Laplace Transform Research Articles

Photothermoelastic interaction in a two-dimensional semiconductor with nonlocal stress theory

The present contribution enlightens a viable simulation which describes the photothermoelastic interactions for a two-dimensional thermoelastic transversely isotropic thick semiconducting plate due to the presence of a varying heat source. The upper surface of the semiconductor is traction-free, subjected to prescribed surface temperature and prescribed carrier density flux while the lower surface of the plate rests on a rigid foundation and is thermally insulated. In this new model, the conventional Fourier law has been modified assimilating the memory-dependent derivative within a slipping interval in association with the nonlocal stress theory. On utilizing Laplace and Fourier transformation mechanism, the governing equations for displacement, carrier density and temperature have been obtained in the Laplace–Fourier transform domain. In order to have the solutions in the space-time domain, the inversion of the double transform have been carried out numerically based on the method of Fourier series expansion. From the graphical representations corresponding to the numerical results, the significance of effective parameters such as nonlocality parameter, delay-time parameter is discussed. Noteworthy differences in the results have been demonstrated for different kernel functions. Also, the superiority of a nonlinear kernel function and its effectiveness is also reported in the study.

PHÂN TÍCH TĨNH VỎ TRỤ FG-CNTRC CHỊU TẢI TRỌNG CƠ NHIỆT VỚI TÍNH CHẤT VẬT LIỆU PHỤ THUỘC VÀO NHIỆT ĐỘ

This article presents a thermo-elastic static analysis of the FG-CNTRC cylinderical shell subjected to a thermo-mechanical load. The temperature is assumed to vary through the thickness, and the temperature distribution is derived from the heat transfer formula. The gorvening equations of thermoelastic analysis are established by using the higher-order shear deformation theory, including the transverse normal stress effect. The material properties of FG-CNTRC are temperature-dependent. Using an analytical solution that used the single trigonometric series and the Laplace transformation to solve these equations of shell with the different boundary conditions. The present approach is validated by comparing the results with published results. The effects of different boundary conditions and temperatures on the displacement and stresses of the FG-CNTRC cylindrical shells are investigated. The results show the jumping variation of the stress components at the boundary position. Besides, when the temperature increases, the maximum absolute value of the axial stress increases while that of the circumferential stress decreases.

Origin and Recovery of Negative VTH Shift on 4H–SiC MOS Capacitors: An Analysis Based on Inverse Laplace Transform and Temperature-Dependent Measurements

SiC power MOSFETs are reported to suffer from both positive and negative threshold voltage shifts. Positive shift is well understood in the literature and attributed to electron trapping in near interface oxide traps (NIOTs). Negative shift is explained by hole generation and trapping in the oxide via impact ionization favored by the strong oxide field. However, studies on negative shift are still ongoing, and the origin and nature of the process are not fully understood. This study advances the comprehension of negative threshold voltage shift by investigating MOS capacitors subject to positive bias stress.We demonstrate that: a) a significant negative threshold shift is observed when the electric field is greater than 7.5 MV/cm; b) the detected shift is not recoverable at zero bias. Recovery can be obtained only by applying a positive gate voltage indicating a field-driven detrapping process. c) Trap-state mapping measurements were carried out to extract the activation energy of the detrapping processes, and a weak thermal activation was observed.Results collected within this paper indicate that holes are trapped at deep centers, and thermal detrapping is not possible. Hole release is only possible when a high field is applied to the insulator, possibly due to the recombination between leaking electrons and trapped holes.

Theoretical estimation of the thermal damages of living tissues caused by laser irradiation in tumor thermal therapy

This article aims to provide theoretical predictions for the thermal reactions of human tissues during tumor thermotherapy when exposed to laser irradiation and an external heat source. For the construction of a theoretical study of bioheat transfer, the selection of a suitable thermal model capable of accurately predicting the required thermal responses is essential. The effect of heat production by heat treatment on a spherical multilayer tumor tissue is evaluated using this approach. Analytical solution for the non-homogenous differential equations is derived in the Laplace domain. The study examines the impact of thermal relaxation time on tissue temperature and the subsequent thermal damage. The numerical findings of thermal damage and temperatures are depicted in a graphical representation. This model explains laser treatment, physical events, metabolic support, and blood perfusion. The numerical outcomes of the recommended model are validated by comparing them to the literatures.

A semi-analytical solution for transient confined-unconfined flow toward the partially penetrating well

In this paper, a semi-analytical solution has been proposed for the transient confined-unconfined flow toward a partially penetrating well in a confined aquifer. It can be used to characterize the dynamic development of drawdown and the transient unconfined region during the transient confined-unconfined conversion. The proposed solution takes into account the delayed response recharge from the unsaturated region by the Neuman's upper surface recharge boundary condition, which is derived by the Laplace and Fourier transformations. The reliability of the semi-analytical solution has been proved by comparisons with numerical solutions from Visual MODFLOW Flex. In addition, a new model for the pumping flow in a pure unconfined aquifer is also developed by degradation analysis of the proposed solution, which has been used to fit well with the measured data from the pumping test in Cape Cod aquifer. The results indicate that both the use of partially penetrating pumping well and aquifer anisotropy have negative effects on the drawdown delayed responses and the development of transient unconfined region, which are disappeared at times >105Srb/Kr. The delayed response is positively related to the gravity storage and vertical hydraulic conductivity.

Two Different Analytical Approaches for Solving the Pantograph Delay Equation with Variable Coefficient of Exponential Order

The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients.

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

Efficient techniques for nonlinear dynamics: a study of fractional generalized quintic Ginzburg-Landau equation

In this research, we explore the solution of the fractional Generalized Quintic Complex Ginzburg-Landau equation (GQCGLE) using the Controlled Picard Transform Method. We solve the equation by incorporating the Laplace transform (LT) and the Caputo-Fabrizio fractional derivative (CF). The fractional GQCGLE is a complex-valued equation that exhibits intricate dynamics and plays a significant role in various physical phenomena, particularly in the realm of nonlinear optics. The suggested method can efficiently address a wide variety of differential equations, encompassing both integer and fractional-order equations. This is achieved by integrating an extra parameter that enhances convergence, making it especially suitable for non-linear differential equations. To confirm the precision and convergence of our approach, we validate it by comparing it with other methods, presenting graphical representations to clarify the effects of different parameters on the solution behaviour, and confirming its precision and convergence. The existence and uniqueness of the solutions are also examined.

Generalized Analytical Well-Test Solutions for Vertically Fractured Wells in Commingled Reservoirs

Abstract Layered zones with vertical fracture are often found in analyzing well-test data of deep/ultra-deep reservoirs and tight reservoirs. Analyzing and modeling the well-test data in a computation-accurate and easy-program manner have been a challenge for these problems due to the lack of suitable solutions. This work thus presents the generalized analytical well-test solutions for vertically fractured wells in infinite and bounded commingled reservoirs with computation accuracy and functional simplicity. These solutions are derived based on the early-time approximate solution of the infinite/finite-conductivity fracture model, Laplace and Fourier cosine transformation, pressure superposition principle, and Duhamel principle. Subsequently, model validation is carried out by comparing the pressure and derivative results with those of commercial saphir software. The results show that the average absolute percent deviation between the presented analytical solutions and saphir for three kinds of outer boundaries is ∼2% for pressure results and ∼4% for pressure derivative results. Finally, a field case in Xinjiang oilfield is interpreted, indicating that the proposed analytical well-test solutions are feasible to interpret the parameters of commingled reservoirs.

Thermomagnetic responses of a thermoelastic medium containing a spherical hole exposed to a timed laser pulse heat source

The current work aims to investigate the interaction between thermal, magnetic, and elastic phenomena in a homogeneous and unbounded material that includes a spherical cavity. A laser beam with a Gaussian distribution was used to expose the solid thermal medium, thereby bringing a source of heat into the system. The study is grounded in the expanded theory of thermoelasticity without energy dissipation, which considers the interconnection between thermal, elastic, and magnetic phenomena. This theoretical framework serves as a basis for understanding the interactions and behaviors of thermal and mechanical processes within an elastic material. The system was analyzed using a direct approach, employing the Laplace transformation method to provide solutions for the system's equations. This methodology enables the computation and presentation of precise equations for several domains of concern, such as displacement, temperature, and thermal stresses. The discussion and analysis focused on a copper-based material, and graphical representations were used to demonstrate the behavior of the researched physical fields. These graphical representations aid in comprehending the development of displacement, temperature, and thermal stresses within the material caused by the laser beam and the interplay between thermal, magnetic, and elastic processes. The results demonstrate that the laser pulse parameters have a significant impact on the temperature and thermal stress distribution, thereby influencing the thermo-mechanical reactions of the solid medium. Ultimately, this study enhances our comprehension of the intricate behavior and interplay of thermal, magnetic, and elastic phenomena in thermoelastic materials.

Evaluation of the thermo-acoustic instability frequency and growth rate via input reflection coefficient measurement for central heating equipment

Experimental measurement methods and theoretical evaluations based on low-order modeling approaches for both the growth rate and frequency at the onset of thermo-acoustic combustion instability are proposed, and their performance is evaluated. The developed techniques are demonstrated through a systematic measurement of the linear growth rate and frequency of evolving oscillations in the laboratory setup and also applied for the thermo-acoustic qualification of an industrial domestic boiler and a heat cell unit (combination of a burner with a heat-exchanger). Generic measurements have been done for a burner deck with premixed surface-stabilized Bunsen-type flames. The industrial domestic boiler and the heat cell unit are equipped with burners of a similar type but differ by their perforation pattern. They have been tested at different conditions and the experimental and theoretical results are compared. Two modeling strategies are tested: 1- in Laplace domain, with estimating a rational function in the complex domain to fit the measured frequency response, 2- exclusively in frequency domain, without estimating a rational function. Both methods include measurement of the frequency response of two reflection coefficients from i) the upstream part of the system, Rup, and ii) burner with flame completed by the downstream part of the appliance, Rin. Within the first approach, a procedure for an analytic continuation of the measured frequency response to the complex domain is applied and complex eigenfrequencies are calculated by solving the corresponding dispersion equation. An alternative approach was proposed by Kopitz and Polifke and allows estimating both the frequency of oscillation and the growth rate from the analysis of the polar plot of the system's characteristic equation in the frequency domain. The comparison shows that the unstable frequencies can be predicted accurately by both tested modeling strategies. This conclusion holds also for the tested industrial applications. The prediction of the instability growth rates is closer to the measured one when the modeling method in the complex domain is used. However, the frequency domain analysis provides less accurate, but still reasonable estimates of the growth rates and frequencies. Moreover, a good overview of thermo-acoustic performance of each industrial boiler/burner at different conditions is obtained via Rin measurements.

Application of the inverse Laplace transform techniques to solve the generalized Bagley–Torvik equation including Caputo’s fractional derivative

This article employs the Laplace transform approach to solve the Bagley–Torvik equation including Caputo’s fractional derivative. Laplace transform is a powerful method for enabling solving integer and non-integer order ODEs and PDEs in engineering and science. Using the Laplace transform for solving integer and non-integer order ODEs and PDEs, however, sometimes leads to solutions in the Laplace domain that are difficult to convert back into the time domain using analytical techniques. Therefore, the best option to convert the Laplace domain solution back into real domain is to use numerical inversion techniques. But it is important to remember that the inverse Laplace transform is not a well-posed problem. There is not a single, general approach that can address every challenge in this matter. Numerous algorithms are devised for numerically inverting the Laplace transform. In this paper, we have evaluated three numerical inverse Laplace transform techniques. These methods include the Fourier series method, the Gauss–Hermite quadrature method, and Zakian’s method. Different test problems are used to evaluate these inversion methods. The effectiveness and efficiency of these methods are demonstrated through graphical representations and tabular data. The comparison of the obtained results with the results of other methods available in literature led us to conclude that the proposed methods exhibit rapid convergence with optimal accuracy without any time instability.

The influences of thermal relaxation time and varying thermal conductivity in thermoelastic media

This study introduces a nonlinear analysis of the transient responses in a thermoelastic medium produced by a thermal shock. The formulations are presented based on a Green-Lindsay (GL) thermoelasticity model with two thermal relaxation times under varying thermal conductivity are included in the governing equations. The finite element process is employed to address the problem without resorting to Kirchhoff transforms, as non-linear formulations pose significant challenges. In the linear case where Kirchhoff's transforms are implemented, the solution to the problem is obtained through Laplace transformations and the eigenvalues technique. The impacts of changing thermal conductivity are examined and contrasted both with and without the use of Kirchhoff transforms. Graphical illustrations of the numerical outcomes display the variations of displacement, temperature, and stress. Compared to the classic thermoelastic (CT) model, the (GL) model excels in accurately depicting the observed waves propagations phenomenon under varying thermal conductivity.

First-exit-time problems for two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses

We study the first-exit-time problem for the two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses which run according to specific rules depending on the processes. We obtain the Laplace Transform of the first-exit-time probability density function and the corresponding moments. For both processes, some computational results on the first-exit-time densities are provided by means of the numerical inversion of the relevant Laplace Transforms. Moreover, we also investigate the asymptotic behaviour of the first-exit-time moments when the ellipse grows. In particular, an asymptotic exponential trend holds for the first-exit-time density of the mean-reverting Ornstein–Uhlenbeck process.

The generalized telegraph equation with moving harmonic source: Solvability using the integral decomposition technique and wave aspects

The paper is devoted to study the frequency shift in the solution of the generalized telegraph equation with a moving point-wise harmonic source. This equation contains the nonlocality in time derivatives which is expressed by the memory functions η(t) and γ(t), where η(t) smears the second time-derivative and γ(t) the first one. Moreover, in the Laplace domain we have ηˆ(s)=γˆ2(s). The generalized telegraph equation with an external source is solved by using the integral decomposition which allows us to write this solution as a product of the solution of the telegrapher equation with harmonic source and fγˆ(ξ,t) which is a function of the Laplace transform of memory function γ. Such obtained solution manifests the frequency shift which is illustrated in three examples of the memory functions γ(t): the localized case, its mixture with power-law, and the power-law case only. We show that only the first two cases have the wave front and the Doppler-like shift. The third example, despite the lack of wave fronts, also manifests the frequency shift. Thus it turns out that the frequency shift occurs regardless of the existence of a wave front, but it is more visible when such a front exists.

Simulation of Readout Electronics for Nuclear Radiation Detector Using MATLAB Program

A computer program was developed using the MATLAB programming language to simulate the electronics readout for a radiation detector system. The function of each stage of this system is described by a mathematical model in the Laplace domain. The electrical signals have been shown and analyzed at two main outputs of the system. They are described according to their related circuit parameters. The obtained results of the simulation can be used to achieve a best design of the concerned circuits and to provide appropriate details about the system operation. Validation of the simulated signals by comparison with available experimental results has been achieved.

Analytical and experimental analyses on cooling performances of radiative SkyCool radiators with various interior flowing channels

This study presents an analytical model for the radiative SkyCool radiators (RSCR) with various interior flowing channels by using Laplace transformations. Moreover, the convolution theory is applied to solve the problem of nonuniform temperature distribution on the RSCR. Then, an outdoor experiment is carried out to verify the proposed model. Subsequently, the influence of different channel geometries, flow rates, tilt angles and wind velocities on the cooling performances of RSCRs are investigated. The results indicate that increasing the RSCR's length, enlarging the pipe spacing, and decreasing the flow rate and thickness of RSCR can definitely intensify the cooling performances. The application suggestions for S-channel RSCR (SRSCR) and I-channel RSCR (IRSCR) are given. When the pipe spacing ratio is smaller than five, the IRSCR is recommended because of lower thermal interferences. By contrast, SRSCR is recommended once the dimensionless pipe spacing is greater than ten. For cases with smaller tilt angles but larger wind velocities, IRSCR is suggested, while SRSCR is prioritized for cases with opposite conditions. This research provides an effective analytical tool in the evaluation of cooling power generation by producing cold water, contributing valuable insights for the optimization of cooling efficiencies of RSCRs under varied configuration scenarios.

Special function form solutions of multi-parameter generalized Mittag-Leffler kernel based bio-heat fractional order model subject to thermal memory shocks.

The primary objective of this research is to develop a mathematical model, analyze the dynamic occurrence of thermal shock and exploration of how thermal memory with moving line impact of heat transfer within biological tissues. An extended version of the Pennes equation as its foundational framework, a new fractional modelling approach called the Prabhakar fractional operator to investigate and a novel time-fractional interpretation of Fourier's law that incorporates its historical behaviour. This fractional operator has multi parameter generalized Mittag-Leffler kernel. The fractional formulation of heat flow, achieved through a generalized fractional operator with a non-singular type kernel, enables the representation of the finite propagation speed of heat waves. Furthermore, the dynamics of thermal source continually generates a linear thermal shock at predefined locations within the tissue. Introduced the appropriate set of variables to transform the governing equations into dimensionless form. Laplace transform (LT) is operated on the fractional system of equations and results are presented in series form and also expressed the solution in the form of special functions. The article derives analytical solutions for the heat transfer phenomena of both the generalized model, in the Laplace domain, and the ordinary model in the real domain, employing Laplace inverse transformation. The pertinent parameter's influence, such as α, β, γ, a0, b0, to gain insights into the impact of the thermal memory parameter on heat transfer, is brought under consideration to reveal the interesting results with graphical representations of the findings.

End-to-end path planning for homogeneous temperature fields in additive manufacturing

This study explores a novel approach to path planning in deposition-based additive manufacturing, integrating the frequently overlooked process-induced temperature fields. Currently, existing approaches either ignore temperature effects entirely or only consider them in small-scale problems due to the high computational cost involved in predicting them and the combinatorial nature of path planning optimization. To address these challenges, the present work proposes an optimization pipeline that involves deriving a reduced order model from a finite volume method model with balanced truncation, using an analytical function to model the heat input and, calculating the steady-state response of the system to an arbitrary path using the Laplace transformation. Then, the optimization is transformed into a sequential decision-making problem and approximated with Monte Carlo tree search. The pipeline is validated through computational and experimental results, demonstrating its efficiency in managing large and complex geometries, as well as its resilience in overcoming the challenges posed by the simulation to reality gap.

Accelerating Analytic-Continuation GW Calculations with a Laplace Transform and Natural Auxiliary Functions.

We present a simple and accurate GW implementation based on a combination of a Laplace transform (LT) and other acceleration techniques used in post-self-consistent field quantum chemistry, namely, natural auxiliary functions and the frozen-core approximation. The LT-GW approach combines three major benefits: (a) a small prefactor for computational scaling, (b) easy integration into existing molecular GW implementations, and (c) significant performance improvements for a wide range of possible applications. Illustrating these advantages for systems consisting of up to 352 atoms and 7412 basis functions, we further demonstrate the benefits of this approach combined with an efficient implementation of the Bethe-Salpeter equation.