Laplace Variational Iteration Method Research Articles

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.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Analytical solution of local fractal continuum traffic flow model

This study developed a continuum traffic flow model with local fractional derivatives. This model overcomes the non-differentiable traffic parameters arising in vehicular traffic flow. The proposed model is solved using the local fractional Laplace variational iteration method (LFLVIM) and is well suited for analyzing the dynamical evolution of non-differentiable traffic density and speed function. Furthermore, the stability of the solution for the continuum model has also been discussed. Illustrative examples are also discussed to show the effectiveness of employing LFLVIM in the suggested model. Additionally, numerical simulations for each instance have been displayed. This research indicates that the utilized iterative approach is efficient and may be used to derive the non-differentiable solution to the proposed continuum traffic model.

COMPARISON OF LAPLACE-ADOMIAN’S DECOMPOSITION METHOD AND LAPLACE VARIATIONAL ITERATION METHOD IN NONLINEAR BOUSSINESQ EQUATION

In this paper, we apply two techniques Laplace Decomposition Method (LADM) and Laplace Variational Iteration Method (LVIM) to determine the analytical solution of Boussinesq Equation.

Applications of the Laplace variational iteration method to fractional heat like equations

The importance of differential equations of integer order and fractional order can be seen in many areas of engineering and applied sciences. The present work involves fractional order heat equations that arise in numerous applications of engineering and aims to find series solutions by the Laplace variational iteration method (LVIM). The method combines the Laplace transform and the variational iteration method. To show the efficiency and validity of LVIM, we have exemplarily considered 1-D, 2-D, and 3-D fractional heat equations and solve them by LVIM. Exact solutions are gained in expressions of the Mittag-Leffler function. The results are also explored through graphs and charts.

Solutions of the Nonlinear Evolution Problems and their Applications

Abstract In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

Analysis of fractional Navier–Stokes equations

AbstractIn this study, we apply the fractional Laplace variational iteration method (FLVIM), a computer methodology for exploring fractional Navier–Stokes equation solutions. In light of the theory of fixed points and Banach spaces, this paper also explores the uniqueness and convergence of the solution of general fractional differential equations obtained by the suggested method. In addition, the fractional Laplace variational iteration method solution's error analysis is covered. The computational technique also clearly demonstrates the validity and dependability of the suggested method for solving fractional Navier‐Stokes equations. Furthermore, the obtained solutions are a perfect fit with previously established solutions.

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.

Semi-analytical solutions of three-dimensional (3D) coupled Burgers’ equations by new Laplace variational iteration method

In this article, New Laplace variational iteration method (NLVIM), which is based upon the combination of Laplace transform and modified variational iteration is used to solve the three dimensional (3D) coupled Burgers’ equations. The proposed method is capable to find the numerical solutions as well as series solution which converges to the exact solution. The numerical results obtained by New Laplace Variational iteration method (NLVIM) after some iterations or terms are very closer to the exact solutions with a negligible absolute error. Some examples are given to illustrate the accuracy of the numerical results by New Laplace variational iteration method (NLVIM).

New Modified Variational Iteration Laplace Transform Method Compares Laplace Adomian Decomposition Method for Solution Time-Partial Fractional Differential Equations

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.

He–Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics

In this article, we suggest an alternative approach for the study of some partial differential equations (PDEs) arising in physical phenomena such as chemical kinetics and population dynamics. He–Laplace variational iteration method (He-LVIM) is a simple but effective way where the variational iteration method (VIM) is coupled with Laplace transforms method. He’s polynomials are obtained by the homotopy perturbation method (HPM) to deal with the nonlinear terms. Fisher equation, the generalized Fisher equation and the nonlinear diffusion equation of the Fisher type are suggested to demonstrate the accuracy and stability of this method.

New iterative approach for the solutions of fractional order inhomogeneous partial differential equations

In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.

Laplace Variational Iteration Method for Modified Fractional Derivatives with Non-singular Kernel

A universal approach by Laplace transform to the variational iteration method for fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering the modified Caputo-Fabrizio fractional derivative and the analogous modifications for the Atangana-Baleanu fractional derivative with non-singular Mittag-Leffler kernel in order to satisfy the initial conditions for some fractional differential equations.

Application of Laplace Variation Iteration Method to Solving the Nonlinear Gas Dynamics Equation

In this work, we use a new analytical technique called Laplace variational iteration method to construct the exact solution of the nonlinear equation of gas dynamics. This method is based on the determination of the Lagrange multiplier in an optimal way. Application of the method to three test modeling problems from mathematical physics leads to a sequence which tends towards the exact solution of the problem. The solution procedure shows the reliability of the method and is high accuracy evident.

Solving Helmholtz Equation with Local Fractional Derivative Operators

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients

In the present study, a new amendment in Laplace variational iteration method for the solution of fourth-order parabolic partial differential equations with variable coefficients is revealed i.e. modified Laplace variational iteration method (ML-VIM). The proposed modification is made by coupling of two methods: one of them is variational iteration method (VIM) and the other one is Laplace transformation (LT). Our modification has an important beauty that one has no need to compute Lagrange multiplier via integration nor by taking convolution theorem and has a simple way to use in the implementation of our proposed scheme. Moreover, homotopy perturbation method (HPM) with He’s polynomials is employed for the computation of nonlinear terms. The main improvement of our proposed scheme is the reduction of the problem to a simple one, which also embraces for solving a nonlinear term. Some illustrated examples are interpreted which unveil the robustness and accuracy of the newly developed scheme.

Analytical approaches to space- and time-fractional coupled Burgers’ equations

We solve the one- and two-dimensional fractional coupled Burgers’ equations (FCBEs) by three different methods. The proposed methods are the Laplace–Adomian decomposition method (LADM), the Laplace-variational iteration method (LVIM) and the reduced differential transform method (RDTM). The solutions are obtained as rapidly convergent series with simply calculable terms. Numerical studies of the application of these approaches for a number of sample problems are given and are illustrated graphically. With these methods, it is possible to investigate the nature of solutions when the fractional derivative parameters are changed. The numerical results reveal the effectiveness and the correctness of the proposed methods.

Modified Laplace Variational Iteration Method for Analytical Approach of Klein–Gordon and Sine–Gordon Equations

This paper reveals an innovative technique built on variational iteration method and Laplace transformation, called modified Laplace variational iteration method. At first, we apply variational iteration method, and then, Laplace transformation is introduced to calculate the Lagrange multipliers. An innovation of this algorithm shows that Lagrange multiplier has carried out in the absence of integration used in variational iteration method and without taking convolution theorem use in Laplace transform. Furthermore, He’s polynomials are calculated by applying homotopy perturbation method to overcome the nonlinear terms arising in the problems. The proposed method points out that the derived solution exists without any linearization, discretization or any kind of hypothesis. Some numerical models are illustrated which shows the compactness and reliableness of this method.

Numerical approaches to system of fractional partial differential equations

In this paper, by introducing the fractional derivative in sense of Caputo, the Laplace- variational iteration method (LVIM) and the Laplace-Adomian decomposition method (LADM) are directly extended to study the linear and nonlinear systems of fractional partial differential equations, as a result the approximated numerical solutions are acquired in the form of rapidly convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when are applied. Compassions are made between the two methods and exact solutions. Figures are used to show the efficiency as well as the accuracy of the achieved approximated results.