Статті в журналах: "Equation laplace"

1

Zaki, Ahmad, Syafruddin Side, and N. Nurhaeda. "Solusi Persamaan Laplace pada Koordinat Bola." Journal of Mathematics, Computations, and Statistics 2, no. 1 (May 12, 2020): 82. http://dx.doi.org/10.35580/jmathcos.v2i1.12462.

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Penelitian ini mengkaji mengenai persamaan Laplace pada koordinat bola dan menerapkan metode pemisahan variabel dalam menentukan solusi persamaan Laplace Persamaan Laplace merupakan salah satu jenis persamaan diferensial parsial yang banyak digunakan untuk memodelkan permasalahan dalam bidang sains. Bentuk umum persamaan Laplace pada dimensi tiga dimana adalah fungsi skalar dengan menggunakan metode pemisahan variable diperoleh persamaan Laplace dimensi tiga pada koordinat bola. Hasil penelitian ini mendapatkan penyelesaian persamaan Laplace pada koordinat bola dalam bentuk variabel terpisah dengan tidak menggunakan nilai batas. Hubungan koordinat kartesian dan koordinat bola pada persamaan Laplace dapat ditentukan dalam persamaan Laplace dan memperoleh solusi dengan menggunakan koordinat bola.Kata Kunci: Koordinat Bola, Pemisahan Variabel, dan Persamaan Laplace. This study examines Laplace equations on spherical coordinates and applies variable separation methods in determining Laplace equation solutions Laplace equations are one type of partial differential equation that is widely used to model problems in the field of science. The general form of the Laplace equation in the third dimension in which u is a scalar function using the separation method of the variable is obtained by the third dimension Laplace equation on spherical coordinates. The result of this research get solution of Laplace equation on spherical coordinate in the form of separate variable by not using boundary value. The relationship of cartesian coordinates and spherical coordinates to the Laplace equation can be determined in the Laplace equation and obtain solutions using spherical coordinates.Keywords: Spherical Coordinat Variabel Separation, and Laplace Equation.

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2

Sanusi, Wahidah, Syafruddin Side, and Beby Fitriani. "Solusi Persamaan Transport dengan Menggunakan Metode Dekomposisi Adomian Laplace." Journal of Mathematics, Computations, and Statistics 2, no. 2 (May 12, 2020): 173. http://dx.doi.org/10.35580/jmathcos.v2i2.12580.

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Abstrak. Penelitian ini mengkaji terbentuknya persamaan Transport dan menerapkan metode Dekomposisi Adomian Laplace dalam menentukan solusi persamaan Transport. Persamaan transport merupakan salah satu bentuk dari persamaan diferensial parsial. Bentuk umum persamaan Transport yaitu: Metode Dekomposisi Adomian Laplace merupakan kombinasi antara dua metode yaitu metode dekomposisi adomian dan transformasi laplace. Penyelesaian persamaan Transport dengan metode Dekomposisi Adomian Laplace dilakukan dengan cara menggunakan tranformasi Laplace, mensubstitusi nilai awal, menyatakan solusi dalam bentuk deret tak hingga dan menggunakan invers transformasi laplace . Metode ini juga merupakan metode semi analitik untuk menyelesaikan persamaan diferensial nonlinier. Berdasarkan hasil perhitungan, metode dekomposisi Adomian Laplace dapat menghampiri penyelesaian persamaan diferensial biasa nonlinear.Kata Kunci: Metode Dekomposisi Adomian Laplace, Persamaan Diferensial Parsial, Persamaan Transport.This research discusses the solving of Transport equation applying Laplace Adomian Decomposition Method. Transport equation is one form of partial differential equations. General form of Transport equation is: Laplace Adomian Decomposition Method that combine between Laplace transform and Adomian Decomposition Method. The steps used to solve Transport equation are applying Laplace transform, initial value substitution, defining a solution as infinite series, then using the inverse Laplace transform. This method is a semi analytical method to solve for nonlinear ordinary differential equation. Based on the calculation results, the Laplace Adomian decomposition method can solve the solution of nonlinear ordinary differential equation.Keywords: Laplace Adomian Decomposition Method, Partial Differential Equation, Transport Equation.

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3

Shabestari, R. Mastani, and R. Ezzati. "The Fuzzy Double Laplace Transforms and their Properties with Applications to Fuzzy Wave Equation." New Mathematics and Natural Computation 17, no. 02 (April 23, 2021): 319–38. http://dx.doi.org/10.1142/s1793005721500174.

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The main focus of this paper is develop of the fuzzy double Laplace transform to solve a fuzzy wave equation. In this scheme, a fuzzy wave equation can be solved without converting it to two crisp equations. Some properties of the fuzzy Laplace transform and the fuzzy double Laplace transform are proved. The superiority and accuracy of the fuzzy double Laplace transform to wave equation are illustrated through some examples.

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4

Abdy, Muhammad, Syafruddin Side, and Reza Arisandi. "Penerapan Metode Dekomposisi Adomian Laplace Dalam Menentukan Solusi Persamaan Panas." Journal of Mathematics, Computations, and Statistics 1, no. 2 (May 19, 2019): 206. http://dx.doi.org/10.35580/jmathcos.v1i2.9243.

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Abstrak. Artikel ini membahas tentang penerapan Metode Dekomposisi Adomian Laplace (LADM) dalam menentukan solusi persamaan panas. Metode Dekomposisi Adomian Laplace merupakan metode semi analitik untuk menyelesaikan persamaan diferensial nonlinier yang mengkombinasikan antara tranformasi Laplace dan metode dekomposisi Adomian. Berdasarkan hasil perhitungan, metode dekomposisi Adomian Laplace dapat menghampiri penyelesaian persamaan diferensial biasa nonlinear.Kata kunci: Metode Dekomposisi Adomian Laplace, Persamaan Diferensial Parsial, Persamaan PanasAbstract. This study discusses the application of Adomian Laplace Decomposition Method (ALDM) in determining the solution of heat equation. Adomian Laplace Decomposition Method is a semi analytical method to solve nonlinear differential equations that combine Laplace transform and Adomian decomposition method. Based on the calculation result, Adomian Laplace decomposition method can approach the settlement of ordinary nonlinear differential equations.Keywords: Adomian Laplace Decomposition Method, Partial Differential Equation, Heat Equation.

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5

Nathiya, N., and C. Amulya Smyrna. "Infinite Schrödinger networks." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 640–50. http://dx.doi.org/10.35634/vm210408.

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Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

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6

Rozumniuk, V. I. "About general solutions of Euler’s and Navier-Stokes equations." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2019): 190–93. http://dx.doi.org/10.17721/1812-5409.2019/1.44.

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Constructing a general solution to the Navier-Stokes equation is a fundamental problem of current fluid mechanics and mathematics due to nonlinearity occurring when moving to Euler’s variables. A new transition procedure is proposed without appearing nonlinear terms in the equation, which makes it possible constructing a general solution to the Navier-Stokes equation as a combination of general solutions to Laplace’s and diffusion equations. Existence, uniqueness, and smoothness of the solutions to Euler's and Navier-Stokes equations are found out with investigating solutions to the Laplace and diffusion equations well-studied.

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7

Kamran, Sharif Ullah Khan, Salma Haque, and Nabil Mlaiki. "On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method." Symmetry 15, no. 6 (June 7, 2023): 1214. http://dx.doi.org/10.3390/sym15061214.

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Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed.

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8

Kogoj, Alessia E., and Ermanno Lanconelli. "On semilinear -Laplace equation." Nonlinear Analysis: Theory, Methods & Applications 75, no. 12 (August 2012): 4637–49. http://dx.doi.org/10.1016/j.na.2011.10.007.

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9

Lu, Guozhen, and Peiyong Wang. "Inhomogeneous infinity Laplace equation." Advances in Mathematics 217, no. 4 (March 2008): 1838–68. http://dx.doi.org/10.1016/j.aim.2007.11.020.

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10

Shokhanda, Rachana, Pranay Goswami, Ji-Huan He, and Ali Althobaiti. "An Approximate Solution of the Time-Fractional Two-Mode Coupled Burgers Equation." Fractal and Fractional 5, no. 4 (November 4, 2021): 196. http://dx.doi.org/10.3390/fractalfract5040196.

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In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into a series of linear equations, which can be effectively and easily solved by the Laplace transform. The solution process is illustrated step by step, and the results show that the present method is extremely powerful for fractional differential equations.

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11

Abdillah, Muhammad Taufik, Berlian Setiawaty, and Sugi Guritman. "The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (July 17, 2023): 631. http://dx.doi.org/10.31764/jtam.v7i3.14193.

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Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used.

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12

Tokibetov, Zh A., N. E. Bashar, and А. К. Pirmanova. "THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (December 29, 2020): 68–72. http://dx.doi.org/10.51889/2020-4.1728-7901.10.

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For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

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13

Wang, Jian, Kamran, Ayesha Jamal, and Xuemei Li. "Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative." Mathematical Problems in Engineering 2021 (February 12, 2021): 1–8. http://dx.doi.org/10.1155/2021/6662808.

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In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

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14

Shim, Sang Oh, Tae Hwa Jung, Sang Chul Kim, and Ki Chan Kim. "Finite Element Model for Laplace Equation." Applied Mechanics and Materials 267 (December 2012): 9–12. http://dx.doi.org/10.4028/www.scientific.net/amm.267.9.

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The mild-slope equation has widely been used for calculation of shallow water wave transformation. Recently, its extended version was introduced, which is capable of modeling wave transformation on rapidly varying topography. These equations were derived by integrating the Laplace equation vertically. Here, we develop a finite element model to solve the Laplace equation directly while keeping the same computational efficiency as the mild-slope equation. This model assumes the vertical variation of the wave potential as a cosine hyperbolic function as done in the derivation of the mild-slope equation, and the Galerkin method is used to discretize it. The computational domain is discretized with proper finite elements, while the radiation condition at infinity is treated by introducing the concept of an infinite element. The upper boundary condition can be either free surface or a solid structure. The applicability of the developed model is verified through example analyses of two-dimensional wave reflection and transmission. Analysis is also made for the case where a solid structure is floated near the still water level.

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15

Kim, Boram, Kwang Seok Yoon, and Hyung-Jun Kim. "GPU-Accelerated Laplace Equation Model Development Based on CUDA Fortran." Water 13, no. 23 (December 4, 2021): 3435. http://dx.doi.org/10.3390/w13233435.

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In this study, a CUDA Fortran-based GPU-accelerated Laplace equation model was developed and applied to several cases. The Laplace equation is one of the equations that can physically analyze the groundwater flows, and is an equation that can provide analytical solutions. Such a numerical model requires a large amount of data to physically regenerate the flow with high accuracy, and requires computational time. These numerical models require a large amount of data to physically reproduce the flow with high accuracy and require computational time. As a way to shorten the computation time by applying CUDA technology, large-scale parallel computations were performed on the GPU, and a program was written to reduce the number of data transfers between the CPU and GPU. A GPU consists of many ALUs specialized in graphic processing, and can perform more concurrent computations than a CPU using multiple ALUs. The computation results of the GPU-accelerated model were compared with the analytical solution of the Laplace equation to verify the accuracy. The computation results of the GPU-accelerated Laplace equation model were in good agreement with the analytical solution. As the number of grids increased, the computational time of the GPU-accelerated model gradually reduced compared to the computational time of the CPU-based Laplace equation model. As a result, the computational time of the GPU-accelerated Laplace equation model was reduced by up to about 50 times.

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16

Yang, Lufeng. "The Rational Spectral Method Combined with the Laplace Transform for Solving the Robin Time-Fractional Equation." Advances in Mathematical Physics 2020 (January 9, 2020): 1–7. http://dx.doi.org/10.1155/2020/9865682.

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In this paper, the rational spectral method combined with the Laplace transform is proposed for solving Robin time-fractional partial differential equations. First, a time-fractional partial differential equation is transformed into an ordinary differential equation with frequency domain components by the Laplace transform. Then, the spatial derivatives are discretized by the rational spectral method, the linear equation with the parameter s is solved, and the approximation Ux,s is obtained. The approximate solution at any given time, which is the numerical inverse Laplace transform, is obtained by the modified Talbot algorithm. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.

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17

Jafarian, Ahmad, Alireza Khalili Golmankhaneh, and Dumitru Baleanu. "On Fuzzy Fractional Laplace Transformation." Advances in Mathematical Physics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/295432.

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Fuzzy and fractional differential equations are used to model problems with uncertainty and memory. Using the fractional fuzzy Laplace transformation we have solved the fuzzy fractional eigenvalue differential equation. By illustrative examples we have shown the results.

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18

MAHALLE, VIDYA N., SWATI S. MATHURKAR, and RAJENDRA D. TAYWADE. "SOME NEW APPLICATIONS OF LAPLACE-WEIERSTRASS TRANSFORM." Journal of Science and Arts 21, no. 1 (March 3, 2021): 15–20. http://dx.doi.org/10.46939/j.sci.arts-21.1-a02.

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This paper is devoted to some new applications of Laplace-Weierstrass transform (i.e., for solving two dimensional diffusion equations). Solution of Cauchy’s linear differential equation is also given. Some results are also given which are required for solving Cauchy’s linear differential equation.

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19

Korpinar, Zeliha. "On numerical solutions for the Caputo-Fabrizio fractional heat-like equation." Thermal Science 22, Suppl. 1 (2018): 87–95. http://dx.doi.org/10.2298/tsci170614274k.

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In this article, Laplace homotopy analysis method in order to solve fractional heat-like equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficient for solving the fractional heat-like equations with variable coefficients.

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20

Bieske, Thomas, and Keller Blackwell. "Generalizations of the drift Laplace equation in the Heisenberg group and Grushin-type spaces." Electronic Journal of Differential Equations 2021, no. 01-104 (December 20, 2021): 99. http://dx.doi.org/10.58997/ejde.2021.99.

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We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations of the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on the p-Laplace-type equation while we primarily concentrate on a generalization of the drift term. For more information see https://ejde.math.txstate.edu/Volumes/2021/99/abstr.html

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21

Abdy, Muhammad, Maya Sari Wahyuni, and Narisa Fahira Awaliyah. "Solusi Persamaan Adveksi-Difusi dengan Metode Dekomposisi Adomian Laplace." Journal of Mathematics Computations and Statistics 5, no. 1 (May 1, 2022): 40. http://dx.doi.org/10.35580/jmathcos.v5i1.32249.

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Artikel ini membahas tentang solusi dari persamaan adveksi-difusi. Persamaan adveksi-difusi merupakan persamaan matematis yang didesain untuk mempelajari fenomena transpor polutan. Pada artikel ini, metode yang digunakan untuk menentukan solusi persamaan adveksi-difusi yaitu metode dekomposisi Adomian-Laplace. Metode dekomposisi Adomian Laplace adalah salah satu metode yang dapat digunakan untuk menyelesaikan persamaan diferensial yang mengkombinasikan metode transformasi Laplace dan metode dekomposisi Adomian. Solusi persamaan adveksi-difusi diperoleh dengan menerapkan tranformasi laplace pada persamaan adveksi-difusi, mensubtitusi syarat awal, menyatakan solusi dalam bentuk deret tak hingga, menentukan suku-sukunya, dan menerapkan invers transformasi Laplace pada suku-suku dari deret tak hingga tersebut. Hasil dari tulisan ini adalah solusi persamaan adveksi-difusi dapat diperoleh dengan metode dekomposisi Adomian Laplace.Kata Kunci: Persamaan Diferensial, Persamaan Adveksi-Difusi, Metode Dekomposisi Adomian Laplace.This paper discusses about the solution of advection-diffusion equation. The advection-diffusion equation is a mathematical equation designed to study the phenomenon of pollutant transport. This paper is using Laplace Adomian Decomposition method to solve the advection-diffusion equation. The Laplace Adomian decomposition method is one of method which can be used to solve a differential equation that combines Laplace transform method and Adomian decomposition method. The solution is obtained by applying the Laplace transform to the advection-diffusion equation, substituting the initial conditions, converting the solution into the form of an infinite series, determining the terms, and applying the inverse Laplace transform to the terms of the infinite series. The results of this paper is the advection-diffusion equation can be solved by using Adomian Laplace decomposition method.Keywords: Differential Equation, Advection-Diffusion Equation, Laplace Adomian Decomposition Method.

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Kılıçman, Adem, and Wasan Ajeel Ahmood. "On matrix fractional differential equations." Advances in Mechanical Engineering 9, no. 1 (January 2017): 168781401668335. http://dx.doi.org/10.1177/1687814016683359.

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The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.

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23

Gan, Jiarong, Hong Yuan, Shanqing Li, Qifeng Peng, and Huanliang Zhang. "A computing method for bending problem of thin plate on Pasternak foundation." Advances in Mechanical Engineering 12, no. 7 (July 2020): 168781402093933. http://dx.doi.org/10.1177/1687814020939333.

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The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

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Mehrazin, Hashem. "Laplace equation and Poisson integral." Applied Mathematics and Computation 145, no. 2-3 (December 2003): 451–63. http://dx.doi.org/10.1016/s0096-3003(02)00499-x.

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25

Brustad, Karl K. "Superposition in thep-Laplace equation." Nonlinear Analysis 158 (July 2017): 23–31. http://dx.doi.org/10.1016/j.na.2017.04.004.

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26

Xiong, Jie. "A stochastic log-Laplace equation." Annals of Probability 32, no. 3B (July 2004): 2362–88. http://dx.doi.org/10.1214/009117904000000540.

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Sedeeg, Abdelilah Kamal, Zahra I. Mahamoud, and Rania Saadeh. "Using Double Integral Transform (Laplace-ARA Transform) in Solving Partial Differential Equations." Symmetry 14, no. 11 (November 15, 2022): 2418. http://dx.doi.org/10.3390/sym14112418.

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The main goal of this research is to present a new approach to double transforms called the double Laplace–ARA transform (DL-ARAT). This new double transform is a novel combination of Laplace and ARA transforms. We present the basic properties of the new approach including existence, linearity and some results related to partial derivatives and the double convolution theorem. To obtain exact solutions, the new double transform is applied to several partial differential equations such as the Klein–Gordon equation, heat equation, wave equation and telegraph equation; each of these equations has great utility in physical applications. In symmetry to other symmetric transforms, we conclude that our new approach is simpler and needs less calculations.

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Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (December 5, 2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.83.

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Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

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29

Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (December 5, 2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.83.

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Анотація:

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

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30

Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (December 2, 2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.91.

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Анотація:

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

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31

Bracken, Paul. "Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the Sinh-Laplace equation." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1393–404. http://dx.doi.org/10.1155/ijmms.2005.1393.

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The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface theory are found to yield a type of geometrically based Lax pair for the system. Given a particular solution of the sinh-Laplace equation, this Lax can be integrated to determine the three fundamental vectors related to the surface. These are also used to determine the coordinate vector of the surface. Some specific examples of this procedure will be given.

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32

Khan, Hassan, Rasool Shah, Poom Kumam, Dumitru Baleanu, and Muhammad Arif. "An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations." Mathematics 7, no. 5 (May 13, 2019): 426. http://dx.doi.org/10.3390/math7050426.

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In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.

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33

Taber, Larry A. "A Theory for Transverse Deflection of Poroelastic Plates." Journal of Applied Mechanics 59, no. 3 (September 1, 1992): 628–34. http://dx.doi.org/10.1115/1.2893770.

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A theory is presented for the bending of fluid-saturated poroelastic plates. The governing equations, based on linear consolidation theory, reduce to a single fourth-order integro-partial-differential equation to be solved for the transverse displacement of the middle surface. This equation resembles the classical plate equation but has an added convolution integral, which represents the viscous losses due to the flow of fluid relative to the solid. Laplace transform and perturbation solution methods are presented. The Laplace-transformed poroelastic plate equation and the first-order equation of the perturbation expansion have the forms of the standard plate equation. Results are given for a simply-supported rectangular plate with a time-dependent surface pressure.

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34

Kiliçman, Adem, and Hassan Eltayeb. "Some Remarks on the Sumudu and Laplace Transforms and Applications to Differential Equations." ISRN Applied Mathematics 2012 (February 1, 2012): 1–13. http://dx.doi.org/10.5402/2012/591517.

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Анотація:

We study the relationship between Sumudu and Laplace transforms and further make some comparison on the solutions. We provide some counterexamples where if the solution of differential equations exists by Laplace transform, the solution does not necessarily exist by using the Sumudu transform; however, the examples indicate that if the solution of differential equation by Sumudu transform exists then the solution necessarily exists by Laplace transform.

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35

Belhallaj, Zineb, Said Melliani, M’hamed Elomari, and Lalla Saadia Chadli. "Solving Intuitionistic Fuzzy Transport Equations by Intuitionistic Fuzzy Laplace Transforms." ITM Web of Conferences 43 (2022): 01021. http://dx.doi.org/10.1051/itmconf/20224301021.

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In this paper, we use an intuitionistic fuzzy Laplace transforms for solving intuitionistic fuzzy hyperbolic equations precisely the transport equation with intuitionistic fuzzy data under strongly generalized H-differentiability concept. For this purpose, the intuitionistic fuzzy transport equation is converted to the intuitionistic fuzzy boundary value problem (IFBVP) based on the intuitionistic fuzzy laplace transform. The related theorems and properties are proved in detail. Finally, we solve an example to illustrate this method.

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36

Seddighi Chaharborj, Sarkhosh, Shahriar Seddighi Chaharborj, Zahra Seddighi Chaharborj, and Pei See Phang. "Theoretical fractional formulation of a three-dimensional radio frequency ion trap (Paul-trap) for optimum mass separation." European Journal of Mass Spectrometry 27, no. 2-4 (July 4, 2021): 73–83. http://dx.doi.org/10.1177/14690667211026790.

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We investigate the dynamics of an ion confined in a Paul–trap supplied by a fractional periodic impulsional potential. The Cantor–type cylindrical coordinate method is a powerful tool to convert differential equations on Cantor sets from cantorian–coordinate systems to Cantor–type cylindrical coordinate systems. By applying this method to the classical Laplace equation, a fractional Laplace equation in the Cantor–type cylindrical coordinate is obtained. The fractional Laplace equation is solved in the Cantor–type cylindrical coordinate, then the ions is modelled and studied for confined ions inside a Paul–trap characterized by a fractional potential. In addition, the effect of the fractional parameter on the stability regions, ion trajectories, phase space, maximum trapping voltage, spacing between two signals and fractional resolution is investigated and discussed.

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37

Yuan, Hong Fen, and Valery V. Karachik. "DUNKL-POISSON EQUATION AND RELATED EQUATIONS IN SUPERSPACE." Mathematical Modelling and Analysis 20, no. 6 (November 23, 2015): 768–81. http://dx.doi.org/10.3846/13926292.2015.1112856.

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Abstract In this paper, we investigate the Almansi expansion for solutions of Dunkl-polyharmonic equations by the 0-normalized system for the Dunkl-Laplace operator in superspace. Moreover, applying the 0-normalized system, we construct solutions to the Dunkl-Helmholtz equation, the Dunkl-Poisson equation, and the inhomogeneous Dunkl-polyharmonic equation in superspace.

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38

Bonazebi-Yindoula, Joseph. "Laplace-SBA Method for Solving Nonlinear Coupled Burger's Equations." European Journal of Pure and Applied Mathematics 14, no. 3 (August 5, 2021): 842–62. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3932.

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Burgerâ€™s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-SomÃ© Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.

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39

Anderson, Michael L., Andrew P. Bassom, and Neville Fowkes. "Exact solutions of the Laplace–Young equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2076 (June 27, 2006): 3645–56. http://dx.doi.org/10.1098/rspa.2006.1744.

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The solution of the Laplace–Young equation determines the equilibrium height of the free surface of a liquid contained in a vessel under the action of gravity and surface tension. There are only two non-trivial exact solutions known; one corresponds to a liquid occupying a semi-infinite domain bounded by a vertical plane wall while the other relates to the case when the liquid is constrained between parallel walls. A technique called boundary tracing is introduced; this procedure allows one to modify the geometry of the domain so that both the Laplace–Young equation continues to be satisfied while the necessary contact condition on the boundary remains fulfilled. In this way, new solutions of the equation are derived and such solutions can be found for certain boundaries with one or more sharp corners and for others that possess small-scale irregularities that can be thought of as a model for roughness. The method can be extended to construct new solutions for a variety of other physically significant partial differential equations.

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40

Moazzam, Ali, Zainab Ijaz, Muhammad Hussain, Nimra Maqbool, and Emad A. Kuffi. "Applications of Fractional-Laplace Transformation in the Field of Electrical Engineering." Journal of Kufa for Mathematics and Computer 10, no. 2 (August 31, 2023): 70–75. http://dx.doi.org/10.31642/jokmc/2018/100211.

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Анотація:

This study examines the various ways that fractional Laplace transform can be used to solve three different kinds of mathematical equations: the equation of analysis of electric circuits, simultaneous differential equations, and the heat conduction equation. This article how to use the fractional Laplace transform to calculate heat flow in semi-infinite solids in the context of heat conduction. The answers that are developed offer important information about how temperatures vary across time and space. The essay also examines how to analyse electrical circuits using the Fractional Laplace transform. This method allows researchers to measure significant electrical parameters including charge and current, which improves their comprehension of circuit dynamics. Practical examples are included throughout the essay to show how useful the Fractional Laplace transform is in various fields. As a result of the answers found using this methodology, researchers and engineers working in the fields of heat conduction, system dynamics, and circuit analysis can gain important new knowledge. In conclusion, this study explains the applicability and effectiveness of the fractional Laplace transform in resolving a variety of mathematical equations. It is a vital tool for researchers because it may be used in a wide range of scientific and engineering areas.

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41

Bosch, Paul, Héctor José Carmenate García, José Manuel Rodríguez, and José María Sigarreta. "On the Generalized Laplace Transform." Symmetry 13, no. 4 (April 13, 2021): 669. http://dx.doi.org/10.3390/sym13040669.

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In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.

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42

Deng, Shuxian, and Xinxin Ge. "Approximate analytical solution for Phi-four equation with He’s fractal derivative." Thermal Science 25, no. 3 Part B (2021): 2369–75. http://dx.doi.org/10.2298/tsci191231127d.

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Анотація:

This paper, for the first time ever, proposes a Laplace-like integral transform, which is called as He-Laplace transform, its basic properties are elucidated. The homotopy perturbation method coupled with this new transform becomes much effective in solving fractal differential equations. Phi-four equation with He?s derivative is used as an example to reveal the main merits of the present technology.

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43

Lazovskaya, Tatyana V., Dmitriy M. Pashkovsky, and Dmitry A. Tarkhov. "SELECTION OF OPTIMAL RBF-NETWORK STRUCTURE FOR APPROXIMATING THE SOLUTION OF THE TWO-DIMENSIONAL LAPLACE EQUATION." SOFT MEASUREMENTS AND COMPUTING 9, no. 70 (2023): 108–22. http://dx.doi.org/10.36871/2618-9976.2023.09.010.

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An algorithm for neural network approximation of the solution of a boundary value problem for the twodimensional Laplace equation on a square domain is proposed. An approximate solution of the Laplace equation is a radial basis neural network with one hidden layer (RBF network). Optimal parameters of the RBF network are obtained from the minimization problem of the quadratic functional for the Laplace equation. In our work we also proposed an algorithm for choosing the minimum number of neurons on the hidden layer of the RBFnetwork for a given accuracy of the approximate solution. Two problems for Laplace equation with different boundary conditions were considered. In the first problem for the Laplace equation, we set discontinuous boundary conditions in the corners of the square, and in the second problem we set regular boundary conditions with error.

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44

Alhefthi, Reem K., and Hassan Eltayeb. "The Solution of Coupled Burgers’ Equation by G-Laplace Transform." Symmetry 15, no. 9 (September 15, 2023): 1764. http://dx.doi.org/10.3390/sym15091764.

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The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientific fields. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of the G-Laplace transform. The G-transform is effective in solving a wide range of non-constant coefficient differential equations, setting it apart from the Laplace, Sumudu, and Elzaki transforms. Consequently, it stands as a powerful tool for addressing differential equations characterized by variable coefficients. By applying this transformative approach, the study provides reliable and exact solutions for both homogeneous and non-homogeneous coupled Burgers’ equations. This innovative technique offers a valuable tool for gaining deeper insights into this equation’s behavior and significance in diverse disciplines.

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45

Johnston, S. J., H. Jafari, S. P. Moshokoa, V. M. Ariyan, and D. Baleanu. "Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order." Open Physics 14, no. 1 (January 1, 2016): 247–52. http://dx.doi.org/10.1515/phys-2016-0023.

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AbstractThe fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

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46

ALmahdi, Mariam. "Fractional partial Differential Equations for Laplace transformation Caputo-Fabrizio and Volterra integration." Academic Journal of Research and Scientific Publishing 5, no. 54 (October 5, 2023): 90–109. http://dx.doi.org/10.52132/ajrsp.e.2023.54.4.

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In this paper we note the numerical methods for solving fractional differential equations, defined in the derivative of the Caputo-Fabrizio fractional operator and Laplace transform of fractional derivatives for integer order, solving differential equation problems using the Laplace transform method, and reducing to Volterra's integral equation, Laplace transform of the Mittage–Leffler function, this problem is not easy to solve analytically because an analytical solution is sometimes not available, even if an analytical solution is available, but it is complected, time-consuming and expensive, so we need to develop a numerical method to address the relevant problem, Analyze a precise result such as the integral or exact expression of a solution to obtain a qualitative answer that shows us what is happening with each variable while numerical methods are more adaptable in the approximate result to obtain quantitative results by iteratively creating an approximate solution sequence for mathematical problems. The method will solve a non-homogeneous linear differential equation directly, following basic steps, without having to solve the integral equation and solutions separately and non-linear differential equations with the rational factor by developing analytical or numerical techniques to find approximate solutions. Finally, we studied some applications, especially for nonlinear differential equations with the rational operator.

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47

Rani, Dimple, Vinod Mishra, and Carlo Cattani. "Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations." Symmetry 11, no. 4 (April 12, 2019): 530. http://dx.doi.org/10.3390/sym11040530.

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This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform, fractional differential equations are firstly converted to system of algebraic equations then the numerical inverse of a Laplace transform is adopted to find the unknown function in the equation by expanding it in a Bernstein series. The advantages and computational implications of the proposed technique are discussed and verified in some numerical examples by comparing the results with some existing methods. We have also combined our technique to the standard Laplace Adomian decomposition method for solving nonlinear fractional order differential equations. The method is given with error estimation and convergence criterion that exclude the validity of our method.

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48

Atangana, Abdon. "A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential Equation." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/769102.

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We introduced a relatively new operator called the triple Laplace transform. We presented some properties and theorems about the relatively new operator. We examine the triple Laplace transform of some function of three variables. We make use of the operator to solve some kind of third-order differential equation called “Mboctara equations.”

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49

Druet, Olivier. "Generalized scalar curvature type equations on compact Riemannian manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 4 (August 2000): 767–88. http://dx.doi.org/10.1017/s0308210500000408.

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50

Azis, Mohammad Ivan. "NUMERICAL SOLUTIONS FOR 2D UNSTEADY LAPLACE-TYPE PROBLEMS OF ANISOTROPIC FUNCTIONALLY GRADED MATERIALS." Mathematical Modelling and Analysis 27, no. 2 (April 27, 2022): 303–21. http://dx.doi.org/10.3846/mma.2022.14463.

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The time-dependent Laplace-type equation of variable coefficients for anisotropic inhomogeneous media is discussed in this paper. Numerical solutions to problems which are governed by the equation are sought by using a combined Laplace transform and boundary element method. The variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation after being Laplace transformed is then written in a boundary-only integral equation involving a time-free fundamental solution. The boundary integral equation is therefore employed to find the numerical solutions using a standard boundary element method. Finally the numerical results are inversely transformed numerically using the Stehfest formula to obtain solutions in the time variable. Some problems of anisotropic functionally graded media are considered. The results show that the combined Laplace transform and boundary element method is accurate and easy to implement.

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