In the present work, a generalized fractional integral operational matrix is derived by using classical Legendre wavelets. Then, a numerical scheme based on this operational matrix and Lagrange multipliers is proposed for solving variational problems with fractional order. This approach has been applied on some illustrative examples. The results obtained for these examples demonstrate that the suggested technique is efficient for solving variational problems with fractional order and gives a very perfect agreement with the exact solution. The results are depicted in graphical maps and data tables. The integral square error, maximum absolute error, and order of convergence have been evaluated to analyze the precision of the suggested method. The present scheme provides better and comparable results with some other existing approaches available in the literature.

在目前的工作中，使用经典的勒让德小波推导了广义分数积分运算矩阵。然后，提出了基于该运算矩阵和拉格朗日乘数的数值方案，用于求解分数阶变分问题。此方法已应用于一些说明性示例。这些示例获得的结果表明，所提出的技术对于解决分数阶变分问题非常有效，并且与精确解非常吻合。结果显示在图形地图和数据表中。已对积分平方误差，最大绝对误差和收敛阶进行了评估，以分析所建议方法的精度。