Article
Mathematics, Applied
Meiramkul Amangaliyeva, Muvasharkhan Jenaliyev, Sagyndyk Iskakov, Murat Ramazanov
Summary: This paper examines the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain, and investigates the general solution representation of a nonhomogeneous integral equation. Estimates for the resolvent and the solution of the boundary value problem are also established in the study.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics
Muslim Malik, Santosh Ruhil, Rajesh Dhayal
Summary: This manuscript investigates an inverse problem for a second-order abstract differential equation in a Banach space, where the parameter is identified using an over-determined condition on a mild solution. The main techniques to find the solution pair of the considered problem are a direct approach using Volterra integral equation for sufficiently regular data and an optimal control approach for less regular data. An application to an inverse problem in elasticity theory is also presented.
INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Darko Volkov, Yulong Jiang
Summary: This paper presents a Lipschitz stability result for a crack inverse problem in half space, focusing on the unknown geometry and location of the crack. By assuming certain conditions on the geometry, it is shown that the inverse problem is uniquely solvable, highlighting the stability of reconstructing the crack plane despite the unknown forcing term.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Aleksandr N. Tynda, Denis N. Sidorov
Summary: This paper presents a new inverse problem statement and numerical method for Volterra integral equations with piecewise continuous kernels. The method has an arithmetic complexity of O(N-3) and first-order convergence, and can be applied to an arbitrary number of discontinuity curves.
Article
Mathematics
Suzan Cival Buranay, Mehmet Ali Ozarslan, Sara Safarzadeh Falahhesar
Summary: This paper aims to numerically solve first kind linear Fredholm and Volterra integral equations using Modified Bernstein-Kantorovich operators. Through discretization, the equations are transformed into systems of algebraic linear equations. By introducing regularization features, stability of the solutions and accuracy of results are improved, especially when high order approximations are used with the operators.
Article
Mathematics
Mohammad Akram, Mohammad Dilshad, Aysha Khan, Sumit Chandok, Izhar Ahmad
Summary: A new generalized Yosida inclusion problem is introduced, which involves an A-relaxed co-accretive mapping. The resolvent and associated generalized Yosida approximation operator are defined, and their characteristics are discussed. The existence result is quantified in q-uniformly smooth Banach spaces. A four-step iterative scheme is proposed and its convergence analysis is discussed. The theoretical assertions are illustrated with a numerical example. Furthermore, an equivalent generalized resolvent equation problem is established, and a Volterra-Fredholm integral equation is examined using the proposed method.
Article
Mathematics, Applied
Alexandr Vatulyan, Pavel Uglich, Vladimir Dudarev, Roman Mnukhin
Summary: This paper focuses on the longitudinal and flexural vibrations of an inhomogeneous rod, considering the variable Young's modulus and density along the longitudinal coordinate. The vibrations are caused by a load applied at the right end. The proposed method enables the consideration of a wider range of inhomogeneity laws compared to other numerical solutions. Sensitivity analysis is performed and a new inverse problem related to the simultaneous identification of the variation laws of Young's modulus and density is addressed. The solution involves an iterative process and the analysis of Fredholm integral equations.
Article
Mathematics, Applied
M. Tadi
Summary: This article presents a computational method for the inverse problem of the Helmholtz equation, aiming to recover subsurface material properties based on boundary data. The major improvement of this method is that it eliminates the need for linearizing the working equations, making it simple and efficient. Using just one set of data is sufficient to obtain a good approximation of the unknown material property, while additional data sets can further improve the quality of the recovered function.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
M. J. Huntul, Muhammad Abbas, Dumitru Baleanu
Summary: This paper investigates the inverse problem of reconstructing the time-dependent potential and displacement distribution in the hyperbolic problem with periodic boundary conditions and nonlocal initial conditions, supplemented by over-determination measurement. The problem, though unstable to noise in the input data, has a unique solution. The Crank-Nicolson-finite difference method along with Tikhonov regularization is used to calculate an accurate and stable numerical solution, with results showing accuracy and stability.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Mathematics, Applied
Antonio Luciano Martire
Summary: In this study, we consider a linear Volterra integral equation and propose a feasible, rapid, and accurate numerical algorithm by exploiting the Lipschitz continuity of the unique unknown solution. The application of this algorithm in risk theory is demonstrated using a Cramér-Lundberg model framework, where we prove the ruin probability to be a Lipschitz function. By employing the proposed algorithm, we evaluate the ruin probability that satisfies the associated Volterra integral equation and demonstrate the reasonable generalizability of the framework by considering a wide range of claim size distributions.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics
Tao Liu, Di Ouyang, Lianjun Guo, Ruofeng Qiu, Yunfei Qi, Wu Xie, Qiang Ma, Chao Liu
Summary: This paper proposes a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in multiphase porous media flow. The combination of the multigrid method with constraint data is utilized and investigated. The results demonstrate the effectiveness of this combination strategy in reducing noise, avoiding local minima, and accelerating convergence.
Article
Mathematics, Applied
M. Moumen Bekkouche, I. Mansouri, A. A. Azeb Ahmed
Summary: In this article, we investigate the existence and uniqueness of the solution of a fractional boundary value problem with a specific type of conformable fractional derivation. By using a new definition of fractional integral, we transform the problem into an equivalent linear Volterra-Fredholm integral equation, and based on the results obtained, we prove the sufficient conditions for the existence and uniqueness of the solution. Additionally, a comprehensive numerical study is conducted.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics, Applied
Y. Estaremi, S. Shamsigamchi
Summary: In this paper, we investigate the Moore-Penrose inverse and characteristic matrix of unbounded WCT operators on the Hilbert space L-2 (mu). We also provide some applications of the Moore-Penrose inverse of unbounded operators on the Hilbert space H to variational regularization problem. Additionally, examples are used to illustrate the applications of these results in linear equations and specially Fredholm integral equations.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Dinh Nguyen Duy Hai
Summary: This paper discusses an ill-posed inverse problem related to practical applications. With two regularization strategies, the paper obtains error estimates for the exact solution and provides numerical examples.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Acoustics
Chapel Rice, Jay Frankel
Summary: This article proposes a novel calibration-based integral formulation for resolving the forcing function in a mass-spring-damper system, applicable to various mechanical systems, and achieved through mathematical modeling and frequency domain analysis.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Engineering, Mechanical
Felicitas Schaefer, Luca Magri, Wolfgang Polifke
Summary: A method is proposed to compute the continuous adjoint of a thermoacoustic network model using the discretized direct equations. It exploits the self-adjoint character of the duct element and derives all jump conditions from the direct scattering matrix, eliminating the need to derive adjoint equations for each network element. This method combines the advantages of discrete and continuous adjoints, achieving the accuracy of the continuous adjoint while maintaining the flexibility of the discrete adjoint. It demonstrates how the obtained adjoint system can be used to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.
JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME
(2022)
Article
Engineering, Mechanical
Naman Purwar, Maximilian Meindl, Wolfgang Polifke
Summary: This article introduces the application of model order reduction (MOR) in large thermo-acoustic models and compares several reduction techniques. The study found that reduction techniques based on preserving transfer behavior are more suitable for thermo-acoustic stability analysis.
JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME
(2022)
Article
Operations Research & Management Science
Barbara Kaltenbacher, Kha Van Huynh
Summary: In this paper, we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Barbara Kaltenbacher, Anna Schlintl
Summary: This paper investigates the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface, relevant in photoacoustic or thermoacoustic tomography. The authors derive and analyze a time stepping method for the numerical solution of the corresponding forward problem, and develop an adjoint-based scheme for gradient computation to efficiently obtain reconstructions by minimizing a Tikhonov regularization functional. Numerical reconstructions in two space dimensions demonstrate the performance of the devised methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Barbara Kaltenbacher, William Rundell
Summary: This study tackles an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high-intensity ultrasound propagation in medical imaging and therapy. The research assumes a more complex physical model and aims to recover an unknown function f from data measurements. The study demonstrates the injectivity of the linearized forward map and introduces several iterative schemes for the recovery of f.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Thermodynamics
Chuhan Wang, Thomas L. Kaiser, Max Meindl, Kilian Oberleithner, Wolfgang Polifke, Lutz Lesshafft
Summary: The response of a 2D laminar premixed slot flame to external forcing is investigated using linear analysis. The study reveals that the mechanisms triggering flame oscillations may involve resonance with intrinsic thermoacoustic (ITA) instability modes.
COMBUSTION AND FLAME
(2022)
Article
Mechanics
Barbara Kaltenbacher
Summary: This paper aims to place the problem of vibroacoustic imaging into the mathematical framework of inverse problems and regularization, specifically regarding coefficient identification in PDEs. A model in frequency domain is presented, the uniqueness of recovering the spatially varying nonlinearity parameter from multiple frequency acoustic pressure measurements is proven, and Newton and gradient based reconstruction methods are derived.
Article
Thermodynamics
Sagar Kulkarni, Camilo F. Silva, Wolfgang Polifke
Summary: This theoretical investigation examines the impact of gas velocity oscillations on droplet number density and evaporation rate. It proposes a mathematical model and analytical formulation to describe the processes involved. The study finds that gas velocity oscillations lead to variations in droplet concentration and modulation of evaporation rate, which in turn affect the equivalence ratio.
INTERNATIONAL JOURNAL OF SPRAY AND COMBUSTION DYNAMICS
(2022)
Article
Thermodynamics
Marcin Rywik, Praveen Kasthuri, Isaac Boxx, Ianko Chterev, Wolfgang Polifke, R. I. Sujith
Summary: This study uses complex network theory to analyze the spatiotemporal dynamics of the PRECCINSTA swirl burner operating on hydrogen-methane fuel blends. Period-1 and period-2 limit cycle oscillations as well as chaotic oscillations were observed. A turbulence network and a heat release rate correlation network were constructed, showing significant differences in their properties.
PROCEEDINGS OF THE COMBUSTION INSTITUTE
(2023)
Article
Thermodynamics
Kah J. Yong, Camilo F. Silva, Guillaume J. J. Fournier, Wolfgang Polifke
Summary: A recent study proposed the use of phasor diagrams to categorize marginally stable modes in an ideal resonator with a compact, velocity-sensitive flame. The method does not rely on any parametric sweep, but on the angle relating the velocity phasors across the flame.
COMBUSTION AND FLAME
(2023)
Article
Thermodynamics
Alex M. Garcia, Sophie Le Bras, Wolfgang Polifke
Summary: This study numerically analyzes the impact of hydrogen addition on the consumption speed of premixed lean methane-air laminar flames under combined strain and heat loss. Different equivalence ratios and fuel compositions are considered. The results indicate that the definition of consumption speed based on heat release rate yields different flame responses compared to the definition based on fuel consumption rate. Strain rate increases the flame speed at first and then decreases it for lean methane-hydrogen mixtures. Heat loss decreases the stretched flame speed and leads to earlier flame extinction. Hydrogen addition and equivalence ratio significantly affect the maximum consumption speed and flame response to strain rate and heat loss. The effect of hydrogen on thermo-diffusive properties of the mixture is also analyzed and related to the consumption speed.
COMBUSTION THEORY AND MODELLING
(2023)
Article
Acoustics
Naman Purwar, Wolfgang Polifke
Summary: Thermoacoustic systems can be modeled using a hybrid approach that combines separate models for acoustic propagation and flame dynamics. Model Order Reduction can be applied to the acoustic subdomains to reduce computational cost. In this study, a frequency-weighted pseudo-optimal rational Krylov algorithm is used along with frequency-domain System Identification and cumulative reduction framework to perform Model Order Reduction. The reduced-order subdomain is then coupled with other acoustic subdomains and a flame model to form a reduced-order thermoacoustic system. Results demonstrate accurate reproduction of thermoacoustic modes by the reduced-order model.
JOURNAL OF SOUND AND VIBRATION
(2023)
Article
Thermodynamics
Alexander J. Eder, Camilo F. Silva, Matthias Haeringer, Johannes Kuhlmann, Wolfgang Polifke
Summary: This study compares the advantages and disadvantages of compressible and incompressible computational fluid dynamics (CFD) formulations for estimating acoustic flame response. By applying system identification (SI) to time series data extracted from large eddy simulation (LES), the flame transfer function of a swirl-stabilized burner is determined. The results show that incompressible simulations have several advantages over compressible simulations in terms of desired statistical properties enhancement, reduced computational costs, and simpler implementation.
INTERNATIONAL JOURNAL OF SPRAY AND COMBUSTION DYNAMICS
(2023)
Article
Thermodynamics
Christopher M. Douglas, Wolfgang Polifke, Lutz Lesshafft
Summary: This paper presents a global nonlinear bifurcation analysis of burner-stabilized laminar premixed conical flames, exploring the dynamics and steady structures of the flame. The analysis reveals saddle-node bifurcations corresponding to spontaneous flash-back and blow-off of the axisymmetric flame, as well as axisymmetry breaking bifurcations associated with transitions to steady three-dimensional polyhedral and tilted flame states.
COMBUSTION AND FLAME
(2023)
Article
Physics, Fluids & Plasmas
Thomas Steinbacher, Wolfgang Polifke
Summary: Convective velocity perturbations (CVPs) play an important role in flame response to acoustic perturbations and thermoacoustic combustion instabilities. This study uses a reduced order flow decomposition approach to model the response of laminar premixed slit flames to low amplitude perturbations of the upstream flow velocity. It analyzes the respective contributions of irrotational and solenoidal flows to the flame response and the effect of flame perturbations on the flow. The results show that convected velocity perturbations are generated by flame-flow feedback, rather than immediate acoustic-to-hydrodynamic mode conversion.
