A wavelet operational method for solving fractional partial differential equations numerically.

*(English)*Zbl 1169.65127Summary: Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult.

In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.

In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.

##### MSC:

65R20 | Numerical methods for integral equations |

26A33 | Fractional derivatives and integrals |

45K05 | Integro-partial differential equations |

65T60 | Numerical methods for wavelets |

##### Keywords:

operational matrix; fractional partial differential equations; Haar wavelets; Lyapunov equation; numerical examples; fractional calculus
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\textit{J. L. Wu}, Appl. Math. Comput. 214, No. 1, 31--40 (2009; Zbl 1169.65127)

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