. Insulators play a crucial role in ensuring the proper functioning of these cables, so it is important to monitor their possible damage, which can be caused by environmental contamination, severe temperature variations, and electrical and mechanical stress. While shunt conductance is a direct health indicator of cable insulation, measuring the cable average shunt conductance is not sufficient for the detection of localized insulator damage, since localized conductance variations are diluted over a long cable length in such measurements. The objective of this paper is to assess the feasibility of reflectometry techniques for the monitoring of insulator damage in electric cables. To this end, the estimation of localized conductance variations is investigated based on electrical measurements made at one end of a cable. To avoid estimating a large number of discretized conductance values along a long cable, the proposed method relies on sparse regression, which automatically focuses on localized conductance variations at unknown positions caused by accidental insulator damage. In order to efficiently apply sparse regression techniques, the telegrapher’s equations describing electric wave propagation in cables are transformed through several steps into a simple linear regression form. Then, Lasso (Least Absolute Shrinkage and Selection Operator) regression is applied to process the voltage and current data collected at a single end of the monitored cable. Numerical simulations show the potential of this method for fast estimation of localized shunt conductance variations. Full article