Double Laplace Transform Research Articles

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.

Exploring the Remarkable Properties of the Double Sadik Transform and Its Applications to Fractional Caputo Partial Differential Equations

The Double Sadik Transform (DST) represents a generalized double integral transform that has emerged as a highly effective analytical technique for solving numerous scientific problems. This study aims to investigate the DST applied to elementary functions and explore its notable properties, including its duality with the Double Laplace Transform and its capability to transform shifting functions, periodic functions, and convolution functions. Furthermore, the DST methodology is employed to resolve prominent linear fractional Caputo partial differential equations with known solutions commonly encountered in diverse mathematical models. The obtained outcomes are expressed in exact closed form, with the most precise results articulated through the Mittag-Leffler function. These results serve to validate the effectiveness and efficiency of the DST approach, establishing it as a valuable tool for addressing scientific problems involving fractional calculus.

A Reliable Combination of Double Laplace Transform and Homotopy Analysis Method for Solving a Singular Nonlocal Problem with Bessel Operator

In this article, we present a numerical iterative scheme for solving a non-local singular initial-boundary value problem by combining two well known efficient methods. Namely, the homotopy analysis method and the double Laplace transform method. The resulting scheme is tested on a set of test examples to illustrate its efficiency, it generates the exact analytical solution for each one of these examples. The convergence of the resulting numerical solutions of these examples is tested both graphically and numerically.

On Fuzzy Conformable Double Laplace Transform with Applications to燩artial Differential Equations

The Laplace transformation is a very important integral transform, and it is extensively used in solving ordinary differential equations, partial differential equations, and several types of integro-differential equations. Our purpose in this study is to introduce the notion of fuzzy double Laplace transform, fuzzy conformable double Laplace transform (FCDLT). We discuss some basic properties of FCDLT. We obtain the solutions of fuzzy partial differential equations (both one-dimensional and two-dimensional cases) through the double Laplace approach. We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

In this paper, we introduce the non-conformable double Laplace transform. Its properties are studied, and it is applied to solve some fractional PDEs involving the nonconformable fractional derivative. Graphical representations of the obtained solutions are shown in figures. The study shows that this transform is effective and easy to apply to create an exact solution for types of fractional PDEs.

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

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

On fractional order multiple integral transforms technique to handle three dimensional heat equation

In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form.

Analytical Solution of Partial Integro Differential Equations Using Laplace Differential Transform Method and Comparison with DLT and DET

Partial Integro Differential Equations (PIDEs) occur naturally in various fields of science and technology. The main purpose of this paper is to study how to solve linear partial integro differential equations with convolution kernel by using the Laplace-Differential Transform Method (LDTM). This method is a simple and reliable technique for solving such equations. The efficiency and reliability of this method is also illustrated with some examples. The result obtained by this method is compared with the result obtained by Double Laplace Transform and Double Elzaki Transform method.

The solution of Poisson partial differential equations via Double Laplace Transform Method

The Poisson’s Partial Differential Equation (PPDEs) is known as the generalization of a famous Laplace’s Equation. The aforementioned differential equation is an elliptic in nature and frequently used in theoretical physics. The consider equation shows linkage between potential difference and volume charge density. The authors, developed the scheme for approximate solution of PPDEs by Double Laplace Transform (DLT). To illustrated the main work, we have steepness some numerical examples. Author’s also provided graphical representation corresponding to desired solutions for the concern class of PDEs.

Solution of One-dimensional Partial Differential Equation with Higher-Order Derivative by Double Laplace Transform Method

Many problems in natural and engineering sciences such as heat transfer, elasticity, quantum mechanics, water flow, and others are modelled mathematically by partial differential equations. Some of these problems may be linear, nonlinear, homogeneous, non-homogeneous, and order greater or equal one. Finding the theoretical solution to these problems with less cumbersome techniques is an active area of research in the aforementioned field. In this research paper, we have developed a new application of the double Laplace transform method to solve homogeneous and non-homogeneous linear partial differential equations (pdes) with higher-order derivatives (i.e order n where n≥2) in science and engineering. We discussed a brief theory of double Laplace transforms that helped in its application. The main advantage of our method is the reduction of computational effort in finding solution to pdes. Another major benefit of our method is solving problems in the form of (21) directly by transforming to an algebraic equation where the inverse double Laplace transform is implemented for analytical solution, unlike other integral transform methods that would first transform to a system of ODEs before they are solved, is it also very effective in solving linear high-order partial differential equations and yield fast convergence. We present a well-simplified solution for easier comprehension by upcoming researchers.

The Double Laplace Transform Expressed in terms of the Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Conformable double Laplace transform method for solving conformable fractional partial differential equations

In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional differential in mathematical physics. The results obtained show that the proposed method is efficient, reliable, and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.

Exact Analytical Solutions of Linear Dissipative Wave Equations via Laplace Transform Method

A wave phenomena evolved day after day, as various concepts regarding waves appeared with the passage of time. These phenomena are generally modelled mathematically by partial differential equations (PDEs). In this research, we investigate the exact analytical solutions of one and two dimensional linear dissipative wave equations which are modelled by second order PDEs with use of some initial and boundary conditions. We use double Laplace transform (DLT) and triple Laplace transform (TLT) methods to determine these exact analytical solutions. We provide examples with figures to test effectiveness of this scheme of Laplace transform

The Fuzzy Double Laplace Transforms and their Properties with Applications to Fuzzy Wave Equation

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.

Stochastic Model of Three Products Inventory System Assuming Seasonal Demand Following Exponential Distribution

In this paper we discuss the production and sales of three different products A, B and C with maximum production size as k sets of them. The production of the products A, B and C are done in order. When one set of A, B, and C is produced, the production of the next set starts. The sales of the sets begin when k sets are produced or when the season for the products starts whichever occurs first.The production times of the products are assumed to have general distributions and the season starts in an exponential time. The double Laplace Transform of production and sale time and their means are obtained. Numerical examples are presented.

DOUBLE LAPLACE TRANSFORM METHOD FOR SOLVING TELEGRAPH EQUATION

Telegraph equation is a partial differential equation which includes the one dimensional wave equation. This equation can be used to solve several problems that physically regulate voltage and current in the electricity transmission line in distance and time. In this study, telegraph equation and partial integrodifferential equation were analyzed and resolved by double Laplace transform. This equation was accompanied by initial values and boundary values solved by Laplace transform. Next, there are some examples of equations to discover a solution. The solution of these examples showed that double Laplace transform is one method that can be used to solve telegraph equation. Then the solutions are simulated in graphical form which shows the waveform of the equation.

On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers’ Equation

In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to obtain the exact solution for nonlinear fractional problems, then we modified the double Laplace transform and combined it with the Adomian decomposition method. Later, we applied the new method to solve regular and singular conformable fractional coupled Burgers’ equations. Further, in order to illustrate the effectiveness of present method, we provide some examples.

Double Laplace Transform Method for Solving Space and Time Fractional Telegraph Equations

Double Laplace transform method is applied to find exact solutions of linear/nonlinear space-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditions. Furthermore, we give illustrative examples to demonstrate the efficiency of the method.

Double Laplace Transform Computing Dynamic Response of the Large Thickness to Span Ratio Beam

In this paper, use double Laplace transform, in the image space deep beam deflection and corner of analytic is obtained,Then use the numerical inversion of Laplace transform ,time domain dynamic response curve is calculated. Also apply to the combined effects of sandwich beam bending and torsion beam dynamic response calculation.