Laplace Transform Method Research Articles

Estimating the Spread of Generalized Compartmental Model of Monkeypox Virus Using a Fuzzy Fractional Laplace Transform Method

The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings.

Homotopy Analysis Laplace Transform Method for Analytical Solution of Porous Medium Equation

Homotopy Analysis Laplace Transform Method for Analytical Solution of Porous Medium Equation

Application of a Machine Learning Algorithm for Evaluation of Stiff Fractional Modeling of Polytropic Gas Spheres and Electric Circuits

Fractional polytropic gas sphere problems and electrical engineering models typically simulated with interconnected circuits have numerous applications in physical, astrophysical phenomena, and thermionic currents. Generally, most of these models are singular-nonlinear, symmetric, and include time delay, which has increased attention to them among researchers. In this work, we explored deep neural networks (DNNs) with an optimization algorithm to calculate the approximate solutions for nonlinear fractional differential equations (NFDEs). The target data-driven design of the DNN-LM algorithm was further implemented on the fractional models to study the rigorous impact and symmetry of different parameters on RL, RC circuits, and polytropic gas spheres. The targeted data generated from the analytical and numerical approaches in the literature for different cases were utilized by the deep neural networks to predict the numerical solutions by minimizing the differences in mean square error using the Levenberg–Marquardt algorithm. The numerical solutions obtained by the designed technique were contrasted with the multi-step reproducing kernel Hilbert space method (MS-RKM), Laplace transformation method (LTM), and Padé approximations. The results demonstrate the accuracy of the design technique as the DNN-LM algorithm overlaps with the actual results with minimum percentage absolute errors that lie between 10−8 and 10−12. The extensive graphical and statistical analysis of the designed technique showed that the DNN-LM algorithm is dependable and facilitates the examination of higher-order nonlinear complex problems due to the flexibility of the DNN architecture and the effectiveness of the optimization procedure.

Analytical solution for horizontal vibration of end-bearing single pile in radially heterogeneous saturated soil

Employing Biot's dynamic equations and Novak's thin-layer theory, an analytical model for the horizontal vibration of a single pile embedded in the radially heterogeneous saturated soil is established. Firstly, the frequency-domain response of surrounding soil is derived by means of the potential function, Laplace transformation and variable separation methods. Secondly, the analytical solution for the horizontal impedance of pile head is deduced by combining the conditions of the displacement continuity and stress equilibrium on the soil-pile interface. Finally, the obtained solution is degenerated to validate its accuracy via independent comparisons with different existing solutions. On this basis, the effect of soil radial heterogeneity on the horizontal vibration of an end bearing pile is further discussed by conducting an extensive parametric analysis.

Modeling of the Resonant Inverter for Wireless Power Transfer Systems Using the Novel MVLT Method

Wireless power transfer (WPT) is a power transfer technique widely used in many industrial applications, medical applications, and electric vehicles (EVs). This paper deals with the dynamic modeling of the resonant inverter employed in the WPT systems for EVs. To this end, the Generalized State-Space Averaging and the Laplace Phasor Transform techniques have been the flagship methods employed so far. In this paper, the modeling of the resonant inverter is accomplished by using the novel Modulated Variable Laplace Transform (MVLT) method. Firstly, the MVLT technique is discussed in detail, and then it is applied to model a study-case resonant inverter. Finally, a study-case resonant inverter is developed and utilized to validate the theoretical results with MATLAB/Simulink.

Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator

In this work, we proposed a new method called Laplace–Padé–Caputo fractional reduced differential transform method (LPCFRDTM) for solving a two-dimensional nonlinear time-fractional damped wave equation subject to the appropriate initial conditions arising in various physical models. LPCFRDTM is the amalgamation of the Laplace transform method (LTM), Padé approximant, and the well-known reduced differential transform method (RDTM) in the Caputo fractional derivative senses. First, the solution to the problem is gained in the convergent power series form with the help of the Caputo fractional-reduced differential transform method. Then, the Laplace–Padé approximant is applied to enlarge the domain of convergence. The advantage of this method is that it solves equations simply and directly without requiring enormous amounts of computational work, perturbations, or linearization, and it expands the convergence domain, leading to the exact answer. To confirm the effectiveness, accuracy, and convergence of the proposed method, four test-modeling problems from mathematical physics nonlinear wave equations are considered. The findings and results showed that the proposed approach may be utilized to solve comparable wave equations with nonlinear damping and source components and to forecast and enrich the internal mechanism of nonlinearity in nonlinear dynamic events.

Application of Fractional Laplace Transform Method in Solving Linear Systems of Fractional Differential Equations

Application of Fractional Laplace Transform Method in Solving Linear Systems of Fractional Differential Equations

Dynamics of heat transport in CNTs based Darcy saturated flow: Modeling through fractional simulations

Nanofluids have a vital role in many industries due to their novelty of heat transfer. Various mathematical techniques are required to simulate such problems. It can seem that traditional partial differential equations are incapable of analyzing and investigating the physical behavior of flow parameters affected by memory effects. This research communicates the implementation of the most interesting analytical method namely Prabhakar fractional derivative regarding the thermal flow of Casson fluid with single and multiwall carbon-nanotubes due to an inclined plate. The water and blood are considered as base particles. slip and Newtonian heating impacts for the thermal flow are also considered. The fractional modal of leading PDE's is attained by Prabhakar fractional derivative with various limiting cases. The generalized solution for the thermal and velocity field is simulated via the Laplace transformation method. The thermal expressions are modeled via Fourier expressions. Graphs are used to illustrate the influence and behaviour of key physical and fractional characteristics. The finding is that the temperature and velocity profiles of SWCNTs are more prominent than those of MWCNTs. Changing the fractional parameter values results in a greater rise in the velocity gradient for blood-based nanofluid than for water-based nanofluid.

Analysis of the Performance Measures of a Non-Markovian Fuzzy Queue via Fuzzy Laplace Transforms Method

Laplace transforms play an essential role in the analysis of classical non-Markovian queueing systems. The problem addressed here is whether the Laplace transform approach is still valid for determining the characteristics of such a system in a fuzzy environment. In this paper, fuzzy Laplace transforms are applied to analyze the performance measures of a non-Markovian fuzzy queueing system FM/ FG/1. Starting from the fuzzy Laplace transform of the service time distribution, we define the fuzzy Laplace transform of the distribution of the dwell time of a customer in the system. By applying the properties of the moments of this distribution, the derivative of this fuzzy transform makes it possible to obtain a fuzzy expression of the average duration of stay of a customer in the system. This expression is the fuzzy formula of the same performance measure that can be obtained from its classical formula by the Zadeh extension principle. The fuzzy queue FM/ FE_k /1 is particularly treated in this text as a concrete case through its service time distribution. In addition to the fuzzy arithmetic of L-R type fuzzy numbers, based on the secant approximation, the properties of the moments of a random variable and Little's formula are used to compute the different performance measures of the system. A numerical example was successfully processed to validate this approach. The results obtained show that the modal values of the performance measures of a non-Markovian fuzzy queueing system are equal to the performance measures of the corresponding classical model computable by the Pollaczeck-Khintchine method. The fuzzy Laplace transforms approach is therefore applicable in the analysis of a fuzzy FM/FG/1 queueing system in the same way as the classical M/G/1 model.

Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform method requires, among other mathematical tools, the use of the Cauchy residue theorem and the computation of an infinite series. Symbolic computation facilitates the whole process, providing solutions that would be unmanageable by hand. The results obtained here emphasize the fact that symbolic computation is a powerful tool for computing analytical solutions for linear delay differential equations. From a computational viewpoint, we found that the computation time is dependent on the complexity of the history function, the number of terms used in the LT solution, the number of intervals used in the MoS solution, and the parameters of the DDE. Finally, we found that, for linear non-neutral DDEs, MATLAB symbolic computations were faster than Maple. However, for linear neutral DDEs, which are often more complex to solve, Maple was faster. Regarding the accuracy of the LT solutions, Maple was, in a few cases, slightly better than MATLAB, but both were highly reliable.

Optimized numerical inverse Laplace transformation

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate-Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest. In this work we propose an NILT procedure that goes beyond this limitation and optimize the accuracy of the NILT utilizing also the transform function and the time point of interest.

Suction effect on MHD flow of Brinkman-type fluid with heat absorption and first-order chemical reaction

Suction/injection is a mechanical effect and used to control the energy losses in the boundary layer region by reducing the drag on the surface. In this study, unsteady MHD flow of Brinkman-type fluid with suction/injection, heat absorption, and chemical reaction is investigated and an analytical solution is established. The corresponding results for temperature, concentration, and velocity fields are obtained with the help of the Laplace transformation method analytically. The physical effects of thermal and mass Grashoff number, Prandtl number, Schmidt number, heat absorption parameter, first-order chemical reaction parameter, suction/injection, Brinkman parameter, and magnetic parameter have been discussed graphically. Finally, it is observed that in the presence of suction effect, fluid’s velocity decreases gradually by increasing the value of suction parameter while show an increasing trend for the increasing value of the injection parameter.

Approximate Analytic Expression for the Time-Dependent Transient Electrophoretic Mobility of a Spherical Colloidal Particle.

The general expression is derived for the Laplace transform of the time-dependent transient electrophoretic mobility (with respect to time) of a spherical colloidal particle when a step electric field is applied. The transient electrophoretic mobility can be obtained by the numerical inverse Laplace transformation method. The obtained expression is applicable for arbitrary particle zeta potential and arbitrary thickness of the electrical double layer around the particle. For the low potential case, this expression gives the result obtained by Huang and Keh. On the basis of the obtained general expression for the Laplace transform of the transient electrophoretic mobility, we present an approximation method to avoid the numerical inverse Laplace transformation and derive a simple approximate analytic mobility expression for a weakly charged particle without involving numerical inverse Laplace transformations. The transient electrophoretic mobility can be obtained directly from this approximate mobility expression without recourse to the numerical inverse Laplace transformation. The results are found to be in excellent agreement with the exact numerical results obtained by Huang and Keh.

Numerical inverse Laplace transformation beyond the Abate–Whitt framework

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate–Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest.In this work we propose an NILT procedure that goes beyond this limitation and optimize the accuracy of the NILT utilizing also the transform function and the time point of interest.

Insight into the dynamics of second-grade fluid subject to inclined magnetic force, newtonian heating, slip flow, and prabhakar-like fractional kind of newtonian heating

Sequel to the significance of the dynamics of second-grade fluid in the industry, nothing is known on the increasing inclined magnetic force, Newtonian heating, slip flow and Prabhakar-like fractional kind of Newtonian heating. In this study, the novel modified and efficient fractional approach, namely, the Prabhakar fractional technique is applied on an unsteady flow of second-grade liquid flowing on an inclined oscillating plate. The flowing fluid is moving under the effects of an applied inclined magnetic field, furthermore, the Newtonian heating effect and slip on the boundaries of the wall are also considered in the boundary conditions. After nondimensionalizing the governing equations, the solution of coupled equations is attained with the help of integral transform i.e., Laplace transformation method, and some numerical approaches are also applied for the Laplace inverse. The results convey that the momentum field decrease with increase in fractional constraints. The thermal change exhibits a declining tendency for increased values of fractional parameters. Furthermore, in comparison of velocity field attained by our applied fractional system with the existing literature, the curves of both Stehfest and Tzou’s methods overlay with each other which signifies the validity of our attained outcomes of momentum profile.

Transient waves in a piezomagnetic half-space subjected to dynamic anti-plane loading

Propagation of transient waves in a piezomagnetic half-space, which is subjected to dynamic anti-plane-concentrated line force, is analyzed theoretically in this study. By employing Laplace transformation and Cagniard de-Hoop method, explicit closed forms of solution for the transverse displacement, shear stress, magnetic potential, and induction are obtained, and contribution from the mechanical and magnetic waves is thus expressed in the transient full-field solution. Propagation of the transient waves in the piezomagnetic half-space is then discussed in detail using numerical calculation. The results show that the magneto-mechanical coupling coefficient and location of the receiver have a significant influence on the transient response of this half-space. The magnitude of the transient shear stress is effectively reduced by the weakening magneto-mechanical coupling coefficient and the arrival time of the transient magnetic induction peak increases with decreasing magneto-mechanical coupling coefficient. In addition, it is found that the smaller the magneto-mechanical coupling coefficient, the higher the static value of the transient magnetic induction.

SOLUTIONS OF THE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS BY DOUBLE LAPLACE TRANSFORM METHOD

SOLUTIONS OF THE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS BY DOUBLE LAPLACE TRANSFORM METHOD

Three-Dimensional Boundary Element Strategy for Stress Sensitivity of Fractional-Order Thermo-Elastoplastic Ultrasonic Wave Propagation Problems of Anisotropic Fiber-Reinforced Polymer Composite Material.

A new three-dimensional (3D) boundary element method (BEM) strategy was developed to solve fractional-order thermo-elastoplastic ultrasonic wave propagation problems based on the meshless moving least squares (MLS) method. The temperature problem domain was divided into a number of circular sub-domains. Each node was the center of the circular sub-domain surrounding it. The Laplace transform method was used to solve the temperature problem. A unit test function was used in the local weak-form formulation to generate the local boundary integral equations, and the inverse Laplace transformation method was used to find the transient temperature solutions. Then, the three-dimensional elastoplastic problems could be solved using the boundary element method (BEM). Initial stress and strain formulations are adopted, and their distributions are interpolated using boundary integral equations. The effects of the fractional-order parameter and anisotropy are investigated. The proposed method’s validity and performance are demonstrated for a two-dimensional problem with excellent agreement with other experimental and numerical results.

Solution of First Order Fuzzy Partial Differential Equations by Fuzzy Laplace Transform Method

In this study, first order fuzzy partial differential equations with negative coefficient on one hand and with both positive and negative coefficients on the other hand are solved using fuzzy Laplace transform method. The results obtained indicate that all the solutions exist and the examples presented illustrate the applicability of the method.

Thermal and concentration diffusion impacts on MHD Maxwell fluid: A generalized Fourier's and Fick's perspective

In this article, a new approach to study the fractionalized Maxwell fluid is described by the Prabhakar fractional derivative near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomena has been described in forms of partial differential equations along with heat and mass transportation effect taken into account. The Prabhakar fractional operator which was recently introduced is used in this work together with generalized Fick's and Fourier's law. The fractionalized model is transfromed into non-dimensional form by using some suitable dimensionless quantities. The non-dimensional developed fractional model for momentum, thermal and diffusion equations based on Prabhakar fractional operator has been solved analytically via Laplace transformation method and calculated solutions expressed in terms of Mittag-Leffler special functions. Graphical demonstration are made to characterized the physical behavior of different parameters and significance of such system parameters over the momentum, concentration and energy profiles. Moreover, for results validation, comparative study among limiting models derived from fractionalized Prabhakar Maxwell fluid such as fractional and classical fluid models for Maxwell and Newtonian are performed. Further, it is observed from the graphs the valocity curves for classical fluid models relatively higher as compared to fractional fluid models, and fractional approach is more realistic and convenient as compared to classical approach.