Laplace Transform Research Articles

Adaptive consensus control of uncertain nonlinear multi-agent systems with communication delays and relative output information

This paper explores the challenges of distributed adaptive consensus control problems in nonlinear multi-agent systems with unknown parameters and communication delays. To address these issues, a novel backstepping-based adaptive state feedback control protocol is proposed by utilizing a reference dynamic system. Furthermore, an output-feedback consensus control algorithm incorporating a state observer is introduced. Through the application of Laplace transformation, matrix properties, and graph theory, it is demonstrated that the closed-loop multi-agent system ensures the boundedness of all signals and achieves output consensus, even in the presence of nonuniform communication delays. Numerical simulations on a group of inverted pendulums illustrate the effectiveness of these algorithms.

Dam control system based on DDPG algorithm

Abstract Dams, regarded as a form of hydraulic engineering, are purposed to intercept river or canal flow to either elevate water levels or regulate discharge rates. The discharge of a dam is subject to influence by various factors, including precipitation, temperature, and regulatory policies. Nevertheless, in real-world scenarios, the precise value of the expected dam flow rate is challenging to ascertain through simple arithmetic operations. In this study, a novel dam flow model has been formulated. Through the utilization of the DDPG algorithm and deep learning techniques applied to data from the current month, the parameters of the dam control differential equations have been determined, and the transfer function after Laplace transformation has been established. Consequently, only the current month’s fluctuations in the water level of Lake Huron need to be determined to derive the corresponding dam flow rate for that month.

Laplace Transform Based Modeling of Drug Delivery with Reversible Drug Binding in a Multilayer Tissue.

Drug delivery from a drug-loaded device into an adjacent tissue is a complicated process involving drug transport through diffusion and advection, coupled with drug binding kinetics responsible for drug uptake in the tissue. This work presents a theoretical model to predict drug delivery from a device into a multilayer tissue, assuming linear reversible drug binding in the tissue layers. The governing mass conservation equations based on diffusion, advection and drug binding in a multilayer cylindrical geometry are written, and solved using Laplace transformation. The model is used to understand the impact of various non-dimensional parameters on the amounts of bound and unbound drug concentrations as functions of time. Good agreement for special cases considered in past work is demonstrated. The effect of forward and reverse binding reaction rates on the multilayer drug binding process is studied in detail. The effect of the nature of the external boundary condition on drug binding and drug loss is also studied. For typical parameter values, results indicate that only a small fraction of drug delivered binds in the tissue. Additionally, the amount of bound drug rises rapidly with time due to early dominance of the forward reaction, reaches a maxima and then decays due to the reverse reaction. The general model presented here can account for other possible effects such as drug absorption within the device. Besides generalizing past work on drug delivery modeling, this work also offers analytical tools to understand and optimize practical drug delivery devices.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load

Considering the importance of understanding the propagation of transient waves in the piezomagnetic solids, the thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load is investigated analytically in this paper. Using one-sided, two-sided Laplace transformation and Cagniard–de Hoop (CH) technique, an efficient and accurate analytical derivation for the solution of the anti-plane displacement, shear stress, magnetic potential, and induction in Laplace domain is presented. The study shows that the thermal stresses developed in x-axis and y-axis directions have significant influence on the transient response of the half-space. The magnetic induction B x increases obviously when the thermal stress is applied in x-axis direction, while it decreases when the thermal stress is applied in y-axis direction. Approaching time of magnetic induction B x and B y will become longer with higher thermal stress in x-axis direction. With the growth of the thermal stress in x-direction, contribution from the electromagnetic–elastic head (EH) wave increases, while the contribution from the shear elastic (SE) wave and the static value of shear stress decrease.

Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function FL(x,Q2), at the leading-order approximation in momentum space, from the structure function F2(x,Q2) and its derivative with respect to lnQ2 in a kinematical region of low values of the Bjorken variable x. Since the x dependence of F2(x,Q2) and its evolution with Q2 are determined much better by the data than FL(x,Q2), this method provides both a direct check on FL(x,Q2) where measured, and a way of extending FL(x,Q2) into regions of x and Q2 where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function F2(x,Q2) [M. M. Block , .]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function FL(x,Q2) and that our obtained results are in line with data from the H1 Collaboration and other results for FL(x,Q2) obtained using Mellin transform method. Published by the American Physical Society 2024

Analysis of a hollow cylinder with variable thermal conductivity and diffusivity by fractional thermoelastic diffusion theory

Based on fractional thermoelastic diffusion theory, an infinite hollow cylinder with variable thermal conductivity and diffusivity is studied. The inner surface of the hollow cylinder is free and subjected to thermal and chemical shocks, while the outer surface is fixed and adiabatic. The closed forms of dimensionless displacement, temperature, chemical potential, stress, and concentration in the Laplace transform domain are given by eigenvalue method. These physical quantities are then numerically obtained through the inverse Laplace transformation. The effects of fractional order parameters, variable thermal conductivity and diffusivity on these quantities are discussed. The present model may be helpful for better understanding the thermoelastic diffusion problems in actual engineering.

Dynamic Thermal Response of Multiple Interface Cracks between a Half-Plane and a Coating Layer under General Transient Temperature Loading.

This paper explores the thermal behavior of multiple interface cracks situated between a half-plane and a thermal coating layer when subjected to transient thermal loading. The temperature distribution is analyzed using the hyperbolic heat conduction theory. In this model, cracks are represented as arrays of thermal dislocations, with densities calculated via Fourier and Laplace transformations. The methodology involves determining the temperature gradient within the uncracked region, and these calculations contribute to formulating a singular integral equation specific to the crack problem. This equation is subsequently utilized to ascertain the dislocation densities at the crack surface, which facilitates the estimation of temperature gradient intensity factors for the interface cracks experiencing transient thermal loading. This paper further explores how the relaxation time, loading parameters, and crack dimensions impact the temperature gradient intensity factors. The results can be used in fracture analysis of structures operating at high temperatures and can also assist in the selection and design of coating materials for specific applications, to minimize the damage caused by temperature loading.

Discharge variations of qanat near an ephemeral stream

Ephemeral streams are the most important sources of groundwater recharge in arid and semi-arid regions of the world. To extract this replenishable water, most of the time, qanats are constructed parallel to the ephemeral streams. Analytical modeling of a stream-qanat system can provide an effective tool for managing such a system in a timely fashion and, thus, is commonly applied in the hydrological communities. However, the available analytical solutions of discharge variation of qanat only consider the distributed recharge from rainfall and cannot be used to investigate the effects of geometric parameters of stream and hydraulic parameters of aquifer on the discharge of qanat near an ephemeral stream. Thus, the goal of this work is to obtain a semi-analytical solution of qanat discharge due to recharge from an ephemeral stream of an arbitrary shape in a three-dimensional anisotropic homogeneous unconfined aquifer. The solution is obtained via Laplace and Fourier transformations along with the principle of superposition. The results of this study are presented in the form of qanat discharge-time curves. The influences of the qanat length, shape, and recharge distribution along the ephemeral stream are explored in great detail. The effects of the hydraulic parameters of the aquifer on the discharge of qanat are also investigated. The results of this study can be used to predict the discharge of qanat due to recharge from an ephemeral stream, to calculate the optimum length of a qanat for maximum extraction of the replenished water, to estimate the adequate recharge from the ephemeral stream and the hydraulic parameters of aquifer using the discharge data of qanat, among other applications.

Annular unidirectional slip Couette flow with heat transfer and variable viscosity by Laplace transform and homotopy perturbation method

This article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux.

A new analytical model for transient heat transfer modeling of geothermal monotube heat exchangers

In this study, we present a new analytical model for the analysis of transient heat transfer (the model is solved after Laplace transformation) in geothermal monotube heat exchangers (GHE). The model accounts for various factors, including the heat exchanger's geometric parameters, the nature of the fluid (air and water), flow characteristics (laminar and turbulent), and heat transfer boundary conditions. Initially, the model's performance is validated by comparing its results with those obtained from computational fluid dynamics simulations using OpenFOAM. This validation process ensures the reliability and accuracy of the model. Additionally, an analysis of the heat transfer mechanisms and key modeling parameters identified from the comparison is provided. In the subsequent section, the analysis is extended by comparing the proposed model to four existing analytical models available in the literature. These reference models employ distinct approaches to simulate heat transfer to the ground. By examining a range of cases, the performance of the model is systematically evaluated against these conventional methods. This work closes by an analysis of the analytical models and the impact on the accuracy of the results highlighting the advantages of the proposed method in geothermal applications.

Numerical analysis of fractional order Black–Scholes option pricing model with band equation method

In recent years, there has been a surge of interest in fractional calculus, primarily due to its enhanced capability in accurately modeling complex systems compared to classical methods. This study delves into the fractional Black–Scholes option pricing model, which adeptly captures the intricate dynamics and memory effects inherent in financial markets. The novelty of this research lies in introducing a novel approach employing band matrix equations for numerically solving the fractional order Black–Scholes option pricing model. We explore the numerical solutions for fractional European options using three distinct methods: Laplace Transform, Hilbert Transform, and Monte Carlo simulation. Each method’s convergence is rigorously proven, and their respective convergence orders are derived. Furthermore, we conduct a comprehensive analysis to discern the factors influencing the convergence rates for each method. Additionally, we observe the effectiveness and efficiency of the proposed methods in solving fractional European options through numerical examples. Our findings offer an in-depth insight into the comparative performance of Laplace transform, Hilbert transform, and Monte Carlo simulation in terms of accuracy, computational complexity, and stability. This research contributes significantly to the comprehension of fractional European option pricing and provides valuable guidance for both practitioners and researchers in selecting the most suitable method for pricing these intricate financial instruments.

Fractional Analysis of Magnetohydrodynamics Maxwell Flow Over an Inclined Plate with the Effect of Thermal Radiation

This current study provides a comprehensive examination of a novel method for studying the dynamics of a fractionalized Maxwell flow near an inclined plate, considering non-uniform mass transfer through a permeable media. Through the use of partial differential equations, incorporating heat and mass movement effects, the study employs a combination of generalized Fick’s and Fourier’s law with the Caputo operator. Transforming the fractionalized model into dimensionless form using appropriate dimensionless values, semi-analytical solutions for the non-dimensional transmitted fractional model are obtained via the Laplace transformation method. Through graphical analysis, the precise contributions of key parameters such as heat generation, radiation, and chemical reactions are elucidated, including their impacts on the calculated heat generation parameter (Qo), radiation parameter (Nr), and others. The study’s significance lies in its implications for the design of efficient heat exchangers, fluid flow systems, and cooling components in complex engineering systems, including nuclear reactors and power generation plants. Furthermore, the fractional derivative approach offers a more accurate representation of the viscoelastic behavior of materials like polymers, crucial for optimizing fabrication processes such as extrusion and molding. The insights gained from this study extend to the realm of miniaturized fluidic devices, including bio-analysis tools, lab-on-a-chip technology, and microfluidic drug delivery systems, where improved performance and control need a grasp of Maxwell fluid dynamics. The physical outcome of this research lays the groundwork for future investigations that will maximize heat transfer efficiency in real-world systems and give insightful information on the behavior of complicated fluids. We compute and display the skin friction, mass and heat transfer rate in tabular form.

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Mathematical Solutions of Free Convection Flow of Casson Fluid in Channel with Accelerated Plate

Mathematical Solutions of Free Convection Flow of Casson Fluid in Channel with Accelerated Plate

Analytical Solution for MHD Casson Nanofluid Flow and Heat Transfer due to Stretching Sheet in Porous Medium

The feature of having a surface that can stretch has garnered attention in numerous industrial and engineering fields because of its advantages. Nevertheless, most fluid mechanics simulations for stretchable surfaces have predominantly relied on numerical solutions, with a notable lack of theoretical investigations into this matter. Consequently, the current research aims to contribute a theoretical exploration of heat transfer and boundary layer flow for Casson nanofluid on a linearly stretching sheet, considering the existence of porosity and magnetic field effects. Two distinct types of water-based nanofluids containing aluminium oxide and silicon dioxide are examined. By employing similarity transformations, the governing momentum and energy equations undergo transformation and subsequent analytical resolution using Laplace transformations. The resulting solutions are graphically presented to examine the influence of key parameters on temperature and velocity distribution. The analysis indicates that heat transfer is improved by the inclusion of nanoparticles, porosity, and a magnetic field. However, the velocity distribution slows down as a result of higher nanoparticle volume fraction, porosity, and magnetic field imposition.

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

Analyzing the impact of time-fractional models on chemotherapy's effect on cancer cells

In this study, we employ the Laplace Variational Iterational Method (LVIM) as a sophisticated mathematical tool to investigate into the complex dynamics of cancer cells under the influence of chemotherapy. The LVIM, a method combining Laplace transformations and variational iteration techniques, is specifically adapted to address a system of time fractional differential equations (FDEs) that characterizes the temporal behavior of cancer cells. To enhance the efficacy of our approach, we introduce a semi-analytic version of LVIM, which proves to be a powerful and versatile tool for solving mathematical problems involving fractional derivatives. The focus of our analysis centers on elucidating the impact of chemotherapy, with a particular emphasis on drug diffusion within cancer cells and the fractality of DNA walks. Through numerical exploration encompassing varying fractional order derivatives, our study exposes nonlinear behaviors that remain secret in systems featuring only integer order derivatives. Notably, the methodology we propose is not only applicable to the specific cases examined in this research but also exhibits broad versatility, making it suitable for exploring the effects of different drugs and types of cancers. This research contributes valuable insights into the dynamics of cancer cells, aiding in the understanding of the implications for therapeutic strategies in the context of cancer treatment.

Photothermal effects induced by laser radiation in a 2D axi-symmetric generalized thermoelastic semiconducting half space using MGL model

In this paper, a novel model based on the strain-temperature rate-dependent theory is investigated to study the coupling influences of thermal, elastic, and plasma waves during the photothermal process in a 2D axi-symmetric semiconducting half space. The surface absorption technique is used to illuminate the surface of the medium by a laser beam whose temporal and spatial profiles are Gaussian, moreover, the surface is also considered traction free. Integral transform method is applied using Laplace and Hankel transformations to obtain a general solution to the considered problem. Inverse Laplace transform is obtained computationally using Rimann-Sum approximation. Silicon element is chosen as an application to show the compatibility of the results of the MGL model with GL and CTE models, moreover the relaxation times effects of the new model were studied.