The Numerical Study of a Hybrid Method for Solving Telegraph Equation

Arslan, Derya

doi:10.2478/amns.2020.1.00027

Cited by 19 publications

(6 citation statements)

References 25 publications

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“…Solving the telegraph equation analytically is not always possible or convenient, especially when the boundary and initial conditions are complicated or nonlinear. Therefore, various numerical and approximate methods have been proposed and used to obtain solutions of the telegraph equation [3,4,5,6,7,8,9,10,11], such as finite difference methods, Runge-Kutta methods, perturbation methods, homotopy methods, and Adomian decomposition method (ADM). Among these methods, ADM is a popular and powerful technique that can handle linear and nonlinear problems without linearization or discretization.…”

Section: Introductionmentioning

confidence: 99%

Solving The Hyperbolic Telegraph Equation Using a Modified Adomian Decomposition Method with an Invertible Partial Differential Operator

Al‐Omari¹,

Hasan²

2023

ARJOM

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In this paper, a modified Adomian decomposition method (MADM) for solving the hyperbolic telegraph equation is proposed. The MADM introduces a new inverse partial differential operator that can speed up the convergence rate of the standard ADM. We also present a technique for converting the equation to a special case form, which makes the MADM easier to implement. The proposed method was tested on six different linear and nonlinear telegraph equations in one and two dimensions. The results show that the method is accurate and efficient for solving the telegraph equation.

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Section: Introductionmentioning

confidence: 99%

Solving The Hyperbolic Telegraph Equation Using a Modified Adomian Decomposition Method with an Invertible Partial Differential Operator

Al‐Omari¹,

Hasan²

2023

ARJOM

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“…Then, a novel strategy built on the Haar wavelet collocation technique was developed to solve one and two-dimensional second-order linear and non-linear hyperbolic telegraph equations [31]. A hybrid method was introduced by Arslan [32] to approximate the telegraph equation, incorporating the finite difference and differential transform method. Recently, Partohaghighi et al, [33] presented a method to solve hyperbolic telegraph equations according to the fictitious time integration (FTI) and group preserving (GP) methods.…”

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solutions for Non-linear Telegraph Equations with Source Term

Abdul Rahman Farhan Sabdin,

Che Haziqah Che Hussin,

Arif Mandangan

et al. 2023

ARASET

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This study proposed a new version of Multistep Modified Reduced Differential Transform Method (MMRDTM) which is applicable for obtaining semi-analytical approximation on Non-linear Telegraph Equations (NLTEs) with source term. Through the experimental approach, we unveiled various advantages of the proposed method, especially its ability to analytically approximate high-speed converging series. Moreover, a significant reduction in the number of calculated terms can also be observed. Before deploying the multistep technique, the substitution of non-linear term in NLTEs with the corresponding Adomian polynomial takes place. Consequently, a simpler technique for approximating NLTEs with source term was discovered. Besides that, the solutions can be estimated more precisely over a longer time. To justify our findings, solutions of NLTEs with source term are obtained using the MMRDTM. The accuracy of the proposed method for solving these equations was tested by comparing the absolute errors between the solutions obtained by the proposed method and the Modified Reduced Differential Transform Method (MRDTM) with the exact solution. The obtained solutions evidenced that the proposed MMRDTM can deliver highly precise approximations for the NLTEs with source term.

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“…Hybrid method is the incorporated form of two or even more approaches used to solve many forms of differential equations, which include the logistics-differential equations [15], partial differential equations [3,4,7,17,24,26,31,33,35], integro-differential equation [5,6,36] and fractional differential equations [28,29,32]. A number of the hybrid method found in the literature feature: Laplace homotopy perturbation method (LHPM) [22,34], which incorporates the Laplace transform and homotopy pertubation method(HPM); RBF-based DQ method [31], which incorporates global radial basis function-based and differential quadrature method; homotopy perturbation and Sumudu transform Method [14], which incorporates the Sumudu transform and homotopy perturbation method; Laplace Adominan decomposition method (LADM) [28,32,36], which incorporates the Laplace transform and Adomian decomposition method (ADM); Laplace differential transform method (LDTM) [8,9], which incorporates Laplace transform and differential transform method (DTM); Laplace Decomposition method [15,17]; Finite difference and differential transform method [7]; Homotopy perturbation and Elzaki transform [12]. However, many of these hybrid approaches entails perturbation, linearization or any other transformation and at the same time entails a ton of computational efforts.…”

Section: Introductionmentioning

confidence: 99%

Laplace Projected Differential Transform Method for Solving Nonlinear Partial Differential Equations

Daniel¹

2022

Preprint

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This study presents a hybrid method that incorporates Laplace transform along with projected differential transform method to solve partial differential equations which may be utilized to describe physical problems emerging in applied scientific research. Laplace transform is introduced to eliminate the demerit of complex estimation of utilization of differential transform method (DTM) and projected differential method (PDTM). The sufficient condition for the convergence of LPDTM is covered and was applied to solve linear and nonlinear partial differential equations to illustrate the efficiency and dependability of the method. The major advantage of this method is that, the computation comes to be much less complicated to solve and the nonlinear term is effortlessly managed through projected differential transform without utilizing Adomian's polynomial and He's polynomial.

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The Numerical Study of a Hybrid Method for Solving Telegraph Equation

Cited by 19 publications

References 25 publications

Solving The Hyperbolic Telegraph Equation Using a Modified Adomian Decomposition Method with an Invertible Partial Differential Operator

Solving The Hyperbolic Telegraph Equation Using a Modified Adomian Decomposition Method with an Invertible Partial Differential Operator

Approximate Analytical Solutions for Non-linear Telegraph Equations with Source Term

Laplace Projected Differential Transform Method for Solving Nonlinear Partial Differential Equations

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