Special Functions in Fractional Calculus and Engineering

Special functions play a crucial role in solving fractional-order differential equations with engineering applications, due to their non-local nature and memory effect. They offer a variety of methods for modeling and solving complex problems in engineering science, and their applicability is justified by their widespread occurrence in the solution of fractional-order equations.

Format: Hardback
Length: 300 pages
Publication date: 29 June 2023
Publisher: Taylor & Francis Ltd

Fractional calculus, a branch of mathematics that deals with functions that are not defined on the entire real number line, has gained significant importance in solving fractional-order differential equations (FODEs) with engineering applications. These equations arise in various fields, including physics, chemistry, biology, and engineering, and require specialized techniques to handle their non-local nature and memory effect.

One of the key methods for solving FODEs is the use of special functions. Special functions are mathematical functions that have specific properties and can be used to model complex phenomena. They have been widely studied in the field of fractional calculus and have found applications in a wide range of engineering problems.

One of the advantages of using special functions is that they can handle non-local effects, which are common in FODEs. Non-local effects occur when the solution of the equation depends on values of the function at points far away from the point where it is being evaluated. Special functions, such as the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, are able to capture these non-local effects and provide accurate solutions to FODEs.

Another advantage of using special functions is that they have memory effects. Memory effects occur when the solution of the equation depends on the values of the function at previous points in time. Special functions, such as the Mittag-Leffler function and the Laguerre function, are able to capture these memory effects and provide accurate solutions to FODEs with long-term behavior.

There are several important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. These methods include the Adomian decomposition method, the fractional differential equation (FDE) method, and the generalized Riemann-Liouville (GRL) method.

The Adomian decomposition method is a powerful tool for solving FODEs with non-linear terms. It involves decomposing the FODE into a sum of two terms: a non-linear term and a linear term. The non-linear term is then solved using a special function, while the linear term is solved using standard numerical methods.

The FDE method is a method for solving FODEs with linear terms. It involves expressing the FODE in terms of a difference equation and then solving it using a special function. The advantage of this method is that it is easy to implement and can handle a wide range of FODEs.

The GRL method is a method for solving FODEs with both linear and non-linear terms. It involves expressing the FODE in terms of a difference equation and then solving it using a special function. The advantage of this method is that it is able to handle a wide range of FODEs and can provide accurate solutions even for complex FODEs.

Illustrating the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations.

Fractional-order differential equations (FODEs) are a class of equations that arise in various fields of engineering, including physics, chemistry, biology, and engineering. These equations are characterized by their non-local nature and memory effect, which make them difficult to solve using traditional numerical methods. Special functions have been widely used in the solution of FODEs due to their ability to handle non-local effects and memory effects.

One of the most important special functions in the solution of FODEs is the Riemann-Liouville fractional integral operator. This operator is used to solve FODEs with non-linear terms and has been widely studied in the field of fractional calculus equations. The Riemann-Liouville fractional integral operator is a powerful tool for solving FODEs with non-linear terms and has been widely studied in the field of fractional equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

Another important special function in the solution of FODEs is the Caputo fractional derivative operator. This operator is used to solve FODEs with linear terms and has been widely studied in the field of fractional differential equations. The Caputo fractional derivative operator is a powerful tool for solving FODEs with linear terms and has been widely studied in the field of fractional differential equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

In addition to these two important special functions, there are several other special functions that have been used in the solution of FODEs. These include the Mittag-Leffler function, the Laguerre function, and the Legendre function. Each of these functions has its own unique properties and can be used to solve a wide range of FODEs.

Fractional-order differential, integral, and integrodifferential equations are a class of equations that arise in various fields of engineering, including physics, chemistry, biology, and engineering. These equations are characterized by their non-local nature and memory effect, which make them difficult to solve using traditional numerical methods. Special functions have been widely used in the solution of FODEs due to their ability to handle non-local effects and memory effects.

One of the most important special functions in the solution of FODEs is the Riemann-Liouville fractional integral operator. This operator is used to solve FODEs with non-linear terms and has been widely studied in the field of fractional equations. The Riemann-Liouville fractional integral operator is a powerful tool for solving FODEs with non-linear terms and has been widely studied in the field of fractional equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

Another important special function in the solution of FODEs is the Caputo fractional derivative operator. This operator is used to solve FODEs with linear terms and has been widely studied in the field of fractional differential equations. The Caputo fractional derivative operator is a powerful tool for solving FODEs with linear terms and has been widely studied in the field of fractional differential equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

In addition to these two important special functions, there are several other special functions that have been used in the solution of FODEs. These include the Mittag-Leffler function, the Laguerre function, and the Legendre function. Each of these functions has its own unique properties and can be used to solve a wide range of FODEs.

Fractional calculus is a branch of mathematics that deals with functions that are not defined on the entire real number line, has gained significant importance in solving fractional-order differential equations (FODEs) with engineering applications. These equations arise in various fields, including physics, chemistry, biology, and engineering, and require specialized techniques to handle their non-local nature and memory effect.

One of the key methods for solving FODEs is the use of special functions. Special functions are mathematical functions that have specific properties and can be used to model complex phenomena. They have been widely studied in the field of fractional calculus and have found applications in a wide range of engineering problems.

One of the advantages of using special functions is that they can handle non-local effects, which are common in FODEs. Non-local effects occur when the solution of the equation depends on values of the function at points far away from the point where it is being evaluated. Special functions, such as the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, are able to capture these non-local effects and provide accurate solutions to FODEs.

Another advantage of using special functions is that they have memory effects. Memory effects occur when the solution of the equation depends on the values of the function at previous points in time. Special functions, such as the Mittag-Leffler function and the Laguerre function, are able to capture these memory effects and provide accurate solutions to FODEs with long-term behavior.

There are several important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. These methods include the Adomian decomposition method, the fractional differential equation (FDE) method, and the generalized Riemann-Liouville (GRL) method.

The Adomian decomposition method is a powerful tool for solving FODEs with non-linear terms. It involves decomposing method is a powerful tool for solving FODEs with non-linear terms. It involves decomposing the FODE into a sum of two terms: a non-linear term and a linear term. The non-linear term is then solved using a special function, while the linear term is solved using standard numerical methods.

The FDE method is a method for solving FODEs with linear terms. It involves expressing the FODE in terms of a difference equation and then solving it using a special function. The advantage of this method is that it is easy to implement and can handle a wide range of FODEs.

The GRL method is a method for solving FODEs with both linear and non-linear terms. It involves expressing the FODE in terms of a difference equation and then solving it using a special function. The advantage of this method is that it is able to handle a wide range of FODEs and can provide accurate solutions even for complex FODEs.

Illustrating the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations.

Fractional-order differential equations (FODEs) are a class of equations that arise in various fields of engineering, including physics, chemistry, biology, and engineering. These equations are characterized by their non-local nature and memory effect, which make them difficult to solve using traditional numerical methods. Special functions have been widely used in the solution of FODEs due to their ability to handle non-local effects and memory effects.

One of the most important special functions in the solution of FODEs is the Riemann-Liouville fractional integral operator. This operator is used to solve FODEs with non-linear terms and has been widely studied in the field of fractional equations. The Riemann-Liouville fractional integral operator is a powerful tool for solving FODEs with non-linear terms and has been widely studied in the field of fractional equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

Another important special function in the solution of FODEs is the Caputo fractional derivative operator. This operator is used to solve FODEs with linear terms and has been widely studied in the field of fractional differential equations. The Caputo fractional derivative operator is a powerful tool for solving FODEs with linear terms and has been widely studied in the field of fractional differential equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

In addition to these two important special functions, there are several other special functions that have been used in the solution of FODEs. These include the Mittag-Leffler function, the Laguerre function, and the Legendre function. Each of these functions has its own unique properties and can be used to solve a wide range of FODEs.

Fractional-order differential, integral, and integrodifferential equations are a class of equations that arise in various fields of engineering, including physics, chemistry, biology, and engineering. These equations are characterized by their non-local nature and memory effect, which make them difficult to solve using traditional numerical methods. Special functions have been widely used in the solution of FODEs due to their ability to handle non-local effects and memory effects.

One of the most important special functions in the solution of FODEs is the Riemann-Liouville fractional integral operator. This operator is used to solve FODEs with non-linear terms and has been widely studied in the field of fractional equations. The Riemann-Liouville fractional integral operator is a powerful tool for solving FODEs with non-linear terms and has been widely studied in the field of fractional equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

Another important special function in the solution of FODEs is the Caputo fractional derivative operator. This operator is used to solve FODEs with linear terms and has been widely studied in the field of fractional differential equations. The Caputo fractional derivative operator is a powerful tool for solving FODEs with linear terms and has been widely studied in the field of fractional differential equations. It is able to capture non-local effects and memory effects and provides accurate solutions to FODEs with long-term behavior.

In addition to these two important special functions, there are several other special functions that have been used in the solution of FODEs. These include the Mittag-Leffler function, the Laguerre function, and the Legendre function. Each of these functions has its own unique properties and can be used to solve a wide range of FODEs.

Weight: 740g
Dimension: 234 x 156 (mm)
ISBN-13: 9781032435008