Time-dependent Source Research Articles

A 3D TLM code for the study of the ELF electromagnetic wave propagation in the Earth's atmosphere

The interest in the study of electromagnetic propagation through planetary atmospheres is briefly discussed. Special attention is devoted to extremely-low-frequency fields in the Earth's atmosphere for its global nature and possible applications to climate monitoring studies among others. In the Earth's case, the system can be considered as a spherical electromagnetic shell resonator in which two concentric and large conducting spheres with a radius around 6300 km are separated by a very small distance of around 100 km, the atmosphere height. A numerical solution using the Transmission Line Method is proposed. The classical spherical-coordinate description is easy to use, however, the important difference in the dimensions along the three coordinate directions causes high numerical dispersion in the results. A Cartesian scheme with equal node size for all directions is used to reduce this undesired dispersion. A pre-processing stage is the key point introduced to lessen the resulting high demand of memory and time calculation and make the solution feasible. A parallelized Fortran code together with pre- and post-processing Python programs to ease the user interface are provided with this work. Details on the Fortran code and the Python modules are included both in the paper and the source codes to allow the use and modifications by other researchers interested in electromagnetic propagation through planetary atmospheres. The program allows calculation of the time evolution of the electromagnetic field at any point in the atmosphere. It includes the possibility of considering multiple time-dependent sources and different homogeneous and inhomogeneous conductivity profiles to model different situations. Profiles to study day-night asymmetries or locally perturbed profiles which have been attributed to earthquakes in the literature are implemented, for instance.

Modelling a simplified device-a wind musical instrument-as an educational tool to study the behavior of a one-dimensional resonator.

The aim of this paper is to show how the analytical modelling of a simplified acoustic device (herein a simplified wind musical instrument) may be a relevant educational way to study several aspects of the behavior of an idealized component [herein a one-dimensional (1-D) resonator], which is the core component of the device considered. A time-dependent source, whether sinusoidal, impulse, or representing approximate valve-effects of the reed, is coupled to the resonator in the analytical modelling. The unavoidable thermo-viscous parietal dissipation, as well as an approximate radiation effect, are accounted for. After a brief presentation of the problem, precisely posed and solved in detail analytically in an approximate manner, results in both the frequency domain and in the time domain are presented. These results include the properties of the transient behaviors (both the starting transient and the transient decay), illustrating the main features of the resonator.

An inverse problem of recovering the heat source coefficient in a fourth-order time-fractional pseudo-parabolic equation

In this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine lsqnonlin tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative

Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed (very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit (ADE) method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $lsqnonlin$ tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.

Solving time-fractional diffusion equations with a singular source term

This article deals with linear time-fractional diffusion equations with time-dependent singular source term. Whether the order of the time-fractional derivative is multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution. As an application, the square integrable space-dependent part and the distributional time-dependent part of the source term of a multi-term time-fractional diffusion equation are simultaneously recovered by partial internal observation of the solution.

An Operator Method for Investigation of the Stability of Time-Dependent Source Identification Telegraph Type Differential Problems

This article is devoted to the study of the stability of time-dependent source identification telegraph type differential problems with dependent coefficients. Time-dependent source identification problems (SIPs) for telegraph differential equations (TDEs) with constant coefficients can be solved by classical integral-transform methods. However, these classical methods can be used, basically, in cases where the differential equation has constant coefficients. We establish the basic theorem of the stability of the time-dependent SIPs for the second-order linear differential equation (DE) in a Hilbert space with a self-adjoint positive definite operator (SAPDO) and damping term. In practice, stability estimates for the solution of the three types of SIPs for one-dimensional and for multidimensional TDEs with dependent coefficients and classic and non-classic conditions are obtained.

An inverse source time-fractional diffusion problem via an input-output mapping

In this paper, we investigate an inverse source problem involving a one-dimensional diffusion equation of a time-fractional RiemannLiouville derivative with 0 < α < 1. First, results on the existence and regularity of the weak solution of the direct problem are obtained. For the determination of the unknown time-dependent source term, we use a monotone and distinguishable input-output mapping defined by the additional over-determination integral data for the considered sub-diffusion problem. Finally, the uniqueness of the solution of the inverse problem is proved.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems

Mikusiński's operational calculus has proven to be a powerful tool for tackling various fractional order differential equations. Notably, it has been extended to encompass the Hilfer-Prabhakar fractional order operator. By employing this operational calculus, we have been able to recover and connect previously derived results pertaining to the Hilfer, Riemann-Liouville, and Caputo operators. Furthermore, we address two inverse problems that involve determining space and time dependent source terms and diffusion concentration. Using our operational calculus results, we obtain series solutions for these inverse problems. It is essential to note that the inverse problems belong to the class of ill-posed problems according to Hadamard's definition. To establish the validity and uniqueness of the solutions to both inverse problems, we rigorously prove the existence and uniqueness results. Additionally, we provide specific examples of inverse problems, thereby illustrating the practical applications and versatility of the operational calculus approach.

Direct and inverse problems for a 2D heat equation with a Dirichlet–Neumann–Wentzell boundary condition

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions.

Direct sampling method to inverse wave-number-dependent source problems: determination of the support of a stationary source

This paper is concerned with a direct sampling method for imaging the support of a frequency-dependent source term embedded in a homogeneous and isotropic medium. The source term is given by the Fourier transform of a time-dependent source whose radiating period in the time domain is known. The time-dependent source is supposed to be stationary in the sense that its compact support does not vary along the time variable. Via a multi-frequency direct sampling method, we show that the smallest strip containing the source support and perpendicular to the observation direction can be recovered from far-field patterns at a fixed observation angle. With multiple but discrete observation directions, the shape of the so-called Θ-convex hull of the source support can be recovered. We even extend the proposed algorithm to the case of near-field data in three dimensions. The frequency-domain analysis performed here can be used to handle inverse time-dependent source problems. Our algorithm has low computational overhead and is robust against noise. Numerical experiments in both two and three dimensions have proved our theoretical findings.

LANDSLIDE TSUNAMI HAZARD ASSESSMENT: A NUMERICAL MODEL FOR THE SIMULATION OF MULTIPLE LANDSLIDE-INDUCED TSUNAMIS SCENARIOS IN A MONTE CARLO FRAMEWORK

Submarine landslides can pose serious tsunami hazard to coastal communities. However, performing a comprehensive landslide tsunami hazard assessment for a given area is in general difficult in view of the large uncertainties associated with tsunamigenic source parameters. These are often known only approximately, on the basis of estimates of the landslide geometry, slide material properties, and kinematics. Here, we present an efficient model to perform such hazard analyses, based on solving the linear mildslope equation with a time-dependent source term representing the seafloor motion as detailed in Iorio et al. (2021). This approach allows carrying out many computations, for a large number of landslide scenarios, in a Monte Carlo (MC) approach framework, at a reduced computational cost compared to other available methods, while still providing physically accurate simulations of most landslide tsunami generation and propagation.

Three-dimensional unsteady state solution of multilayered saturated anisotropic finite media subjected to point source

The three-dimensional (3D) unsteady state solution of a saturated anisotropic finite porous medium due to a point source is presented in this paper. First, Laplace transform is introduced to address the governing equations of anisotropic poroelasticity. Next, the eigen equation method and the modified pseudo-Stroh formalism are used to obtain the transient solution for homogeneous saturated finite media. Then, the propagation matrix method is used to address the corresponding multilayered porous elastic media, and the 3D unsteady state solution of the layered porous medium subjected to a point source is obtained. The solution for a finite porous foundation is presented to demonstrate the use of the closed form solution. The numerical results of a homogeneous foundation due to an internal time-dependent point source are given. Two different point sources are used to show the transient responses of the proposed method. By comparing with those obtained by the finite element method, the accuracy of the proposed method has been proved. Meanwhile, the influence of the depth of the point source on the hydromechanical responses of the homogeneous foundation is discussed. Finally, the transient response of an anisotropic sandwich foundation due to a point source at different depths is investigated. These numerical examples prove the feasibility of the transient 3D solution. The closed form solution presented in this paper can provide help for 3D transient fluid-deforming coupling analysis of laminated porous media.

Probabilistic landslide tsunami modeling of the 2018 Palu Bay event

On September 28, 2018, a Mw 7.5 earthquake triggered near Central Sulawesi generated a highly destructive tsunami within Paul Bay (Indonesia). Field surveys and various studies conducted after the event showed that, as a result of the earthquake, several large submarine landslides were triggered along the shores of the bay, that significantly contributed to tsunami generation. The estimated geometry and other parameters for these slides, however, were affected by a large uncertainty. Here, we present a probabilistic tsunami hazard analysis of this event, based on Monte Carlo simulations using a linear Mild Slope Equation (MSE) model combined with a Green’s function approach, that allow efficiently simulating a large number of stochastic landslide tsunami generation and propagation scenarios within Palu Bay, for each of the identified landslides. In the MSE model, a space and time-dependent source term is used to represent the seafloor motion associated with each landslide scenario. In the Green’s function approach, a large database of elementary solutions and their tsunami elevation at a large number of coastal save points is pre-computed for tsunamis generated by a unit seafloor acceleration specified over a small area. Then, given an actual submarine landslide scenario, with a specific acceleration function, the tsunami elevation at the save points is simply and efficiently computed as a weighed linear superposition of the elementary solutions. Tsunami runup is finally obtained using a semi-empirical method, based on results computed at save points for each landslide scenario. The model is applied to the 2018 Palu Bay tsunami event, allowing to investigate how the uncertainty in landslide parameters affects tsunami hazard. A comparison with observed runups is made, which shows that these fall within the range of uncertainty of the simulated probabilistic runups.

Reconstruction of a time-dependent wave source and the initial condition in a hyperbolic equation

This article is concerned with determination of a time-dependent wave source together with the initial condition in a hyperbolic equation. Based on the given data including the flux tensions of a string at end points and a nonlocal integral observation along with the wave displacement at two distinct instants of time, we prove the unique solvability of the inverse problem. Moreover, we estimate the modulus of continuity of the inverse problem under special conditions. The numerical solution of the problem is also presented by means of Bernstein spectral method, which converts the problem to a linear system of algebraic equations and then Tikhonov regularization technique is employed to derive reliable solutions. Regarding the perturbed boundary conditions, we use a regularized numerical technique to obtain stable derivatives. Numerical simulations are provided to show the effectiveness of the proposed scheme.

Time-dependent source identification problem for a fractional Schrödinger equation with the Riemann–Liouville derivative

UDC 517.9 We consider a Schrödinger equation i ∂ t ρ u ( x , t ) - u x x ( x , t ) = p ( t ) q ( x ) + f ( x , t ) , 0 < t ≤ T , 0 < ρ < 1 , with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u ( x , t ) , a time-dependent factor p ( t ) of the source function is also unknown. To solve this inverse problem, we use an additional condition B [ u ( ⋅ , t ) ] = ψ ( t ) with an arbitrary bounded linear functional B . The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method make it possible to study a similar problem by taking, instead of d 2 / d x 2 , an arbitrary elliptic differential operator A ( x , D ) with compact inverse.

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

Inverse problems for fractional Schrödinger and subdiffusion equations

The inverse problems of determining the right-hand side of the Schrödinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrödinger equation , in a Hilbert space is investigated. To solve this inverse problem, we take the additional condition with an arbitrary bounded linear functional . In the problem 2, we consider the subdiffusion equation with a fractional derivative of order , and take the аbstract operator as the elliptic part. The right-hand side of the equation has the form , where is a given function and the inverse problem of determining element is considered. The condition is taken as the over-determination condition, where is some interior point of the considering domain and is a given element. Obtained results are new even for classical diffusion equations. Existence and uniqueness theorems for the solutions to the problems under consideration are proved.

Very accurate time propagation of coupled Schrödinger equations for femto- and attosecond physics and chemistry, with C++ source code

In this article, I present a very fast and high-precision (up to 33 decimal places) C++ implementation of the semi-global time propagation algorithm for a system of coupled Schrödinger equations with a time-dependent Hamiltonian. It can be used to describe time-dependent processes in molecular systems after excitation by femto- and attosecond laser pulses. It also works with an arbitrary user supplied Hamiltonian and can be used for nonlinear problems. The semi-global algorithm is briefly presented, the C++ implementation is described and five sample simulations are shown. The accompanying C++ source code package is included. The high precision benchmark (long double and float128) shows the estimated calculation costs. The presented method turns out to be faster and more accurate than the global Chebyshev propagator. Program summaryProgram Title: SemiGlobalCppCPC Library link to program files:https://doi.org/10.17632/429rszyc65.1Developer's repository link:http://gitlab.com/cosurgi/SemiGlobalCppLicensing provisions: GNU General Public License 2Programming language: C++Nature of problem: The femto- and attosecond chemistry requires fast and high precision computation tools for quantum dynamics. Conventional software has problems with providing high precision calculation results (up to 33 significant digits), especially when the computation has to be as fast as possible.Solution method: This software fills in the gap by providing the semi-global algorithm [1–3] for arbitrary number of coupled electronic states for the time dependent Hamiltonian and nonlinear inhomogeneous source term. It is implemented in a way that allows computation with precision of 15, 18 or 33 significant digits, where the computation speed can be directly controlled by setting the required error tolerance. The semi-global algorithm [1–3] is implemented in C++ providing a 10× speed boost compared to original publication of semi-global algorithm implemented in Matlab/Octave [1–3].