Abstract
The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.
Highlights
The increasing interest in applications of fractional calculus, together with the difficulty of finding analytical solutions of fractional differential equations (FDEs), naturally forces researchers to study, devise and apply numerical methods to solve a large range of ordinary and partial differential equations with fractional derivatives.The investigation of computational methods for fractional-order problems is a very active research area in which, each year, a large number of research papers are published.The task of finding efficient and reliable numerical methods for handling integrals and/or derivatives of fractional order is a challenge in its own right, with difficulties that differ in character but are no less severe than those associated with finding analytical solutions
The numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties
In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results
Summary
The increasing interest in applications of fractional calculus, together with the difficulty of finding analytical solutions of fractional differential equations (FDEs), naturally forces researchers to study, devise and apply numerical methods to solve a large range of ordinary and partial differential equations with fractional derivatives.The investigation of computational methods for fractional-order problems is a very active research area in which, each year, a large number of research papers are published.The task of finding efficient and reliable numerical methods for handling integrals and/or derivatives of fractional order is a challenge in its own right, with difficulties that differ in character but are no less severe than those associated with finding analytical solutions. The increasing interest in applications of fractional calculus, together with the difficulty of finding analytical solutions of fractional differential equations (FDEs), naturally forces researchers to study, devise and apply numerical methods to solve a large range of ordinary and partial differential equations with fractional derivatives. The scientific literature is rich with examples of methods that are inappropriate for fractional-order problems. In most cases these are just methods that were devised originally for standard integer-order operators applied in a naive way to their fractional-order counterparts; without a proper knowledge of the specific features of fractional-order problems, researchers are often unable to understand why unexpected results are obtained
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