2. Basic results

Following the ordinary differential equations with boundary value condition

dt<sup>n</sup> <sup>¼</sup> f t; <sup>x</sup>;

u0

u ∂x∂y

> c Dα

The following fractional differential equation will boundary value condition.

uð Þ¼ 0 0, uð Þ¼ 1

Dα

<sup>þ</sup> <sup>C</sup>∂<sup>2</sup> u <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>D</sup> <sup>∂</sup><sup>u</sup> ∂x

dx dt ;…;

highest derivative). This will be discussed. Existence and uniqueness of solution for initial

ðÞ¼ t f tð Þ ; u tð Þ

Differential equations contains derivatives with respect to two or more variables is called a

where u is dependent variable and A, B, C, D, E, F and G are function of x, y above equation is

This will be discussed. Existence of solution for semilinear elliptic equation. Consider a func-

�Δu ¼ f uð Þ in Ω u ¼ u<sup>0</sup> on ∂Ω

where <sup>f</sup> : <sup>R</sup><sup>m</sup> ! <sup>R</sup><sup>m</sup> is a typically nonlinear function. And fractional differential equations. This will be discussed. Fractional differential equations are of two kinds, they are Riemann-Liouville fractional differential equations and Caputo fractional differential equations with

> <sup>t</sup> u tðÞ¼ Bu tð Þ; t > 0 uð Þ¼ 0 u<sup>0</sup> ∈ X

<sup>t</sup> is the Caputo fractional derivative of order α ∈ð Þ 0; 1 , and t ∈½ � 0; τ , for all τ > 0.

<sup>0</sup>þu tð Þþ f tð Þ¼ ; u tð Þ <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>t</sup> <sup>&</sup>lt; <sup>1</sup>, <sup>1</sup> <sup>&</sup>lt; <sup>α</sup> <sup>≤</sup> <sup>2</sup>

ð1 0 u sð Þds,

u tð Þ¼ <sup>0</sup> u0:

� �

dn�<sup>1</sup> x dtn�<sup>1</sup>

ð Þ¼ <sup>x</sup><sup>1</sup> <sup>c</sup>1, …, yð Þ <sup>n</sup>�<sup>1</sup> ð Þ¼ xn�<sup>1</sup> cn�<sup>1</sup> the positive integer <sup>n</sup> (the order of the

<sup>þ</sup> <sup>E</sup> <sup>∂</sup><sup>u</sup> ∂y

þ Fu ¼ G

dnx

where y xð Þ¼ <sup>0</sup> 0, y<sup>0</sup>

value problem (IVP).

partial differential equation (PDEs). For example,

4 Differential Equations - Theory and Current Research

1. Elliptic equation if <sup>B</sup><sup>2</sup> � <sup>4</sup>AC � � <sup>&</sup>lt; <sup>0</sup>,

2. Hyperbolic equation if <sup>B</sup><sup>2</sup> � <sup>4</sup>AC � � <sup>&</sup>gt; <sup>0</sup>,

3. Parabolic equation if <sup>B</sup><sup>2</sup> � <sup>4</sup>AC � � <sup>¼</sup> 0.

tion <sup>u</sup> : <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> that solves,

boundary value.

Dα

where <sup>c</sup>

A ∂2 u <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>B</sup> <sup>∂</sup><sup>2</sup>

classified according to discriminant <sup>B</sup><sup>2</sup> � <sup>4</sup>AC � � as follows,

Throughout the rest of the chapter unless otherwise stated ð Þ X; d stands for a complete metric space.
