Applications of Double Laplace Transform to Boundary Value Problems

Dhunde, Ranjit R.

doi:10.9790/5728-0925760

Cited by 3 publications

(2 citation statements)

References 7 publications

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“…Researchers extended the concept of the single Laplace transform method to the double Laplace transform method to find the solution to some kinds of differential equations and fractional differential equations, such as space-time fractional telegraph equations and functional, integral, and partial differential equations, because the Laplace transform method is a crucial technique in solving mathematical problems arising in various fields of science [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Application of Iterative Three-Dimensional Laplace Transform Method for 2-Dimensional Nonlinear Klein-Gordon Equation

Deresse

2023

Trends Sci

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In the present study, the exact analytical solutions of the 2-dimensional nonlinear Klein-Gordon equation (NLKGE) is investigated using the 3-dimensional Laplace transform method in conjunction with the Daftardar-Gejji and Jafari Method (Iterative method). Through this method, the linear part of the problem is solved by using the 3-dimensional Laplace transform method, while the noise terms from the nonlinear part of the equation disappear through a successive iteration process of the Daftardar-Gejji and Jafari Method (DJM), where a single iteration gives the exact solution to the problem. Three test modeling problems from mathematical physics nonlinear Klein-Gordon equations are taken to confirm the performance and efficiency of the presented technique. For each illustrative example, the convergence of the novel iterative approach is shown. The findings also suggest that the proposed method could be applied to other types of nonlinear partial differential equation systems. HIGHLIGHTS Partial differential equations play a crucial role in the formulation of fundamental laws of nature and the mathematical analysis in different fields of studies The Klein-Gordon equation (KGE) is a nonlinear hyperbolic class of partial differential equations that occur in relativistic quantum mechanics and field theory, both of which are crucial to high-energy physicists The 3-dimensional Laplace transform method provides rapid convergence of the exact solution without any restrictive assumptions about the solution for linear differential equations The Laplace Transform method is often combined with other methods like the Daftardar-Gejji and Jafari (the new Iterative) method to solve complicated nonlinear differential equations The new iterative technique yields a series that can be summed up to get an analytical formula or used to build an appropriate approximation with a faster convergent series solution GRAPHICAL ABSTRACT

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Section: Introductionmentioning

confidence: 99%

Application of Iterative Three-Dimensional Laplace Transform Method for 2-Dimensional Nonlinear Klein-Gordon Equation

Deresse

2023

Trends Sci

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“…Recently, the concept of single Laplace transform is extended to double Laplace transform to solve some kind of differential equations and fractional differential equations such as linear/nonlinear space-time fractional telegraph equations, functional, integral, and partial differential equations [26][27][28]. Dhunde and Waghmare [29] applied the double Laplace transform method for solving a one-dimensional boundary value problems. Through this method, the boundary value problem is solved without converting it into ordinary differential equation; therefore, there is no need to find complete solution of ordinary differential equation.…”

Section: Introductionmentioning

confidence: 99%

Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

Deresse

Mussa

Gizaw

2021

Journal of Applied Mathematics

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This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

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