Laplace Transform Research Articles

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.

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.

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.

A fast quadrupole-based analytical model for solving transient heat transfer in ventilated double-skin walls and assessing their energy saving potential

This research proposes a fast and accurate method for modeling transient heat transfer in ventilated double-skin façades (VDFs) with forced ventilation to predict their energy-saving potential. Using the thermal quadrupole formalism, the method solves the VDF problem in the Laplace domain while considering the full transient nature of the heat transfer; the only approximation is in space. The solution involves obtaining a general transfer function that predicts heat transfer rates or air temperatures at ventilated cavity's exit. The quadrupole method is validated against computational fluid dynamic numerical simulations conducted under similar meteorological, design and boundary conditions. A very good agreement was found (less than 3 % average deviation) with a significant reduction in computation time (less than 3s against 6h for CFD calculations). The model is computationally efficient and can consider important factors such as the opacity of the walls, construction materials, and air cavity design parameters. Finally, the model allows the assessment of the energy-saving potential of VDFs under various scenarios compared to conventional systems, which helps contributing to more sustainable building design practices. The energy efficiency of a VDF configuration was compared against a conventional wall system. Energy savings of 8.4 and 5.2 % were obtained, in cold and hot climates respectively.

Pricing Vulnerable Options With Variance Gamma Systematic and Idiosyncratic Factors by Laplace Transform Inversion

Pricing Vulnerable Options With Variance Gamma Systematic and Idiosyncratic Factors by Laplace Transform Inversion

Stochastic Resolution of Identity to CC2 for Large Systems: Ground State and Triplet Excitation Energy Calculations.

An implementation of stochastic resolution of identity to the CC2 (sRI-CC2) ground state energy followed by triplet excitation energy calculations is presented. A set of stochastic orbitals is introduced to further decouple the expensive 4-index electron repulsion integrals on the basis of RI approximation. A Laplace transformation of the orbital energy difference denominators into numerical summations is adopted to obtain a third-order overall scaling. We select a series of hydrogen dimer chains with nearly thousands of electrons, as well as some other molecules, for sRI-CC2 energies and test the accuracy and time consumption in comparison with those of RI-CC2. Our sRI-CC2 results reproduce a modest agreement with the RI-CC2 in Q-Chem program package and it allows a steep scaling reduction from O(N5) to O(N3). Besides, the unrestricted sRI-CC2 calculations also fit well with the restricted results. Thus, our sRI-CC2 implementation of ground state energy and triplet excitation energy provides a cost-efficient alternative approach, especially for some large-sized systems.

Fitting Quality of NMR Relaxation Data to Differentiate Asphalt Binders

Asphalt binder performance grades (PGs) are important metrics in designing pavements for effective transportation infrastructure. The PG system relies on the binder’s stiffness and is determined through energy- and time-intensive physical testing. Physical properties, like stiffness, can also be determined by spin–lattice NMR relaxometry, a non-destructive chemical method. NMR relaxometry can quantify the molecular mobility of materials by determining relaxation times from exponential decays of excited nuclear magnetization. While relaxation times have been used to determine physical properties of materials, a quantitative relation to the PG grades of asphalt binder is yet to be established. In this study, T1 NMR relaxation analyses were used to differentiate between solid asphalt binders and determine the fastest yet still-reliable method of modeling exponential decay data. Algorithms that fit exponential decay relaxation data using one, two, or three independent relaxation times were compared with a 128-coefficient discrete inverse Laplace transformation to determine the best mathematical fit for a comparative analysis. The number of data points was then reduced from 256 to 64 to 16 and finally to 8 data points on a relaxation curve to reduce the testing time and determine the minimum number of data points needed for comparison. Two batches of PG 64-22 asphalt binder, along with samples of PG 76-22 and 94-10 binders, were investigated. The best compromise between measuring time and data reliability was found by acquiring 64 data points and then using a biexponential model to fit the experimental data. The PG 64-22 sources provided similar results, indicating similar physical properties. The PG 64-22 and PG 76-22 binders could also be compared via monoexponential data fits, but the PG 94-10 samples required an additional relaxation parameter for comparison. To differentiate all three binder grades, the primary relaxation times, along with their relative ratios, were utilized.

A new solution approach to proportion delayed and heat like fractional partial differential equations

The importance of fractional partial differential equations (FPDEs) may be observed in many fields of science and engineering. On the same hand their solutions and the approaches for the same are also very important to notice due to the effectiveness of the methods and accuracy of the results. This work discusses the diverse estimated analytic description of fractional partial differential equations (with proportion delay and heat like equation), applying the Iterative Laplace Transform Method. The specified method represents a significant advancement in the tool case of applied mathematicians and scientists. Its ability to efficiently and accurately solve complex differential equations, especially FPDEs. Here in this work, the solution of four test problems of FPDEs related to proportion delay and heat like equations is obtained for testing the validity and asset of the Iterative Laplace Transform Method. Further their numerical and graphical interpretations are also mentioned.

Estimation of the surface impedance based on the discrete complex image source method and array processing

This paper introduces a method for estimating the sound absorption coefficient of a large panel of absorbing material from measurements using a microphone array. The method relies upon describing the pressure field as the set of monopoles obtained when the reflection coefficient is represented as the Laplace Transform of an image source distribution. This formulation results in a well-behaved integral equation, which is discretized using Gaussian quadrature to yield an inverse problem. It is possible to determine the regularized solution to the inverse problem and reconstruct the surface impedance of the sample by conducting measurements of the sound pressure at multiple points of observation. This sound field representation is adequate for sound sources in the close range of the material and the array. The paper presents simulated and experimental investigations to delineate the efficacy of the proposed measurement technique against the well-known image source method, thereby highlighting its significant potential to advance acoustic characterization methodologies.

Exact closed-form solution for buckling and free vibration of pipes conveying fluid with intermediate elastic supports

In this paper, the exact closed-form solution is given to investigate the influence of the intermediate elastic support on the buckling and free vibration of an elastically supported pipe. According to the Euler–Bernoulli beam theory, the mechanical model of the pipe is established. The exact equilibrium configuration is derived using the generalised function method without enforcing continuity conditions. A simple solution to the eigenvalue problem is formulated using the methods of complex mode superposition and Laplace transformation. The comparative study shows the differences in the supercritical vibration characteristics and highlights the limitations of previous studies. Parametric studies are carried out to investigate the influence of elastic support and intermediate support conditions on the equilibrium configuration, critical flow velocity, and natural frequency. The results demonstrate that the proposed closed-form solution can determine the support conditions that lead to the maximum critical flow velocity and natural frequency of a pipe with multiple intermediate supports. The maximum values are required to adjust the support conditions leading to the nodes of higher-order equilibrium configurations and complex modes. Furthermore, the natural frequencies of the pipe conveying supercritical fluid no longer satisfy the monotonicity for the support stiffness, the symmetry for the support position, and the ‘zero-point’ property for the support number.

Fault Location Method for MMC-HVDC Systems Based on Numerical Laplace Transform

Fault Location Method for MMC-HVDC Systems Based on Numerical Laplace Transform

Application of Laplace and Sumudu transforms to the modeling of unsteady rotating MHD flow in a second-grade tetra hybrid nanofluid in a porous medium

Understanding and managing the complex rheological behavior of fluid models is crucial for many real-world applications, such as industrial fluid dynamics, biomedical engineering, and environmental management. This study aims to predict the unsteady magnetohydrodynamic (MHD) flow of a second-grade tetra hybrid nanofluid in a porous medium containing uniformly dispersed dust particles. To address this challenge, we develop a mathematical model utilizing a non-fractional approach with ramping conditions. Our methodology incorporates Laplace and Sumudu transforms to analyze the velocity fields of both the fluid and dust particles. Additionally, we examine the heat transfer characteristics and the underlying dynamics of the magnetized second-grade tetra hybrid nanofluid flow. The momentum transfer of the dust particles is modeled using a separate equation. Key outcomes of this research include visual representations of novel findings, facilitating comparison with existing literature. Numerical simulations based on Sumudu and Laplace transformations produce velocity profiles for both fluid and dust particles, as well as temperature distributions, aligning with theoretical expectations. These results demonstrate the effectiveness of Laplace and Sumudu transforms in analyzing and simulating the turbulent MHD flow of nanofluids through porous media with dispersed dust particles.

Hybrid finite element and laplace transform method for efficient numerical solutions of fractional PDEs on graphics processing units

Fractional Partial Differential equations (FPDEs) are essential for modeling complex systems across various scientific and engineering areas. However, efficiently solving FPDEs presents significant computational challenges due to their inherent memory effects, often leading to increased execution times for numerical solutions. This study proposes a highly parallelizable hybrid computational approach that combines the Finite Element Method (FEM) for spatial discretization with Numerical Inversion of the Laplace Transform (NILT) for time-domain solutions, optimized for execution on Graphics Processing Units (GPUs). The NILT method’s high parallelizability, stemming from the independence of its series terms, combined with the robust spatial discretization provided by FEM, enables the efficient and accurate solution of FPDEs on GPUs, demonstrating substantial performance improvements over traditional CPU-based implementations. We observe a generalized pattern in execution time behavior that accounts for both the number of nodes and the number of NILT terms. Specifically, execution time scales quadratically with the number of nodes, while showing only a logarithmic marginal increase with the number of NILT terms These behaviors not only enables consistent performance assessment but also highlights potential areas for algorithm optimization. Validation against exact solutions of fractional diffusion and wave equations, employing Caputo, modified Caputo-Fabrizio, and modified Atangana-Baleanu derivatives, demonstrates the accuracy and convergence of the hybrid FEM-NILT method. Notably, the exact solutions of wave equation are novel in literature. The results highlight the method’s potential for enabling high-precision, large-scale simulations in fractional calculus applications, thereby advancing computational capabilities and efficiency in the field.

An Analytical Solution for Characterizing Mine Water Recharge of Water Source Heat Pump in Abandoned Coal Mines

Due to tremendous mining operations, large quantities of abandoned mines with considerable underground excavated space have formed in China during the past decades. This provides huge potential for geothermal energy production from mine water in abandoned coal mines to supply clean heating and cooling for buildings using heat pump technologies. In this study, an analytical model describing the injection pressure of mine water recharge for water source heat pumps in abandoned coal mines is developed. The analytical solution in the Laplace domain for the injection pressure is derived and the influences of different parameters on the injection pressure are investigated. This study indicates that a smaller pumping rate results in a smaller injection pressure, while smaller values of the hydraulic conductivity and the thickness of equivalent aquifer induce larger injection pressures. The well distance has insignificantly influenced the injection pressure at the beginning, but a smaller well distance leads to a larger injection pressure at later times. Additionally, the sensitivity analysis, conducted to assess the behavior of injection pressure with concerning changes in each input parameter, shows that the pumping rate and the hydraulic conductivity have a large influence on injection pressure compared with other parameters.

Multi-Channel Signals in Dynamic Force-Clamp Mode of Microcantilever Sensors for Detecting Cellular Peripheral Brush.

The development of numerous diseases, such as renal cyst, cancer, and viral infection, is closely associated with the pathological changes and defects in the cellular peripheral brush. Therefore, it is necessary to develop a potential new method to detect lesions of cellular peripheral brush. Here, a piecewise linear viscoelastic constitutive model of cell is established considering the joint contribution of the peripheral brush and intra-cellular structure. By combining the Laplace transformation and its inverse transformation, and the differential method in the temporal domain and differential quadrature method (DQM) in the spatial domain, the signal interpretation models for quasi-static and dynamic signals of microcantilever are solved. The influence mechanisms of the peripheral brush on the viscoelastic properties of cells and quasi-static/dynamic signals of microcantilever are clarified. The results not only reveal that the peripheral brush has significant effects on the complex modulus of the cell and multi-channel signals of the microcantilever, but also suggest that an alternative mapping method by collecting multi-channel signals including quasi-static and higher frequency signals with more brush indexes could be potentially used to identify cancerous cells.