In this paper, a method is implemented to a one-dimensional inverse problem with a parabolic differential equation of fractional order in which the fractional derivative is in the Caputo sense. The considered inverse problem involves a time-dependent source control parameter . In order to numerically solve the problem, first, the main problem is converted to a homogeneous problem by Lagrange interpolation. Consequently, a new problem is derived by a practical technique that verifies all the conditions of the main problem. Finally, a system of nonlinear algebraic equations is solved by Newton's method to obtain the unknown coefficients. It is notable that all the needed computations are done in MATHEMATICA . In this work, operational matrices of Bernoulli polynomials are stated and applied to approximate functions. Illustrative examples are included to prove the efficiency and applicability of the proposed methods. In the numerical tests, a low amount of polynomials is needed to acquire a precise estimate solution. For demonstrating the low running time of this method, CPU time for all examples is exhibited.

中文翻译：

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一维分数阶逆问题的伪谱方法

在本文中，对分数阶抛物线微分方程的一维逆问题实施了一种方法，其中分数阶导数为卡普托意义。所考虑的逆问题涉及与时间相关的源控制参数 。为了数值求解该问题，首先通过拉格朗日插值将主问题转化为齐次问题。因此，通过验证主要问题的所有条件的实用技术推导出一个新问题。最后，通过牛顿法求解非线性代数方程组，得到未知系数。值得注意的是，所有需要的计算都是在 MATHEMATICA 中完成的。在这项工作中，伯努利多项式的运算矩阵被陈述并应用于近似函数。包括说明性示例以证明所提出方法的效率和适用性。在数值测试中，需要少量多项式来获得精确的估计解。为了证明此方法的低运行时间，展示了所有示例的 CPU 时间。