scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Approximate Analytical Solution of One-Dimensional Beam Equations by Using Time-Fractional Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Dessalegn Mekonnen Yadeta ◽  
Ademe Kebede Gizaw ◽  
Yesuf Obsie Mussa
Keyword(s):  
Initial Conditions ◽  
Convergent Series ◽  
Differential Transform Method ◽  
Beam Equation ◽  
Transform Method ◽  
One Dimensional ◽  
Beam Equations ◽  
Differential Transform ◽  
Convergent Method ◽  
Reduced Differential Transform Method

In this paper, a recent and reliable method, named the fractional reduced differential transform method (FRDTM) is employed to solve one-dimensional time-fractional Beam equation subject to the appropriate initial conditions. This method provides the solutions very accurately and efficiently in convergent series form with easily computable coefficients. The efficacy and accuracy of this method are verified by means of three illustrative examples which indicate that the present method is very effective, simple, and easy to implement. Finally, it is observed that the FRDTM is the prevailing and convergent method for the solutions of linear and nonlinear fractional-order partial differential equations.

scholarly journals Analytical solutions of nonlinear Klein-Gordon equations using Multistep Modified Reduced Differential Transform Method

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Hussin Che ◽  
Ahmad Ismail ◽  
Adem Kilicman ◽  
Amirah Azmi
Keyword(s):  
Approximate Analytical Solution ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Transform Method ◽  
Adomian Polynomials ◽  
Differential Transform ◽  
Non Linear ◽  
Klein Gordon ◽  
Time Region ◽  
Reduced Differential Transform Method

This paper explores the approximate analytical solution of non-linear Klein-Gordon equations (NKGE) by using multistep modified reduced differential transform method (MMRDTM). Through this proposed strategy, the non-linear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM.

scholarly journals Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Salah Abuasad ◽  
Ishak Hashim ◽  
Samsul Ariffin Abdul Karim
Keyword(s):  
Initial Conditions ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Diffusion Equations ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Fractional Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

In this study, we introduce a new modification of fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for nonhomogeneous linear multiterm time-fractional diffusion equations (MT-TFDEs) of constant coefficients in a bounded domain with suitable initial conditions. Different applications in two and three fractional order terms are given to illustrate our new modification. The approximate solutions are given in the form of series solutions. The results show that the m-FRDTM for MT-TFDEs is a powerful method and can be generalized to other types of multiterm time-fractional equations.

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