Laplace Transform Research Articles

Piezothermoelastic interaction in piezoelectric rods induced by a timed laser pulse without energy dissipation

The main aim of this article is to study the piezo-thermoelastic interaction in piezoelectric rods without energy dissipation due to a laser heat source with a timed pulse. Laplace transforms are used to represent the analytical solutions in the transformations domain. The governing equations of piezo-thermoelasticity in the Green and Naghdi model (GNII model without energy dissipation) in the presence of a laser heat source with a timed pulse are solved exactly in the Laplace domain, calculating the temperature, the stress, the displacement, and the electric potential. Numerical Laplace inversion is then used to find the general solution of the variables under study, which are then graphically shown.

Exact solution to a class of problems for the Burgers’ equation on bounded intervals

In this study, we consider Burgers’ equation with fixed Dirichlet boundary conditions on generic bounded intervals. By employing the Hopf-Cole transformation and a recently established exact operational solution for linear reaction–diffusion equations, an exact solution in the time domain is derived through inverse Laplace transforms. In the event that analytic inverses do in fact exist, they can be obtained in closed form through the use of Mellin transforms. Nevertheless, highly efficient algorithms are available, and numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. Two illustrative tests demonstrate that the results align closely with those of classical exact solutions. In comparison to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent a novel alternative, offering new insights and perspectives. The exact solution via the inverse Laplace transform is shown to be more computationally efficient, providing a reference point for numerical and semi-analytical methods.

Research on temperature rise prediction model for circuit breaker based on numerical Laplace transform

Abstract The temperature rise in circuit breakers’ thermal paths is a critical indicator influencing their tripping characteristics, directly affecting response speed, reliability, and safety. Existing models for predicting circuit breaker temperature rise are often structurally limited, lack generality, and exhibit low predictive accuracy. Addressing this issue, this paper investigates an efficient, precise, and widely applicable theoretical model for predicting circuit breaker temperature rise. This model focuses on the thermal conduction mechanisms within small-scale circuit breakers, establishing a fundamental numerical heat transfer model based on Newton’s cooling formula and Fourier’s heat conduction law. By applying numerical Laplace transformation, the model is simplified into the complex frequency domain, and employing residue theorem for solving, it is transformed back into the time domain using Laplace inverse transformation to derive the predictive model. Experimental validation using a 32 A rated current circuit breaker demonstrates the model’s high predictive accuracy, suitability across various circuit breaker structures, with an average deviation below the permissible deviation in industry standards. This research holds significant implications for enhancing the thermal tripping characteristics of circuit breakers.

Well-Test Interpretation Model of Water-Injection Well in a Low-Permeability Reservoir and Its Application

For low-permeability reservoirs, water-flooding development is usually adopted, which leads to induced fractures near the wellbore, increasing reservoir heterogeneity, and making water-flooding development more complex. This paper focuses on low-permeability reservoirs, considering the characteristics of induced fractures and elliptic-flow composite, and the well-test model for injection wells is established. The mathematical model in Laplace space is obtained through dimensionless transformation and Laplace transformation. Subsequently, the Mathieu function is introduced to obtain the bottom hole pressure, and the pressure response curve is drawn. The six flow stages of the curve are defined, and the sensitivity of parameters such as half-length of induced fractures, range of lateral-swept area, permeability in unswept area, and outer boundary distance at constant pressure are analyzed. The results show that the half-length of the fracture mainly affects the linear flow of the fracture, the range of the lateral wave-affected area mainly affects the radial flow of the swept area, the permeability of the unswept area mainly affects the radial flow of the unswept area, and the outer boundary distance at constant pressure mainly affects the boundary flow. Based on the production performance of a certain injection well in J Oilfield, a series of key parameters are obtained through analytical solution model inversion, including the induced-fracture half-length of 10.32 m, the lateral-swept range of elliptic partition flow of 128.95 m, the permeability of the swept area of 6.87 mD, and the mobility ratio of 119.92, which show the superiority of the analytical solution model.

Application of Triple- and Quadruple-Generalized Laplace Transform to (2+1)- and (3+1)-Dimensional Time-Fractional Navier–Stokes Equation

In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match with the exact solutions of the problems, which proves that, as the terms of the series increase, the approximate solutions are closer to the exact solutions of each problem. To verify the appropriateness of these methods, some examples are offered. The TGLTDM and FGLTDM results indicate that the suggested methods have higher evaluation convergence as compared to the ADM and HPM.

Generalized Laplace Transform with Adomian Decomposition Method for Solving Fractional Differential Equations Involving ψ-Caputo Derivative

In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the ψ-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various ψ-fractional differential equations.

Transient analysis of functionally graded axisymmetric plates via finite element method in the Laplace domain

The principal goal of this research article is to examine the forced vibration of Functionally Graded Material (FGM) axisymmetric plates with the aid of the finite element method in the Laplace space. The material properties are assumed to be isotropic, linear viscoelastic, or elastic and vary continuously in the direction of the thickness of the plate. The governing equations of motion are transferred to the Laplace domain and solved for a set of Laplace parameters. Then, a modified Durbin’s inverse Laplace method is employed to retransfer the obtained results back to the time domain. A convergence analysis is conducted to optimize the number of Laplace parameters and time steps. An 8-node quadratic quadrilateral element featuring two degrees of freedom per node is employed to generate the model of the axisymmetric plates. Different loads such as step load, square wave, and sawtooth wave loads are used to observe the impact of load type on the transient response of FGM plates. In viscoelastic analysis, the Kelvin damping model is preferred, and the effect of the damping ratio is investigated. The obtained results are validated with the aid of the available literature and the accuracy of the suggested model is confirmed. It is carried out that the material variation significantly affects the dynamic behavior of the plates, for instance as the material coefficient increases, displacement amplitudes decrease, and periods increase. The novelty of this paper is the employment of the finite element method together with Laplace transformation for addressing the present class of problems for the first time.

Unsteady flow rate in transient, incompressible pipe flow

AbstractKnowledge regarding current flow rates through pipes and other components is crucial for most hydraulic systems. The hydraulic power can be computed in combination with the pressure, which may be used in many applications, such as predictive maintenance. Most flow rate sensors used in the field of fluid power operate invasively. Therefore, the measurement process itself alters the flow rate. Furthermore, most sensors operate accurately for stationary flow but produce inaccurate measurements for transient flow. A well‐known method of determining the flow rate is to measure the pressure difference between two points along a pipeline and calculate the flow rate based on the law of Hagen and Poiseuille. However, since the relation mentioned above only applies to laminar, steady, and incompressible flow, its usefulness for transient flows is limited. This paper derives a system equation based on the fundamental laws of fluid mechanics, which describe transient, incompressible pipe flow. As a result, the fundamental equations are solved in the Laplace domain and subsequently transformed back into the time domain. The four‐pole theorem relates the pressure difference and the volumetric flow rate. The analytical solution consists of a convolution integral containing a weighting function and the pressure difference. Compared to a simulation, the novel equation displays high accuracy for transient and stationary incompressible pipe flow. This equation paves the way for a soft sensor, which allows the noninvasive measurement of arbitrary volumetric flow rates within pipes.

Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK α,γ-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

Presentation of the efficient scheme for solving fractional order telegraph problems

The Asymptotic Homotopy Perturbation Transform Method AHPTM is presented in this work. It is combined version of the Asymptotic Homotopy Perturbation Method AHPM and Laplace transformation. The focus of the work is the introduction of a new fast convergent scheme to obtain the solution of the fractional partial differential equations. Therefore, the first demonstration of the AHPTM is present for the solution of space-fractional telegraph equation (SFTE) in this work. The Caputo version of fractional derivatives has been utilized. Three test problems of the important fractional telegraph model were solved by this proposed scheme. The scheme of AHPTM worked without exploiting Ji. Huan He polynomials or Adomian polynomials. This application was elaborated by providing error estimates, a graphical presentation and tabulation of the results obtained by AHPTM. The comparison of results obtained by AHPTM with exact results is provided which indicated the accuracy of the scheme.

Coupled of Semi Analytic Approach Associated with Laplace Transform First Step for Solving Matrix Differential Equations Quadratic Form when the Time-Delay in Noise Term

في هذا البحث تناولنا طريقة فعالة وجديدة وهي الدمج بين طريقتي الادوميان والهوموتوبي مع إستخدام طريقة الخطوات لتسهيل المسألة والتي تخص المعادلات التفاضلية الاعتيادية التباطئية لحل معادلة المصفوفات التباطئية التربيعية الغير خطية . كلتا الطريقتين على درجة عالية من التأثير والفعالية. جزء التكلمل الكلي لطريقة الهوموتوبي سيستخدم بدلا من جزء التكامل الخاص ب الادوميان . الميزة الرئيسية لهذه التقنية هي الحصول على نتائج اكثر دقة و لفترة و منطقة اوسع واطول ولمعرفة دقة هذه النتائج تحت تأثير التأخير. الجزء الخاص بالتأخير يختفي بعد استخدام طريقة الخطوات. تم حساب الخطأ المتبقي . لتقليل الوقت والعمليات الحسابية المعقدة تم أستخدام تحويلات لابلاس. أخيرا النتائج التي تم الحصول عليها بينت ان التقنية فعالة وسريعة التقارب للحل المظبوط ولفترة ومنطقة اوسع . يمكن استخدام هذه التقنية لحل مسائل غير خطية مختلفة. طريقة الادوميان هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية ؛ جزئية ؛ كسرية و تباظؤية . هذه الطريقة تطورت بواسطة جورج ادوميان. هي سريعة التقارب للحل المظبوط وتستخدم للخطية و غير الخطية و المتجانسة و غيلر المتجانسة. متعددة ادوميان تسمح للحل التقارب للحل المظبوط للمسألة قيد الدراسة دون الحاجة الى أي تحوير. طريقة الهوموتوبي هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية؛ جزئية ؛ كسرية و تباطؤية وانواع مختلفة. هذه الطريقة تطورت عن طريق العالم ليو . هي سريعة التقارب للحل المظبوط وتستخدم للخطية وغير الخطية و المتجانسة وغير المتجانسة. جاء مفهوم الهوموتوبي من مفهوم التبلوجي في توليد متسلسلة متقاربة للحل المظبوط. هذه الطريقة تم ايجادها من قبل ليو خلال اطروحته. تحتوي الطريقة على متقير او معلمة خلاله تمكننا التقارب.

Large time solution for collisional breakage model: Laplace transformation based accelerated homotopy perturbation method

The behavior of several particulate processes, such as cell interaction, blood clotting, bubble formation, grain breakage, and cheese formation from milk, have been studied using coagulation and fragmentation models (Fogelson and Guy, 2008 [1]; Pazmiño et al., 2022 [2]; Chen et al., [3]). Various studies utilize the linear fragmentation model to simplify the underlying physics. However, in real-life scenarios, particles form due to the collision of two particles, leading to a non-linear collisional breakage model. Unfortunately, the collisional breakage model is less explored due to its complex behavior. While analytical solutions are difficult to compute and are still missing in the literature, this article proposes an approximate solution for the model using the Laplace-based accelerated homotopy perturbation method. Further, coupling with Padé approximant, the accuracy of the solution is extended for the longer time. Considering various physically relevant kernels, the approximate series solutions are compared with the well known finite-volume solutions to measure the accuracy in terms of qualitative and quantitative errors. The article also encompasses theoretical convergence analysis and error estimations to enhance comprehension of the proposed formulation.

The Heat Transfer in Skin Tissues Under The General Two-Temperature Three-Phase-Lag Model of Heat Conduction with a Comparative Study

In this paper, a new and more general model of heat conduction that depends on the drift velocity due to thermomass motion assumption will be established and will be applied to the skin tissue. Four different heat conduction models will be incorporated into a unified equation of heat conduction: the Pennes, Vernotte-Cattaneo, dual-phase-lag of Tzou, and the general two-temperature three-phase-lag of Youssef. The governing partial differential equations of the general two-temperature three-phase-lag model of bioheat conduction will be implemented and solved directly in the domain of the Laplace transformation. The numerical solutions of the Laplace transform will be calculated by executing the Tzou iteration formula. The ramp-type heat on the surface of the skin tissue will be considered as thermal loading. The conductive and dynamic temperature increment reactions have been studied and discussed with different values of ramp-time heat, characteristic length, and drift velocity parameters. The novelty of this work is to introduce some comparisons of the four under-studied bioheat conduction models and show the differences between them in the figures. The numerical results show that the ramp-time heat, drift velocity, and characteristic length parameters have major impacts on the increment of both dynamical and conductive temperature distributions.

TD-EFIE Method-of-Moments Solution Using the Numerical Inversion of the Laplace Transform With Time-Stepping Re-Initialization

TD-EFIE Method-of-Moments Solution Using the Numerical Inversion of the Laplace Transform With Time-Stepping Re-Initialization

A Local Radial Basis Function Method for Numerical Approximation of Multidimensional Multi-Term Time-Fractional Mixed Wave-Diffusion and Subdiffusion Equation Arising in Fluid Mechanics

This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously incorporating both wave-diffusion and subdiffusion terms. Our proposed method follows three basic steps: (i) The given equation is transformed into a time-independent form using the Laplace transform (LT); (ii) the LMM is then used to solve the transformed equation in the LT domain; (iii) finally, the time domain solution is obtained by inverting the LT. We use the improved Talbot method and the Stehfest method to invert the LT. The LMM is used to circumvent the shape parameter sensitivity and ill-conditioning of interpolation matrices that commonly arise in global mesh-free methods. Traditional time-stepping methods achieve accuracy only with very small time steps, significantly increasing the computational time. To overcome these shortcomings, the LT is used to provide a more powerful alternative by removing the need for fine temporal discretization. Additionally, the Ulam–Hyers stability of the considered model is analyzed. Four numerical examples are presented to illustrate the effectiveness and practical applicability of the method.

Unsteady Free Convection of Nanofluid Past an Infinite Vertical Sheet Under the Effects of Inclined Magnetic Field and Cogley Radiation

Background: An infinite vertical sheet is the medium of inquiry for studying the copper- water nanofluid behavior with an inclined magnetic field and Cogley radiation. The research contains significant aspects. The use of this nanofluid is of great practical importance, especially in the areas of energy efficiency, thermal management, radiative cooling for space technology, environmental impact reduction, and improving heat transfer efficiency in electronic device cooling operations. Moreover, the research enhances the understanding and manipulation of sophisticated materials. The uttermost significance of this resides in its capacity to uncover issues inherent in the stages of planning, developing, analyzing, and improving manufacturing processes, possibly securing new solutions via patent protection. Objective: This research aims to provide a comprehensive analysis of the unsteady natural convection of a nanofluid along an infinite vertical sheet. The study focuses on understanding how an inclined magnetic field, Cogley radiation, heat source, and varying nanoparticle volume concentrations impact the velocity, thermal, and concentration profiles of the nanofluid. Additionally, the investigation seeks to evaluate the Nusselt number, Sherwood number, and skin friction coefficient across different concentration profiles to derive meaningful insights. Methods: The Laplace transform (L.T.) method, in combination with the MATLAB symbolic computing program, is used to develop numerical solutions for the main boundary value issues. The Laplace transform technique is used to clarify dimensionless partial differential equations (PDEs) and their limiting situations. An in-depth study is conducted on the profiles of momentum, energy, and concentration for various kinds of nanofluids with changing volume concentrations of nanoparticles. This research provides a comprehensive examination and also identifies prospective prospects in this field that might be protected by patents. Result: The work provides an in-depth understanding of how inclined magnetic fields, Cogley radiation, heat generation, and nanoparticle volume concentration affect the velocity, energy, and mass profiles of the nanofluid. Graphical representations depict these effects, offering a visual comprehension of the system's behavior. The Nusselt number and skin friction coefficient are determined by analytical and numerical methods. These findings demonstrate that the Nusselt number and skin friction coefficient have an inverse relationship with changes in concentration profiles. Conclusion: The work improves our comprehension of the fluctuating natural convection of nanofluids across an unbounded vertical surface, taking into account variables such as an inclined magnetic field, heat source, Cogley radiation, and nanoparticle volume concentration. The practical implications of the discoveries, together with the analytical and numerical results, enhance our comprehension of the dynamics of nanofluid systems. Ensuring the protection and widespread distribution of patented technology is essential for promoting sustainable advancements.

A New General Integral Transform of Modified Laguerre’s Polynomial 𝐿𝐚,𝐛,𝐜,𝐧(𝑡)

Objectives: The core objective of this paper is to introduce the explicit Modified Laguerre polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) and obtain a new general integral transform of Modified Laguerre’s Polynomial, by using this general transform, some transforms of Laguerre’s Polynomial are identified as special cases. Method: We derive transforms of the polynomials 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡), 𝐿𝑛 (𝑡) by using this new integral transform approach and found the transforms like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_transform, Kamal transforms and their relationship for the same. Findings: A new general integral transform of the Modified Laguerre’s Polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) has been obtained, and a special case for the polynomial 𝐿𝑛 (𝑡) has been derived by using this transform. Novelty: This is a novel general integral transform for this polynomial and by applying this transform to the Modified Laguerre’s polynomial as special transform cases like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_, Kamal, Sumudu, Elzaki, Natural, and Pourreza are obtained. Keywords: Modified Laguerre polynomial; Laguerre polynomial; A New Integral Transform; Laplace Transform; Sumudu Transform

Modelling pressure dynamics of oil–gas two-phase flow in double-porosity media formation with permeability-stress sensitivity

Pressure dynamics can reflect the basic characteristics of fluid flow through underground porous formation. In this research, the oil–gas two-phase flow model for a double-porosity media formation with permeability-stress sensitivity was first established for three kinds of outer boundaries. The unified governing equation was deduced by utilizing the H function. The nonlinear mathematical model in consideration of permeability-stress sensitivity was linearized by implementing the canonical perturbation transformation and then solved by using the Laplace transformation. After that, a sequence of typical log–log curves of pressure dynamics influenced by various model parameters were plotted and analyzed. These curves reflect the typical characteristic of a V shape caused by the inter-porosity fluid flow from matrix toward natural fractures. Oil saturation and permeability-stress sensitivity coefficient have much influence on the pressure dynamics. Ultimately, the established model of oil–gas two-phase flow was validated through a well-test fitting interpretation for a real condensate gas well in a sandstone formation. This research can offer insights into the pressure dynamics dominated by the oil–gas two-phase flow in naturally fractured formations and the permeability-stress sensitivity effect.

A Note on Fractional Third-Order Partial Differential Equations and the Generalized Laplace Transform Decomposition Method

This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations.

Slope stability/instability analysis of advanced nanocomposite reinforced column structure under mechanical loading via coupled Carrera unified formulation, layer-wise and Laplace transform approaches

This study presents a comprehensive investigation into the slope stability and instability of column structures made of Triply Periodic Minimal Surfaces (TPMS) under mechanical loading. TPMS structures, known for their high strength-to-weight ratio, are increasingly used in engineering applications, particularly in lightweight structures. However, their stability behavior under complex loading conditions remains largely unexplored. To address this gap, we employ a coupled approach integrating the Carrera Unified Formulation (CUF), the Layer-Wise (LW) theory, and Laplace Transform techniques. The CUF framework, known for its versatility in modeling structural behavior across different geometrical and loading configurations, is utilized to capture the mechanical response of the TPMS-based columns. The LW theory further enhances the model by accurately representing the through-thickness behavior, particularly crucial for layered or composite TPMS structures. Finally, the Laplace Transform approach is applied to efficiently solve the governing differential equations, reducing the computational complexity of time-dependent mechanical analyses. A parametric study investigates the influence of various geometrical parameters, material properties, and loading conditions on the stability of the structures. The results highlight the critical factors influencing slope stability, including the interplay between the TPMS geometry and material distribution. Moreover, the findings offer insights into failure mechanisms, providing a basis for optimizing the design of TPMS-based columns for enhanced mechanical performance. This work contributes to advancing the theoretical understanding of TPMS structures, offering novel methodologies for slope stability analysis in complex mechanical systems.