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<p>In this paper, we present a hybrid numerical scheme to solve linear and nonlinear advection-diffusion type partial differential equations. We propose a hybrid procedure that couples the Haar wavelet (HW) with a time integrator solver based on Runge–Kutta method of order four (RK-4). The current procedure is designed in such a way that it tackle the non-linear terms of the underlying problem without the need of linearization techniques. Initially, the spatial derivatives are discretized via HW basis using a collocation approach transforming the model into a system of ordinary differential equations (ODEs). Then, the RK-4 technique is employed to solve the resultant system of ODEs. The efficiency and accuracy of the proposed solution strategy is demonstrated by computing different error estimates for various test models. The computed results are compared with exact solutions and with the state-of-the-art schemes. Simulations demonstrate the performance of the proposed method for various test problems. The computational stability of the proposed solution strategy is also discussed theoretically and computationally.</p>

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Solving one- and two-dimensional advection-diffusion type initial boundary value problems with a wavelet hybrid scheme

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月4日

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  • Aslam Khan,
  • Abdul Ghafoor,
  • Emel Khan,
  • Kamal Shah,
  • Thabet Abdeljawad,
  • Manar Alqudah

摘要

In this paper, we present a hybrid numerical scheme to solve linear and nonlinear advection-diffusion type partial differential equations. We propose a hybrid procedure that couples the Haar wavelet (HW) with a time integrator solver based on Runge–Kutta method of order four (RK-4). The current procedure is designed in such a way that it tackle the non-linear terms of the underlying problem without the need of linearization techniques. Initially, the spatial derivatives are discretized via HW basis using a collocation approach transforming the model into a system of ordinary differential equations (ODEs). Then, the RK-4 technique is employed to solve the resultant system of ODEs. The efficiency and accuracy of the proposed solution strategy is demonstrated by computing different error estimates for various test models. The computed results are compared with exact solutions and with the state-of-the-art schemes. Simulations demonstrate the performance of the proposed method for various test problems. The computational stability of the proposed solution strategy is also discussed theoretically and computationally.