In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by computers, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems.
Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations.
By default, the Caputo definition of fractional calculus is used by dsolve(). If you want to use the Riemann defintion, use the Laplace transform solver lasolve().
lasolve(y(-0.5,x)=1) give nonzero.
dsolve(y(-0.5,x)=1) give zero.
there are 4 way to input derivative of y: y', y(1,x), ds(y,x), d(y(x))
there are 3 way to input second order derivative of y: y(2,x), ds(y,x,2), d(y(x),x,2)
there are 3 way to input the 0.5 order derivativeof y: y(0.5,x), ds(y,x,0.5), d(y(x),x,0.5)
there are 5 way to input the 0.5 order integral of y: integral(y,x,0.5), ints(y,x,0.5), y(-0.5,x), ds(y,x,-0.5), d(y(x),x,-0.5)
Input y(1,x) - 2y = 0 as first order differential equation
`dy/dx - 2y=0`
A second order differential time equation, e.g. a wave equation.
Input ds(y,t,2) = ds(y,x,2)-ds(y,x)-cos(x)-exp(t)-sin(x) as second order partial differential time t in the left hand side of the equation,
Integral equation can be converted to differential equation by differentiating both sides, but not every integral equation can.
Property of a fractional differential equation is the same as a differential equation.
Solution of linear fractional differential equation = general solution + parcular solution = gsolution()+psolution(),
similar to linear differential equation. So method to solve fractional differential equation is similar to differential equation.
Solve (fractional) differential equation for y by dsolve() or lasolve(), e.g.
dsolve( y'=2y )
dsolve( `d^0.5/dx^0.5 y=2y` )
lasolve( `d^0.5/dx^0.5 y=2y` )
Integral equation can be converted to differential equation by differentiating both sides. Some fractional integral equation aslo can be converted to fractional differential equation, but not every fractional integral equation can.
By default, the unknown function are x(t) and y(t) and their variable is t, their initial value are x(0) and y(0).
x(0):=1, y(0):=1, lasolve(x(1,t)=x-t,y(1,t)=t-x)
The order change is similar to a change of the n-order fractional derivative `d^n/dx^n x` in below picture.
The order changes from -1 to 1.
|Order||name||equation||general solution||parcular solution|
|i||complex order differential equation||`d^i/dx^i y - 2y=exp(x)`||`C_1*exp(1/2^i*x)`||-exp(x)|
|2||second order differential equation||`d^2/dx^2 y - 2y=exp(x)`||`C_1*exp(sqrt(2)*x)`||-exp(x)|
|1.5||1.5 order differential equation||`d^1.5/dx^1.5 y - 2y=exp(x)`||`C_1*exp(2^(2/3)*x)`||-exp(x)|
|1||differential equation||`d/dx y -2y=exp(x)`||`C_1*exp(2*x)`||-exp(x)|
|0.5||semi differential equation||`d^0.5/dx^0.5 y - 2y=exp(x)`||`C_1*exp(4*x)`||-exp(x)|
|-0.5||semi integral equation||`d^-0.5/dx^-0.5 y - 2y=exp(x)`||`C_1*exp(1/4*x)`||-exp(x)|
|-1||integral equation||`int y\ dx-2y=exp(x)`||`C_1*exp(1/2*x)`||-exp(x)|
All of their solutions are in the same format exp(k x). When the n-order differential equation decreased from 2 to 0.5, its general solution increased from exp(sqrt(2)*x) to exp(4x). When the n-order differential equation decreased from -0.5 to -1, its general solution increased from exp(1/4 x) to exp(1/2 x). but their parcular solutions are unchanged.