Fractional Differential Equations: Mathematics and Its Applications

Fractional differential equations are a generalization of classical differential equations to arbitrary non-integer Free Downloads. They have been used to model a wide variety of phenomena in science and engineering, including fluid flow, heat transfer, viscoelasticity, and finance.

This book provides a comprehensive to the theory and applications of fractional differential equations, with a focus on mathematical methods. It is suitable for graduate students and researchers in mathematics and applied sciences.

**Chapter 1: **

Methods of Mathematical Modelling: Fractional Differential Equations (Mathematics and its Applications)

by Noel Streatfeild

4.8 out of 5

Language	:	English
Hardcover	:	264 pages
Item Weight	:	1.05 pounds
Dimensions	:	6.14 x 0.93 x 9.52 inches
File size	:	15232 KB
Screen Reader	:	Supported
Print length	:	256 pages

FREE

Fractional differential equations are a generalization of classical differential equations to arbitrary non-integer Free Downloads. They have been used to model a wide variety of phenomena in science and engineering, including fluid flow, heat transfer, viscoelasticity, and finance.

In this chapter, we will provide a brief to fractional calculus and fractional differential equations. We will also discuss the applications of fractional differential equations in various fields.

1.1 Fractional Calculus

Fractional calculus is the study of derivatives and integrals of arbitrary Free Download. The fractional derivative of a function f(x) of Free Download α is defined as

$$D_x^\alpha f(x) = \frac{d^\alpha}{dx^\alpha}f(x) = \frac{1}{\Gamma(n-\alpha)}\int_0^x \frac{f^{(n)}(t)}{(x-t)^{1-\alpha}}dt,$$

where n is the smallest integer greater than or equal to α and Γ(x) is the gamma function.

The fractional integral of a function f(x) of Free Download α is defined as

$$I_x^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_0^x \frac{f(t)}{(x-t)^{1-\alpha}}dt.$$

1.2 Fractional Differential Equations

A fractional differential equation is an equation that contains fractional derivatives or integrals of the unknown function. The general form of a fractional differential equation is

$$a_0D_t^{\alpha_0}y(t) + a_1D_t^{\alpha_1}y(t) + \cdots + a_nD_t^{\alpha_n}y(t) = f(t),$$

where α0, α1, ..., αn are arbitrary real numbers and a0, a1, ..., an are constants.

1.3 Applications of Fractional Differential Equations

Fractional differential equations have been used to model a wide variety of phenomena in science and engineering, including:

• Fluid flow
• Heat transfer
• Viscoelasticity
• Finance
• Control theory
• Signal processing
• Image processing

1.4 Numerical Methods for Fractional Differential Equations

There are a variety of numerical methods for solving fractional differential equations. These methods can be divided into two main categories:

• Direct methods: These methods solve the fractional differential equation directly, without first converting it to a system of ordinary differential equations.
• Indirect methods: These methods convert the fractional differential equation to a system of ordinary differential equations, which can then be solved using standard numerical methods.

1.5 Open Problems and Future Directions

There are a number of open problems and future directions in the study of fractional differential equations. These include:

• Developing new and more efficient numerical methods for solving fractional differential equations.
• Investigating the stability and convergence of numerical methods for fractional differential equations.
• Developing new analytical methods for solving fractional differential equations.
• Applying fractional differential equations to new problems in science and engineering.

Fractional differential equations are a powerful tool for modeling a wide variety of phenomena in science and engineering. This book provides a comprehensive to the theory and applications of fractional differential equations, with a focus on mathematical methods. It is suitable for graduate students and researchers in mathematics and applied sciences.

Methods of Mathematical Modelling: Fractional Differential Equations (Mathematics and its Applications)

by Noel Streatfeild

4.8 out of 5

Language	:	English
Hardcover	:	264 pages
Item Weight	:	1.05 pounds
Dimensions	:	6.14 x 0.93 x 9.52 inches
File size	:	15232 KB
Screen Reader	:	Supported
Print length	:	256 pages

FREE