Laplace Transform Method 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.

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.

Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel

Objectives: To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. Methods: Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. Findings: By solving a series of precise and understandable examples, the double Laplace transform clearly transforms the fractional partial integro-differential equation into an algebraic equation that can be solved easily. Novelty: The double Laplace transform is an effective instrument for solving fractional partial integro-differential equations, which are commonly used in fields such as physics, fluid mechanics, gas dynamics, and signal processing. Mathematics Subject Classifications: 35R09, 35R11, 44A10, 44A30. Keywords: Fractional PIDE, Weakly singular kernel, Double Laplace transform, Inverse double Laplace transform, Exact solution

Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

Exact Analysis of MHD Casson Fluid Flow Past an Exponentially Accelerated Vertical Plate in a Porous Medium with Radiation Absorption, Heat Generation, and Diffusion-Thermo Effects with Thermal and Solutal Ramped Conditions

The current investigation aims at to examine the effect of radiation absorption, heat generation and Dufour number on MHD Casson fluid flow past an exponentially accelerated vertical plate in a porous medium with chemical reaction. The governing equations for momentum, energy and concentration are solved by implementing the Laplace transformation method. Skin friction, rate of heat transfer and rate of mass transfer expressions are also extracted and depicted in tabular form. Investigation simulates that Casson parameter diminished the fluid velocity, whereas energy flux due to a mass concentration gradient improves the temperature field of the flow problem. In addition to this, temperature field is observed to be developed under the influence of radiation absorption and heat generation. Furthermore, the effects of different non-dimensional parameters on velocity field, temperature fluid and species concentration are exhibited graphically.

A fast and reliable semi-analytical method for assessing energy replenishment from fracturing-flooding in low-permeability and tight oil reservoirs

The development of low-permeability and tight oil reservoirs is challenged by insufficient natural energy and rapid production decline. Fracturing-flooding is a technique that relies on high-pressure and large-volume fluid injection to replenish reservoir energy, making it a significant method for rapidly boosting formation energy. To evaluate the energy replenishment effect of fracturing-flooding technology in low-permeability and tight reservoirs, this study proposes a semi-analytical method for quick calculation. This approach employs dimensionless simplification, Pedrosa's substitution, Laplace transformation, and Stehfest inversion methods to derive pressure solutions for both the stimulation region and the external matrix region, each with varying flow capacities. The average formation pressure (AFP) of the reservoir is determined using the area-weighted average method, and numerical verification is performed using a commercial simulator. A case study from the Binnan area, along with a sensitivity analysis, demonstrates that after 30 days of fracturing-flooding, the AFP of the reservoir increases to 46.97 MPa, the corresponding reservoir pressure coefficient rises from 1.2 to 1.68, and reservoir energy increases by 40%. The factors influencing energy replenishment are ranked as follows: reservoir thickness, injection rate, stress sensitivity coefficient, matrix permeability, stimulation region radius, and mobility ratio. This study provides theoretical guidance for optimizing fracturing-flooding development schemes in low-permeability and tight oil reservoirs and offers valuable reference for the industry.

Analytical Solution of Fractional Order Diffusion Equations Using Iterative Laplace Transform Method

In this present article, by using the Iterative Laplace Transform Method (ILTM), the diffusion equation of fractional order is solved. The ILTM, which works as a combination of two methods, the iterative method and the other is the Laplace transform method, is applied to several diffusion equations to obtain analytical solutions. The proposed method gives the closed-form of series solutions in terms of the Mittag-Leffler function, which is a queen of functions in fractional calculus. The main aim of this work is to present a simple but reliable algorithm for the solution of diffusion equations of the multi-dimensional type, which clearly describes the materials of density dynamics in the diffusion process. The results obtained by using the ILTM approach indicate that this approach is attractive computationally and implemented easily. Due to its straightforward approach and comfortable way of solving problems, the ILTM can be utilized to solve nonlinear fractional problems in various applied and engineering sciences.

Influence of chemical reaction on transient natural convective flow in an infinite vertical cylinder packed with porous material

A study of mass and heat transport on fully developed laminar and transient free convective flow in a vertical cylinder containing a porous substance inside it. The Laplace Transform Method (LTM) is used to solve the governing equations with suitable boundary conditions. A compact form solution represented by using Bessel and modified Bessel functions. The numerical results were thoroughly scrutinized using graphs and tables to illustrate the effects of newly discovered fluid characteristics, including time, Grashof, mass Grashof, Schmidt, Prandtl, Darcy numbers, viscosity ratio on the concentration and the velocity profiles. The velocity and concentration profiles progressively increase with varying time values until reaching a steady state. Furthermore, it is found that the velocity profiles are decreased by the Prandtl, Schmidt number, and viscosity ratio. In contrast, the Darcy, Grashof, and mass Grashof numbers exhibit an opposite effect. The rate of mass transport, skin friction, and mass flux are presented in tables. The major finding of the current research is that the viscosity ratio parameter decelerates the skin friction and mass flux. This study has potential applications in innovative fields such as solar energy systems, chemical engineering processes, etc.

A Critical Look at the Laplace Transform Method in Engineering Education

A Critical Look at the Laplace Transform Method in Engineering Education

A Novel Approach to Solve Nonlinear Higher Order VFIDE Using the Laplace Transform and Adomian Decomposition Method

A Novel Approach to Solve Nonlinear Higher Order VFIDE Using the Laplace Transform and Adomian Decomposition Method

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.

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

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

Semi-analytical solutions for forced and free vibration of damped fluid-conveying pipe systems based on complex modal superposition method

In this paper, the response solutions of the damped fluid-conveying pipe system with elastic torsion constraints at both ends are analyzed. The pipe system considering gyroscopic effect induced by internal flow and damping effect is a typical damped gyroscopic system. This system cannot be decoupled in the modal space by the traditional modal analysis, and then the semi-analytical response solutions cannot be obtained by the classical modal superposition method. In order to remedy this problem, this paper proposes a new method based on complex modal superposition method and state-space method to give the semi-analytical response solutions of the damped fluid-conveying pipe system. The adjoint system is introduced through the concept of adjoint operator, and the orthogonality conditions of the damped fluid-conveying pipe system are derived in the state-space by using the eigenvalue relationship between the original system and the adjoint system. The Laplace transformation method is used to solve the eigenvalue problems of the original system and the adjoint system, and the implicit function equations about the eigenvalues and analytical eigenfunctions are obtained. According to the obtained orthogonality conditions, the semi-analytical response solutions of the system under arbitrary initial conditions and excitations are given, and the obtained solutions include transient and steady-state response solutions. In the numerical discussion part, the reliability and accuracy of the present method are verified. This research has positive significance for the dynamic analysis of fluid-conveying pipe systems, and the obtained orthogonality conditions have potential value for the nonlinear dynamics research.

Application of Caputo fractional approach to MHD Casson nanofluid with alumina nanoparticles of various shape factors on an inclined quadratic translated plate

The present study aims to examine the effect of convective heat transfer of nanofluid past an inclined plate. This paper scrutinizes heat transfer over an inclined surface of alumina nanofluids under the influence of radiation and magnetic fields with temperature-dependent thermophysical properties. The mathematical findings of velocity and temperature distributions are examined and visualized using non-dimensional flow parameters. The core of the research is to know the impact of using shape effects of alumina nanoparticles with a variety of base fluids like water, ethylene glycol, and engine oil. The outcomes concluded that the friction of the non-Newtonian viscous fluid decreased with increasing the inclination of the surface. The governing equations of the flow have been developed by using a few approximations such as Boussinesq's, and Roseland's. Using the dimensional analysis, the initial governing equations are simplified to dimensionless partial differential equations. A fractional model is developed using the Caputo fractional derivative. The solution of the proposed fractional partial differential equations is obtained in terms of special functions, Wright function, and Mittag-Leffler function by using the Laplace transformation method. Based on the interpretation, it is found that a reduction in velocity is observed with an increase in the angle of inclination i.e., γ = π/3 to γ = π/2. From the analysis, it is found that alumina/engine oil can carry higher temperatures than compared to other nanofluids and thus is a good option for heat transfer devices.

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

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

Approximate Solution of Linear Fuzzy Random Ordinary Differential Equations Using Laplace Variational Iteration Method

In this article, the Laplace transformation method in connection with the variational iteration method will be used to solve approximately fuzzy random ordinary differential equations. After that, the sequence of approximated closed form iterated solutions is derived based on the general Lagrange multiplier evaluated using the well-known convolution theorem of the Laplace transformation method. In addition, two examples are given and solved to illustrate the reliability, efficiency and applicability of the proposed method, they are simulated using computer programs with two different generations of stochastic processes, namely the Wiener process or Brownian motion, which are 1000 and 10000, respectively.

Effect of hydrocephalus in an unsteady cerebrospinal fluid influencing diffusivity

The present study proposes a mathematical framework based on in vivo inputs to compare the dynamic interactions of cerebrospinal fluid (CSF) through an asymmetric channel due to hydrocephalus. This study’s primary goal is to determine the significance of diffusivity in cerebrospinal fluid concerning the rate of fluid velocity for the cardiac cycle. The model of governing equations was simulated by applying a suitable number of dimensionless parameters. The dimensionless governing equations are solved through the classical Laplace transformation method. The impacts of numerous variables over velocity, temperature, and concentration are examined. These variables comprise the porosity parameter, resistance parameter, buoyancy force, and Schmidt number. These plots are sketched for the considerable magnitude of these variables through MATLAB software, and the plots are discussed in detail. The present study shows a decent justification for the neurological papers showing an increase in pressure reflects the velocity for an increase in resistance, porosity, and the Schmidt number shows an inverse in diffusivity, as elaborated in the tabular outline. It was justified that rigorous cerebrospinal fluid velocity for hydrocephalus proved more than the result in ref. [1]. That is, according to ref. [1] the velocity of hydrocephalic patient is 10 mm/s; we proved it is more than 10 mm/s using analytical Laplace transform method.

Application of machine learning method to estimate bending properties of advanced composite elastic system subjected to external mechanical loading

This work introduces the novel analysis of the bending behavior of a composite doubly curved panel made of graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) subjected to external mechanical force. The motivation for this study stems from the many applications of such structures in the field of aeronautics. These materials have mechanical and structural properties that continuously vary. This study is the first to examine the effects of an instantaneous external shock on the bending response of a composite doubly curved panel made of GOEAMs due to the extraordinary properties of these advanced materials. The mathematical governing equations are obtained from the higher-order shear deformation theory and are solved using the Laplace transform method (LTM) and the analytical solution procedure (ASP). After solving the equations, the results of the current system are compared to those of a previously published work, revealing a significant level of concurrence between the two sets of data. This study presents a machine learning approach that utilizes a deep neural network (DNN) with input, hidden, and output layers, as well as independent variables and other relevant factors. The aim is to provide an efficient computational technique for solving engineering issues. This is achieved by the use of mathematical modeling and the verification of the current output’s outcomes. Additionally, a database is supplied for an in-depth examination of this structure’s suitability for aeronautical purposes. The database contains comprehensive bending information, including normal, shear, and displacement fields in various directions, specifically for the GOEAMs subjected to external shock loading.

Numerical treatment for time fractional order phytoplankton-toxic phytoplankton-zooplankton system

<abstract><p>The study of time-fractional problems with derivatives in terms of Caputo is a recent area of study in biological models. In this article, fractional differential equations with phytoplankton-toxic phytoplankton-zooplankton (PTPZ) system were solved using the Laplace transform method (LTM), the Adomain decomposition method (ADM), and the differential transform method (DTM). This study demonstrates the good agreement between the results produced by using the specified computational techniques. The numerical results displayed as graphs demonstrate the accuracy of the computational methods. The approaches that have been established are thus quite relevant and suitable for solving nonlinear fractional models. Meanwhile, the impact of changing the fractional order of a time derivative and time $ t $ on populations of phytoplankton, toxic-phytoplankton, and zooplankton has been examined using graphical representations. Furthermore, the stability analysis of the LTM approach has been discussed.</p></abstract>