# A new operational approach for solving fractional calculus and frational differential equations numerically

Jiunn Lin Wu, Chin Hsing Chen

Research output: Contribution to journalArticle

12 Citations (Scopus)

### Abstract

Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.

Original language English 1077-1082 6 IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A 5 Published - 2004 May

### Fingerprint

Fractional Calculus
Fractional Differential Equation
Differential equations
Differential equation
Numerical Methods
Mathematical operators
Operational Matrix
Numerical methods
Orthogonal Functions
Operator
Orthogonal functions
Analytical Solution
Unstable
Engineering
Demonstrations

### All Science Journal Classification (ASJC) codes

• Signal Processing
• Computer Graphics and Computer-Aided Design
• Electrical and Electronic Engineering
• Applied Mathematics

### Cite this

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title = "A new operational approach for solving fractional calculus and frational differential equations numerically",
abstract = "Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.",
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In: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E87-A, No. 5, 05.2004, p. 1077-1082.

Research output: Contribution to journalArticle

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T1 - A new operational approach for solving fractional calculus and frational differential equations numerically

AU - Wu, Jiunn Lin

AU - Chen, Chin Hsing

PY - 2004/5

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N2 - Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.

AB - Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.

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