Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical techniques for FPDEs with boundary conditions, although some methods do exist to address such problems. In this paper, a modified technique based on the Adomian decomposition method with Laplace transformation is presented, which effectively treats initial-boundary value problems. The non-linear term has been controlled by Daftardar-Jafari polynomials. Our proposed technique is applied to several initial and boundary value problems and the obtained results are presented through graphs. The differing behavior of the solutions for the suggested problems is observed by using various fractional orders. It is found that our proposed technique has a high rate of convergence towards the exact solutions of the problems. Moreover, while implementing this modification, higher accuracy is achieved with a small number of calculations, which is the main novelty of the proposed technique. The present method requires a new approximate solution in each iteration that adds further accuracy to the solution. It demonstrates that our suggested technique can be used effectively to solve initial-boundary value problems of FPDEs.

翻译：针对具有初始条件的分数阶偏微分方程（FPDEs）及其方程组，已发展出多种解析与数值求解技术。然而，尽管存在部分方法可用于处理此类问题，为具有边界条件的FPDEs开发解析或数值技术仍面临更大挑战。本文提出一种基于Adomian分解法与拉普拉斯变换的改进技术，能有效处理初边值问题。非线性项通过Daftardar-Jafari多项式进行控制。我们将所提技术应用于若干初边值问题，并通过图表展示所得结果。通过采用不同分数阶次，观察到所研究问题解的行为差异。研究发现，所提技术对问题精确解具有较高的收敛速度。此外，在实施该改进方法时，仅需少量计算即可获得更高精度，这是本技术的主要创新点。该方法在每次迭代中都需要新的近似解，从而进一步提升了解的精度。结果表明，我们提出的技术可有效用于求解FPDEs的初边值问题。