This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with and error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

中文翻译：

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一维和二维非线性广义Benjamin-Bona-Mahony Burgers方程的局部分数阶导数逼近


本研究提出了一种求解广义非线性 Benjamin-Bona-Mahony-Burgers 方程的数值方法，具有变阶局部时间分数阶导数。该导数被表示为两个函数的乘积，即通常的整数阶时间导数和具有小数指数的时间函数。然后，使用前向差分近似来求时间导数。使用 Haar 小波近似 (HWA) 估计微分问题的未知解和相应的导数。然后在HWA中实现配置过程，将给定模型转换为线性代数方程组，以确定Haar小波级数的未知常数系数，更新导数和数值解。为所提出技术的稳定性建立了充分条件，然后进行了计算验证。为了检查该方案的性能，还给出了一些一维和二维的说明性示例以及误差范数。除此之外，还计算了两种类型方程的计算收敛率。此外，将计算的解决方案与文献中的可用结果进行比较。模拟和图形数据表明，所建议的方案对于此类复杂问题非常有效。