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# Fractional Differential Equation

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1. ## Differential equation

there are 3 way to input derivative y: y', y(1,x), or ds(y,x,1)
there are 3 way to input second order derivative y: y'', y(2,x), or ds(y,x,2)
Input y(1,x) as first order differential equation dy/dx - 2y=0

2. ## Fractional differential equation

Differential equation is extend to fractional differential equation by fractional order. e.g. input ds(y,x,0.5) as semi differential equation d^0.5/dx^0.5 y = 2y by fractional calculus.

3. ## Fractional integral equation

Integral equation is extend to fractional integral equation by minus fractional order. e.g. input ds(y,x, -0.5) as semi integral equation d^-0.5/dx^-0.5 y = 2y

4. ## Solve (fractional) differential equation

Solution of linear fractional differential equation = general solution + parcular solution,
similar to linear differential equation. So method to solve fractional differential equation is similar to differential equation.

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.

Solve (fractional) differential equation for y by the dsolve(), e.g.
dsolve( y'=2y )
dsolve( d^0.5/dx^0.5 y=2y )

5. ## Test solution for (fractional) differential equation

The odetest() is ode test to test solution for (fractional) differential equation, e.g.
odetest( exp(4x), d^0.5/dx^0.5 y=2y )
f:= 2y+1, eq:=y'-f=0, s:=dsolve(eq), odetest(s, eq)
f:= 2y+1, eq:=ds(y,x,0.5)-f=0, s:=dsolve(eq), odetest(s, eq)

6. ## Comparison of differential equations

Ordernameequationgeneral solutionparcular solution
nfractional differential equation d^n/dx^n y - 2y=exp(x)C_1*exp(2^(1/n)*x)-exp(x)
2second order differential equation d^2/dx^2 y - 2y=exp(x)C_1*exp(sqrt(2)*x)-exp(x)
1.51.5 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)
0.5semi differential equation d^0.5/dx^0.5 y - 2y=exp(x)C_1*exp(4*x)-exp(x)
-0.5semi integral equation d^-0.5/dx^-0.5 y - 2y=exp(x)C_1*exp(1/4*x)-exp(x)
-1integral 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 of the differential equation decreased from 2 to 0.5, its general solution increased from exp(sqrt(2)*x) to exp(4x). When the n order of the 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. Their change are similar to a change of the n order in fractional derivative d^n/dx^n x in below picture.

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