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In this paper, we apply the Laplace decomposition and the conformable variational iteration methods to derive new numerical solutions of time and space-conformable fractional Wu-Zhang system. Some graphs for the obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the application of the Laplace decomposition method (LDM) to find approximate solutions for some classes of time and space-conformable fractional evolution equations. Some illustrative examples are presented showing that the LDM is an efficient method for finding solutions of nonlinear conformable fractional problems. [ABSTRACT FROM AUTHOR]

This paper comprehensively studies effective numerical methods for solving the simplest chaotic circuit model. We introduce a novel scheme for the Atangana–Baleanu Caputo fractional derivative (ABC-FD), coupled with the Laplace decomposition method (LDM). Furthermore, we rigorously compare the performance of these proposed methods with the Runge–Kutta fourth-order method. Using two mathematical techniques, we have discovered effective and highly convergent solutions to the chaotic model. We gave different values to the parameters to plot the chaos and create a phase portrait of the system. Therefore, the provided methods can be applied to more sophisticated examinations of different models. This study advances numerical techniques for understanding chaotic dynamics in complex systems. By introducing a novel scheme for the Atangana–Baleanu Caputo fractional derivative and the Laplace decomposition method, we provide a robust framework for effectively solving the simplest chaotic circuit model. This framework enhances accuracy and efficiency in unraveling chaotic behaviors, contributing to a broader understanding of chaotic dynamics across scientific domains in the future. [ABSTRACT FROM AUTHOR]

The present manuscript gives an overview of how two-dimensional heat diffusion models underwent a fractional transformation, system coupling as well as solution treatment. The governing diffusion models, which are endowed with Caputo's fractional-order derivatives in time t, are suitably coupled using the (1) convection phenomenon, (2) interfacial coupling by considering the mechanism of a double-layered bar, and the (3) nonlinear coupling due to temperature-dependent thermal diffusivities. Semi-analytical and analytical methods are considered for the solution treatment. Moreover, we seek a computational environment to graphically illustrate the systems' response to different fractional orders in each case through the determined diffusional fields. Besides, we supply certain concluding notes at the end.

In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems. [ABSTRACT FROM AUTHOR]

The present manuscript gives an overview of how two-dimensional heat diffusion models underwent a fractional transformation, system coupling as well as solution treatment. The governing diffusion models, which are endowed with Caputo's fractional-order derivatives in time t, are suitably coupled using the (1) convection phenomenon, (2) interfacial coupling by considering the mechanism of a double-layered bar, and the (3) nonlinear coupling due to temperature-dependent thermal diffusivities. Semi-analytical and analytical methods are considered for the solution treatment. Moreover, we seek a computational environment to graphically illustrate the systems' response to different fractional orders in each case through the determined diffusional fields. Besides, we supply certain concluding notes at the end. [ABSTRACT FROM AUTHOR]

The subject of this study is the solution of a fractional Bernoulli equation and a chaotic system by using a novel scheme for the fractional derivative and comparison of approximate and exact solutions. It is found that the suggested method produces solutions that are identical to the exact solution. We can therefore generalize the strategy to different systems to get more accurate results. We think that the novel fractional derivative scheme that has been offered and the algorithm that has been suggested will be utilized in the future to construct and simulate a variety of fractional models that can be used to solve more difficult physics and engineering challenges. [ABSTRACT FROM AUTHOR]

In this work, two analytical solutions for the 3D steady-state fractional advection–diffusion equation are obtained in order to study the dispersion of air pollutants. The first solution was obtained considering Caputo's derivative, and the other considered a conformable derivative. The Laplace decomposition method (LDM), which generates a fast convergence series solution, was used to obtain the solutions. The solution generated from the Caputo derivative (nonlocal derivative) is represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable derivative (local derivative) generates an exponential function. The solutions were only identical when they were of integer order (α = 1). In addition, sensitivity analyses were performed on two explicit parameters obtained in the solutions: fractional parameter α , which controls the order of the fractional derivative, and parameter φ , responsible for maintaining the correct dimensionality of the equations. When Caputo and conformable solutions utilized data from the traditional Copenhagen experiment with φ = 1 m and α = 0.98, the values of statistical indexes NMSE = 0.10, COR = 0.90, and FAT2 = 0.91 were shown in both with only small differences in FB and FS, and these results are better than some of the more complex integer-order equation models already existing in the literature. In practical terms, using experimental data moderately convective, it is clear that the models did not show statistically significant differences in the concentrations of pollutants simulated at ground level under conditions of low fractionality, even when considering the dimensional parameter ranging from unity to the smallest scale of atmospheric turbulence ( φ ≃ 10 - 3 m, Kolmogorov length scale). However, when fractionality increased, the microscale dimensional parameter became more effective, and the results quickly deteriorated. [ABSTRACT FROM AUTHOR]

This work presents an analytical series solution of a two-dimensional advection–diffusion equation of fractional order, in the Jumarie fractional derivative sense, for the dispersion of pollutants within the planetary boundary layer. The series solution is obtained by applying the Laplace decomposition method considering the vertical turbulent diffusion dependent on the longitudinal distance from the source. The fractional parameter is used not only in the advective term but also in the vertical diffusive term to account for anomalous dispersion. This modification makes the fractional solution more general than the traditional, and the proposed model can give new prospects for simulation of the pollutant dispersion in the atmosphere. Numerical simulations were conducted to compare the performance of the fractional solution method to that of the traditional solutions approaches using the strongly convective Prairie Grass experimental dataset. Statistical indices show that the best results of ground-level concentration were obtained with the fractional-order derivatives. [ABSTRACT FROM AUTHOR]

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We consider a fractional‐order nerve impulse model which is known as FitzHugh–Nagumo (F–N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional‐order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave‐type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers. [ABSTRACT FROM AUTHOR]

In this paper, we consider mostly the inhomogeneous Fitzhugh-Nagumo-Huxley equation with its initial value. Adomian Decomposition Method (ADM), Modified Decomposition Method (MDM) and Laplace Decomposition Method (LDM) are applied to this equation to obtain the solution that satisfies the given initial condition. It can be easily seen that each method gives the exactly same solution. And then, the inhomogeneous Boussinesq equation and another nonlinear partial differential equation subject to given initial values are solved by using LDM. In applications of the methods, it is demonstrated that we get the solution with a fast convergence by using the advantage of the noise terms. Moreover, we do not need to turn the nonlinear terms into linear ones since Adomian polynomials are used. Therefore, it is deduced that these methods are, indeed, effective and suitable for the nonlinear partial differential equations with the initial values.

The analytical approximate solutions of the wave equation with local fractional derivative operators (LFDOs) are utilized in this manuscript. The reduced differential transform method (RDTM) and Laplace decomposition method (LDM) are investigated in the LFDOs sense. The results obtained by the local fractional RDTM are compared with the results obtained by local fractional LDM. The efficiency of the considered methods is illustrated by some examples. [ABSTRACT FROM AUTHOR]

In this work, a combined form of the Laplace transform method and the Adomian decomposition method is implemented to give an approximate solution of nonlinear systems of differential equations such as fish farm model with three components nutrient, fish and mussel. The technique is described and illustrated with a numerical example.

In this paper, the Laplace decomposition method (LDM) is proposed to solve the two-dimensional nonlinear Burgers’ differential equations. Two test problems are considered to illustrate the accuracy of the proposed algorithm. It is shown that the numerical results are in good agreement with the exact solutions for each problem.

In this study, we consider the inhomogeneous Benney-Luke equation with its initial conditions. Laplace Decomposition Method and Adomian Decomposition Method are applied to this equation. Then, the solution yielding the given initial conditions is gained. [ABSTRACT FROM AUTHOR]

The principal purpose of this work is to present an efficient analytical approach based on the q -homotopy analysis transform technique in order to analyze a fractional model of the vibration equation for large membranes. The Laplace decomposition technique is also employed to obtain the solution of the fractional-order vibration equation. Numerical examples are provided to examine the efficiency of the proposed schemes and the results derived are demonstrated graphically. Our examples illustrate that the suggested computational approaches are straightforward to perform and computationally very striking. [ABSTRACT FROM AUTHOR]

In applications where partial differential equations are used to model populations, there is frequently a critical density threshold below which the population cannot be detected in practice and the corresponding position is often termed the leading edge of the distribution. Historically this position has been investigated for large time problems, but little attention has been afforded to understanding its short term dynamics. In this work we describe a novel approach, utilizing the Laplace decomposition method, that generates algebraic expressions for the initial kinematic properties of the leading edge in terms of the initial data and model parameters. The method is demonstrated on two well-studied partial differential equations and two established systems of equations (representing the growth of fungal networks), all of which display travelling fronts. The kinematics of these advancing fronts are determined using our method and are shown to be in excellent agreement with both exact solutions and numerical approximations of the model equations. [ABSTRACT FROM AUTHOR]

In this paper, Haar wavelets method (HWM) is applied to compute the numerical solutions of the foam drainage equation. The mathematical base of the method is presented. First, the time derivative is discretized by a forward difference scheme and then a quasilinearization technique is used to linearize the foam drainage equation. Finally, we solve a system of linear equations which is obtained by applying the Haar wavelet method for discretizing the space derivatives. Obtained results by HWM are very similar to the exact solutions. Further, a comparison between our results and results which are obtained by HPM, HPTM, LDM and ADM is presented. Numerical results show that our method works better than the other methods. [ABSTRACT FROM AUTHOR]

Laplace Decomposition Method (LDM), an efficient numerical method has been successfully tested for analytic solution of various nonlinear partial differential equations (PDE). In this paper, we discuss the application of LDM to find analytic and approximate solution of a nonlinear gas dynamic equation. Examples with different initial profiles are considered to compare the results and illustrate the effectiveness of the method. In this paper, LDM is used to find analytic and approximate solutions of nonlinear gas dynamic equation which arises in modeling of diverse physical phenomena. [ABSTRACT FROM AUTHOR]

In this study we considered the initial value problem for the sine-Gordon equation by using Laplace Decomposition Method (LDM). The advantage of this method was its ability and flexibility to provide the analytical or approximate solutions to linear and nonlinear equations without linearization or discretization which makes it reliable for solving sine-Gordon equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the Laplace decomposition method (LDM) is proposed to solve the two-dimensional nonlinear Burgers’ differential equations. Two test problems are considered to illustrate the accuracy of the proposed algorithm. It is shown that the numerical results are in good agreement with the exact solutions for each problem. [Copyright &y& Elsevier]

The aspire of this article is to bring in a new approximate method, that is to say the Laplace Padé decomposition method which is a mixture of Laplace decomposition and Padé approximation to offer an analytical approximate way out to magnetohydrodynamics flow over a nonlinear porous stretching sheet. This new iteration approach provides us with a convenient way to approximate solution. A closed agreement between the obtained solution and some well-known results has been established. The proposed procedure can be applied to handle other nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract: The chief aim of the present article is to introduce a new analytical technique, two-step Laplace decomposition method (TSLDM) for nonlinear Volterra integral equations. The newly proposed method efficiently find exact solution with less computation as compared with standard Laplace decomposition method (LDM). The proposed algorithm is used to solve nonlinear Volterra integral equations effectively and overcome the deficiencies in Laplace decomposition method (LDM) easily. [Copyright &y& Elsevier]

Abstract: The aim of this article is to introduce a new approximate method, namely homotopy analysis transform method (HATM) which is a combination of homotopy analysis method (HAM) and Laplace decomposition method (LDM) to provide an analytical approximate solution to Blasius flow equations. The proposed scheme finds the solutions of nonlinear problems without any discretization, restrictive assumptions and avoids the rounding-off errors. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). A good agreement between the obtained solution and some well-known results has been demonstrated. [Copyright &y& Elsevier]

Abstract: In this paper, the Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion–wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method. [Copyright &y& Elsevier]

Abstract: In this paper, the Laplace Adomian Decomposition Method is implemented to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for HIV infection of cells. The technique is described and illustrated with numerical example. Some plots are presented to show the reliability and simplicity of the methods. [ABSTRACT FROM AUTHOR]

In this article, Laplace decomposition method (LDM) is applied to obtain series solutions of classical Blasius equation. The technique is based on the application of Laplace transform to nonlinear Blasius flow equation. The nonlinear term can easily be handled with the help of Adomian polynomials. The results of the present technique have closed agreement with series solutions obtained with the help of Adomian decomposition method (ADM), variational iterative method (VIM) and homotopy perturbation method (HPM). [ABSTRACT FROM AUTHOR]

This paper suggests a novel modified Laplace decomposition method for nonlinear equations by suitable choice of an initial solution. The modification overcomes the noise oscillation during the iteration procedure by the standard Laplace decomposition method. The Klein-Gordon equation is used as an example to show the effectiveness of the method [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the numerical comparison of methods applied to solve an integro-differential equation. Four numerical methods are compared, namely, the Laplace decomposition method (LDM), the Wavelet–Galerkin method (WGM), the Laplace decomposition method with the Pade approximant (LD–PA) and the homotopy perturbation method (HPM). [Copyright &y& Elsevier]

This paper carries out the numerical analysis of perturbed optical solitons that are governed by Sasa–Satsuma equation. Laplace–Adomian decomposition is the integration scheme that is implemented to recover bright and dark optical solitons to the model. The surface plots of bright and dark optical solitons are exhibited. The contour plots and error plots are also displayed. The error measure that emerged from the integration scheme is O (1 0 − 7) and is an impressive measure! [ABSTRACT FROM AUTHOR]

In this paper, we designed a reliable recipe of homotopy analysis method and Laplace decomposition method namely homotopy analysis transform method to solve fuzzy fractional heat and wave equations. This method overcomes the difficulties arise in other analytical method and removes the restrictive condition of nonlinearity and assumptions of small and large parameters. [ABSTRACT FROM AUTHOR]

The aim of the article is to apply an efficient semi-analytical algorithm for the approximate solution of a fractional differential equation that governs ray tracing through the crystalline lens. A Laplace decomposition method (LDM) is implemented for the solution of the nonlinear ray equation of fractional order that is complemented with initial conditions. This method is a combination of Laplace transform and a decomposition method. The nonlinear term is decomposed and a recursive algorithm is constructed for the determination of the infinite series solution. Numerical tests are conducted, and the results confirm that this strategy converges rapidly with high accuracy using only few iterates of the recursive scheme. [ABSTRACT FROM AUTHOR]

• We propose a new analytical series solution for the fractional 3D advection–diffusion equation. • Use of the GITT and LDM, for the first time, combined with the concept of fractional derivatives. • The memory effect is depending on both the source distance and the fractional derivative. • The fractional parameter (memory effect) compensates the inhomogeneous turbulence in the vertical direction. • The 3D solution allows to adjust the lateral dispersion parameter for use in the Gaussian model. In atmospheric environments, traditional differential equations do not adequately describe the problem of turbulent diffusion because the usual derivatives are not well defined in the non-differentiable behaviour introduced by turbulence, where the fractional calculation has become a very useful tool for studying anomalous dispersion and other transportation processes. Considering a new direction, this paper presents an analytical series solution of a three-dimensional advection–diffusion equation of fractional order, in the Caputo sense, applied to the dispersion of atmospheric pollutants. The solution is obtained by applying the generalised integral transform technique (GITT), solving the transformed problem by the Laplace decomposition method (LDM), and considering the lateral and vertical turbulent diffusion dependence on the longitudinal distance from the source, as well as a fractional parameter. The fractional solution is more general than the traditional solution in the sense that consideration of the integer order of the fractional parameter yields the traditional solution. The solution considers the memory effect in eddy diffusivity and in the fractional derivative, and it is simple, easy to implement, and converges rapidly. Numerical simulations were conducted to compare the performance of the proposed fractional solution to the traditional solution using an experimental dataset and other models, which also made it possible to find a better parametrisation for use in Gaussian models. The best results are obtained with the fractional order of the derivative. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to suggest an efficient hybrid computational technique, namely the q-homotopy analysis transform method (q-HATM) to find the solution of the nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. The uniqueness and convergence analysis of the nonlinear time-FZK equation is presented. The Laplace decomposition method (LDM) is also employed to get the approximate solution of the nonlinear FZK equation. We implemented these techniques on two numerical examples, plotted the solution and compared the absolute error with the variational iteration technique and homotopy perturbation transform technique to show the efficiency of these techniques. [ABSTRACT FROM AUTHOR]