DrHuang.com, Sydney, Australia

## Introduction

The science of physics is built fundamentally upon differential equations. Many of the most useful differential equations appearing in physics are of partial differential equations, known as math physical equations, so that understanding the solutions that can arise and their interconnections has much to say about possible systems amenable to analyze physical problems.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDE 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 (ODE), which deal with functions of a single variable and their derivatives.

PDE 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 (FDE) 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 [1-5].

In this paper, it is shown that analytical solution of fractional differential equation are solved by MathHand.com. MathHand.com is the online math calculator, its former is SymbMath [6-7]. The examples included fractional differential equation, fractional partial differential equation, fractional integral equation, equation of mixed Fractional differential and integral orders, system of fractional differential equation, complex order differential equation, and a variable order differential equation. By default, the Caputo definition of fractional calculus [8] is used here.

## Differential equation

By default, the unknown function is y(x) and its initial value is y(0). An independent variable is x for ordinary differential equation. For a partial differential equation, the unknown function is y( ) and its two independent variables are x and t. For a system of equations, two unknown functions are x(t) and y(t) with an independent variable t.There are 6 way to input derivative of y: y(1,x), ds(y,x), d(y(x),x), ds(y), d(y(x)), y'

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 4 way to input the 0.5 order integral of y: y(-0.5,x), ds(y,x,-0.5), d(y(x),x,-0.5), ints(y,x,0.5)

Table 1. Comparison of different orders

Order name equation y( ) ds( ) d( ) ints( ) i complex order `(d^i y)/dx^i - 2y = exp(x)` y(i,x) ds(y,x,i) d(y(x),x,i) 2 second order `(d^2 y)/dx^2 - 2y = exp(x)` y(2,x) ds(y,x,2) d(y(x),x,2) 1.5 1.5 order `d^1.5/dx^1.5 y - 2y = exp(x)` y(1.5,x) ds(y,x,1.5) d(y(x),x,1.5) 1 first order `dy/dx - 2y = exp(x)` y(1,x) ds(y,x) d(y(x),x) 0.5 semi differential order `d^0.5/dx^0.5 y - 2y = exp(x)` y(0.5,x) ds(y,x,0.5) d(y(x),x,0.5) -0.5 semi integral order `d^-0.5/dx^-0.5 y - 2y = exp(x)` y(-0.5,x) ds(y,x,-0.5) d(y(x),x,-0.5) ints(y,x,0.5) -1 integral order `int y\ dx-2y = exp(x)` y(-1,x) ds(y,x,-1) d(y(x),x,-1) ints(y,x) -2 double integral order `int int y\ (dx)^2 - 2y = exp(x)` y(-2,x) ds(y,x,-2) d(y(x),x,-2) ints(y,x,2) x variable order `(d^x y)/dx^x - 2y = exp(x)` y(x,x) ds(y,x,x) d(y(x),x,x) Input your equation into mathHand.com, click the "

**dsolve**" button for solution, then click the "**test**" button to test its solution. e.g.

input**y(1,x) - 2y = exp(x)**as first order differential equation for

`y^((1))(x) - 2y = exp(x)`

Or input the dsolve( ) function, click the "**=**" button for solution, then click the "**test**" button to test its solution. e.g.

input dsolve( ds(y) - 2y = exp(x) ) for

dsolve `dy/dx - 2y = exp(x)`

Its solution is in Table 2.Table 2. Comparison of differential equations with different orders

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 y)/dx^2 - 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) -2 double integral equation `int int y\ (dx)^2 - 2y = exp(x)` `C_1*exp(1/sqrt(2)*x)` -exp(x) cos(x) variable order equation `(d^cos(x) y)/dx^cos(x) - 2y = exp(x)` `C_1*E_cos(x)(2x^cos(x))` -exp(x) ### Solve graphically

Some differential equations cannot be solved symbolly, but can be solved numerically and graphically by the ODE plot function odeplot( ), e.g. input sin(x)-cos(y) for y' = sin(x)-cos(y), then tick the**y'=**or**y"=**checkbox to solve first or second order differential eq.## Partial differential equation

Input your equation, click the**dsolve**button, then click the**plot2D**button to show its curve graph where you can change a time t value, or click the**plot3D**button to show its 3D graph where you can spin graph.### A first order differential time equation

Input ds(y,t,1) = ds(y,x,2)-ds(y,x)-exp(x)-exp(t) as a first order partial differential time t in the left hand side of the equation,

`(dy)/dt = (d^2y)/dx^2-dy/dx-exp(t)-exp(x)`

e.g. a diffusion equation in one dimemtion:

`(dy)/dt = (d^2y)/dx^2 + dy/dx`

e.g. a diffusion equation in 2 dimemtions:

`(dy)/dt = (d^2y)/dx^2 + (d^2y)/(du^2) `

e.g. a spherical diffusion equation:

`(dy)/dt = (d^2y)/(dx^2) + 1/x*dy/(dx)-y `

e.g. a diffusion equation in 3 dimemtions:

`(dy)/(dt) = (d^2y)/dx^2 + (d^2y)/(du^2) + (d^2y)/(dv^2) `### A second order differential time equation,

Input ds(y,t,2) = ds(y,x,2)-ds(y,x)-exp(x)-exp(t) as second order partial differential time t in the left hand side of the equation,

`(d^2y)/dt^2 = (d^2y)/dx^2-dy/dx-exp(t)-exp(x)`

e.g. a wave equation in one dimemtion:

`(d^2y)/dt^2 = (d^2y)/dx^2+dy/dx `

e.g. a wave equation in two dimemtions:

`(d^2y)/(dt^2) = (d^2y)/dx^2 + (d^2y)/(du^2) `

e.g. a wave equation in three dimemtions:

`(d^2y)/(dt^2) = (d^2y)/dx^2 + (d^2y)/(du^2) + (d^2y)/(dv^2) `## Integral equation

Integral equation is equation with the -1 differential order. e.g.

Input ds(y,x, -1) - 2y = exp(x) for

`d^-1/dx^-1 y - 2y = exp(x)`

Intput integrates(y) = 2y for

`int y` dx - 2y = exp(x)Its solution is in Table 2. Integral equation can be converted to differential equation by differentiating both sides, and then it is solved.

### Double Integral equation

Double integral equation is equation with the -2 differential order. e.g.

Intput ds(y,x,-2) - 2y = exp(x) for

`int int y` `(dx)^2` - 2y = exp(x)

Its solution is in Table 2.## Fractional differential equation

Differential and integral equations can be extend to fractional differential and integral equations by fractional order of fractional calculus. e.g.

Input ds(y,x,0.5) = 2y as semi differential equation with the 0.5 differential order for

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

dsolve( `d^0.5/dx^0.5 y = 2y` )Its solution is in Table 2. 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( )**It is similar to linear differential equation, so method to solve fractional differential equation is similar to differential equation [3-5].

## Fractional partial differential equation

Consider a time fractional partial differential equation with fractional order differential time. e.g.

Input ds(y,t,0.5)=ds(y,x,2)-ds(y,x)-exp(x)-exp(t) as semi partial differential time t in the left hand side of the equation,

`(d^0.5y)/dt^0.5 = (d^2y)/dx^2-exp(x)-exp(t)`

`(d^0.5y)/dt^0.5 = (d^2y)/dx^2-dy/dx-exp(t)-exp(x)`## Fractional integral equation

Integral equation is extend to fractional integral equation by minus fractional order. e.g.

Input ds(y,x, -0.5) - 2y = exp(x) as semi integral equation with the -0.5 differential order for

`d^-0.5/dx^-0.5 y - 2y = exp(x)`

Intput integrates(y,x,0.5) - 2y = exp(x) for

`int y` `(dx)^0.5` - 2y = exp(x)Its solution is in Table 2. 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( )**. Difference between Caputo definition and the Riemann-Liouville (R-L) definition are in section 6 of fractional calculus [8]. e.g.dsolve(y(-0.5,x)=1) give zero.

lasolve(y(-0.5,x)=1) give nonzero.Integral equation can be converted to differential equation by differentiating both sides. In the same way, fractional integral equation aslo can be converted to fractional differential equation, then it is solved.

## Equation of mixed Fractional differential and integral orders

A mix of differential and integral equation, e.g.

`d^(pi)/dx^(pi) y - int y(x) (dx)^pi` = exp(x)## System of Fractional differential equations

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) in a system of differential equations.

dsolve(x(1,t)=x-t, y(1,t)=t-x)

A system of differential equations is extend to system of fractional differential equation by fractional order. e.g.

`(d^0.5x)/dt^0.5 = 2x, (d^0.5y)/dt^0.5 = x+y`## Complex order differential equation

Differential equation is extend to complex order differential equation by extension of order to complex number. e.g.

Input ds(y,x,i) - 2y -exp(x) = 0 as complex order differential equation for

`(d^(i)y)/dx^(i)- 2y - exp(x) = 0`

Its solution is in Table 2.## Variable order differential equation

When the order is variable, it is a variable order differential equation. e.g. the order is continually changing between 1 and -1 in the cos(x) order differential equation. The order change is similar to a change of the n-order fractional derivative `d^n/dx^n x` in below animation Fig.1.

`(d^cos(x) y)/dx^cos(x) - 2y = exp(x)`

Its solution is in Table 2, where `E_cos(x)( x )` is Mittag-Leffler function.

Fig. 1. Animation of the order changes between 1 and -1.## Comparison of differential equations with different orders

It is interesting to compare differential equations with different orders in Table 2. It is shown in Table 2 that all of solutions of the constant order equations are in the same format exp(k x). When the order of differential equations decreased from 2 to 0.5, their general solution increased from exp(sqrt(2)*x) to exp(4x). When the order of integral equations decreased from -0.5 to -2, their general solution also increased from exp(1/4 x) to exp(1/sqrt(2) x), but all of their parcular solutions are unchanged.More examples are found in website http://drhuang.com/index/example/ [9], and the examples included the differential equations [10] which other software cannot, e.g. Wolfram Bugs [11].

## Test solution

Test solution for (fractional) differential equation by the solutions back into equation to check if result is zero. There sre three ways to check solutions:- Click the
**"test"**button to test solution, if it is not 0, then click the plot2D button to show a line of 0. - Use the test( ) to test solution by
**test(solution, equation)**, e.g.

`test( exp(4x), d^0.5/dx^0.5 y = 2y )`

f:= 2y+1, eq:=y'-f=0, s:=dsolve(eq), test(s, eq)

f:= 2y+1, eq:=ds(y,x,0.5)-f=0, s:=dsolve(eq), test(s, eq)

- the
**ntest( )**is numeric test to put a random number into independent variable back to the equation by**ntest(solution, equation)**.

## References

- K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London (1974).
- B. Ross. fractional calculus and its applications. Springer, Berlin, Heidelberg, 1975.
- K.B. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).
- I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
- Y. Hu, Y. Luo, Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method, Volume 215, Issue 1, 15 May 2008, p 220-229.
- W. Huang, SymbMath: A Program for Symbolic Mathematics, Int. J. Math. Edu. Sci. Technol., 1992, 23(1), 160-165.
- W. Huang, SymbMath Update to Version 2.0, Abs. Amer. Math. Soc., 1992, 13(6), 535.
- Fractional calculus , http://drhuang.com/science/mathematics/fractional_calculus/, last accessed 2021/04/21.
- Examples of Fractional Calculus Computer Algebra System, http://drhuang.com/index/example/, last accessed 2021/04/21.
- Special differential equation, https://jingyan.baidu.com/article/19020a0a6bb358529d284293.html, last accessed 2021/04/21.
- Wolfram bugs, http://drhuang.com/index/bugs/, last accessed 2021/04/21.

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