Time-dependent Source Term Research Articles

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

IDENTIFICATION OF A TIME-DEPENDENT SOURCE TERM IN A NONLOCAL PROBLEM FOR TIME FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data.

Reconstruction of the time-dependent source in thermal grooving by surface diffusion

In hot polycrystalline materials, when a vertical flat grain boundary meets a horizontal surface the grain boundary forms a groove in the surface. Mathematically modelling features of such thermal grooving mechanism is therefore very important in characterizing polycrystalline materials composed of tiny grains intersecting an external free surface. With this aim in mind, we formulate and investigate a novel inverse problem of reconstructing the unknown time-dependent source term entering the fourth-order parabolic equation of thermal grooving by surface diffusion from a given integral observation. We formulate and prove in Theorems 2.3–2.7 that this linear inverse problem is well-posed. However, in practice, the ideal regularity of data under which the inverse source problem is stable is never satisfied due to the inherent non-smoothness of the measurement. Consequently, this leads to the inverse problem with raw data becoming ill-posed. In order to obtain accurate and stable solutions, we develop and compare two numerical methods, namely, a time-discrete method and an optimization method. We obtain error estimates and convergence rates for the time-discrete method. For the optimization method, an objective functional, which is proved to be Fréchet differentiable, is introduced and the conjugate gradient method (CGM), regularized by the discrepancy principle, is developed to compute the minimizer yielding the source term. The results of two numerical tests illustrate the performance of the two methods for both exact and noisy measured data.

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.

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 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.

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.

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.

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].

Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation

Abstract The determination of an unknown time-dependent source term is investigated in a Kuramoto–Sivashinsky equation from given additional integral-type measurement. Based on Schauder’s fixed point theorem, the existence and uniqueness of such inverse problem are obtained under certain assumptions on the input data. In order to calculate the unknown source term, a time-discrete system is established, and its solution shall be applied to approximate the unknown quantity. The existence, uniqueness and some estimates to the time-discrete system are derived, and the convergence rates are deduced rigorously for both exact and noisy observation, respectively. Finally, the theoretical convergence rate results are verified, and accurate and stable solutions to the inverse problem are computed numerically by two numerical experiments.

Molecular Photothermal Effects on Time-Resolved IR Spectroscopy: Solute-Solvent Intermolecular Energy Transfer.

Time-resolved IR pump-probe (IR-PP) and two-dimensional IR (2D-IR) spectroscopy are valuable tools for studying ultrafast chemical and biological processes in solutions. However, the corresponding signals at long times are obscured by the molecular photothermal effects resulting from the heat dissipation of vibrationally photoexcited molecules to the surroundings. Recently, a phenomenology model was used to describe molecular photothermal effects on IR-PP signals and the diagonal and cross-peaks of 2D-IR spectra at long pump-probe delay times. Here, we consider the thermal diffusion equation with a time-dependent heat source term to describe the solute-solvent energy transfer process. An approximate solution to the nonhomogeneous differential equation shows that the molecular photothermal effect is determined by the mean intermolecular distance between IR-absorbing molecules. We show that the time profile of heat dissipation from a vibrationally excited molecule to the surroundings, which provides information about the mechanisms involved in the solute-solvent intermolecular energy transfer process in solutions, can be directly measured by analyzing the molecular photothermal IR-PP and 2D-IR signals. We anticipate that the present work can be used to interpret local heating-induced time-resolved IR spectroscopic signals and understand the rate of and the mechanisms involved in the conversion from high-frequency molecular vibrational energy to solvent kinetic energy in condensed phases.

A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term

In this paper, a novel spatial-temporal radial Trefftz collocation method (STRTCM) is proposed to solve 2D and 3D backward heat conduction equations with time-dependent source term. Unlike the traditional time-domain discretization strategies (Laplace/Fourier transformation and time-stepping methods), the proposed spatial-temporal meshless collocation method employs the derived semi-analytical spatial-temporal radial Trefftz basis function as basis function, which naturally satisfies the governing equation with zero source term in advance. Due to this property, only the final conditions and boundary conditions need to be discretized. For non-zero source term, the extended multiple reciprocity method (E-MRM) is introduced to transform the original governing equation with several specific source terms to the high-order governing equation without the source term. By deriving the high-order semi-analytical spatial-temporal radial Trefftz basis function, the STRTCM can also avoid the node discretization of governing equation. For solving the backward heat conduction equations, the present numerical scheme uses the final temperature distribution and the related boundary conditions to obtain the unknown initial temperature field. Finally, the accuracy and efficiency of the proposed method is numerically verified by four benchmark examples about backward heat conduction equations with time-dependent source term.

Existence and uniqueness results for a multi-parameters nonlocal diffusion equation

This paper is devoted to identifying a time-dependent source term for a multi-term time-fractional diffusion equation with a nonlocal dynamic boundary and integral type over-determination condition. The time-fractional derivatives are considered in Caputo's sense. By applying Fourier's method we obtained multi-term ordinary fractional order differential equation which has been reduced to an algebraic equation by using Laplace transform. Inverse Laplace transform is used to obtain the solution of multi-term ordinary fractional order differential equation which involves multinomial Mittag-Leffler functions. Under some regularity and consistency conditions on the data, unique existence and stability of the regular solution of the inverse problem is proved.

Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation

The main purpose of this paper is to identify simultaneously a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation from two points observed data. First of all, using the fixed point theorem, we prove the existence and uniqueness of the solution for the direct problem. Secondly, the stability of the inverse problem is proved and the uniqueness is a direct result of the stability estimate. In addition, we illustrate the ill-posedness of the inverse problem and use a non-stationary iterative Tikhonov regularization method to recover numerically the time dependent potential coefficient and source term. At the same time, we give the existence of the minimizer for the minimization functional. In order to solve the minimization problem, we apply an alternating minimization method to find the minimizer and prove the solving sub-problems are stable on noisy data as well as prove the data fidelity item decreases monotonously with the iterative running. Finally, some numerical examples are provided to shed light on the validity and robustness of the numerical algorithm.

Physics Informed Neural Networks for Power Transformer Dynamic Thermal Modelling

The emerging methodology of Physics Informed Neural Networks (PINNs) promises to combine available data and physical knowledge to achieve high accuracy and fast evaluation. Dynamic thermal modelling of power transformers is an application specifically set to benefit from these characteristics. Data collected during typical operation is not representative of extreme loading scenarios and the number of thermal sensors is limited. The detailed geometry is often not known by the asset owner which creates high uncertainty for physics-based simulation models. In this study, the transformer is modeled by the one-dimensional heat diffusion equation. PINN is constructed with a loss function including both data-based and physics-based terms. A time-dependent source term from a time series of measurement is also part of the PINN. The result is compared with a finite volume solution demonstrating good agreement. The PINN approach will be useful for further development in thermal modelling for power transformers.

Identification of time-varying source term in time-fractional diffusion equations

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.

A numerical model for the efficient simulation of multiple landslide-induced tsunamis scenarios

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 uncertainty associated with tsunamigenic source parameters, which are often only approximately defined, based on estimates of the landslide geometry, slide material properties, and resulting kinematics. Therefore, a Probabilistic Tsunami Hazard Analysis (PTHA) should be performed by considering a large number of cases, which is computationally demanding. Here, we present an efficient model based on solving the linear Mild-Slope Equation with a time-dependent source term representing the seafloor motion. 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 processes. To further speed-up the MC simulations, a database of elementary solutions is first developed, for many landslide sources of unit amplitude motion over a small seafloor area within the possible landslide footprint. For each unit source, the resulting tsunami elevations are computed and saved at many locations of interest. In the MC simulations, a large number of landslide scenarios are defined by randomly selecting slide parameters within their statistical distributions and each is then simulated for their specific bottom motion using a linear combination of the pre-computed unit sources. Hence, each resulting tsunami is quickly computed at the locations of interest by linear superposition. The paper presents the model validation against two tests cases and describes its novel methodology to perform multiple landslide tsunami scenarios.

Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation

The aim of this paper is to find the time-dependent heat source term numerically in the third-order pseudo-parabolic equation with initial and boundary conditions supplemented by nonlocal integral observation. This is a very interesting and challenging nonlinear inverse source problem with important applications in various fields of mechanics and physics. Unique solvability theorem of this inverse problem is supplied. However, since the problem is still ill-posed (small errors in the integral input data cause large errors in the output force) the solution needs to be regularized. Therefore, to obtain a stable solution, a regularized objective function is minimized to retrieve the unknown heat source coefficient. The third-order pseudo-parabolic problem is discretized using the finite difference method (FDM) and recast as nonlinear least-squares minimization of the Tikhonov regularization function with simple bounds on the unknown force coefficient. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both analytical and perturbed (noisy) data are inverted. Numerical results for three benchmark test examples are presented and discussed. In addition, the von Neumann stability analysis is also discussed.