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.

中文翻译：

---

二维Sine-Gordon方程的解析解

本文成功地采用了简化的差分变换方法（RDTM）来求解具有适当初始条件的二维非线性正弦-Gordon方程。证明了一些引理，可以帮助我们用所提出的方法解决治理问题。该方案的优点是可以生成具有方便确定成分的收敛幂级数形式的解析近似解或精确解。该方法考虑使用适当的初始条件，并且在没有任何离散，变换或限制性假设的情况下找到了解决方案。我们的四个测试问题证明了所提方法的准确性和效率，并使用表格和图表显示了测试问题的解决方案行为。更多，发现数值结果与文献中可用的精确解和数值解非常吻合。我们已经证明了所提出方法的收敛性。而且，所得结果表明，所介绍的方法有望用于解决科学和工程领域中的其他类型的非线性偏微分方程（NLPDE）。