Differential equation

Differential equations describe exchanges of matter, energy, information or any other quantities, often as they vary in time and/or space. Their thorough analytical treatment forms the basis of fundamental theories in mathematics and physics, and they are increasingly applied in chemistry, life sciences and economics. Actually, it enlightens the basic inherent mechanism of any biological or any physical system. Basically it is a backbone of any system through which we can easily understand the underlying dynamics of that system.  

Differential equations are solved by integration, but unfortunately, for many practical applications in science and engineering, systems of differential equations cannot be integrated to give an analytical solution, but rather need to be solved numerically. 

Depending on the problem, mathematical formalization may consist of ordinary differential equations (ODE), partial differential equations (PDE), differential algebraic equations (DAE), or delay differential equations (DDE). In addition, a distinction is made between initial value problems (IVP) and boundary value problems (BVP). All of these differential equation described above represents the deterministic counterpart of any system i.e. there does not exist any sorts of environmental influence i.e. these kind of system exhibits their deterministic counterpart. Now, when natural phenomena affects any system then that system  becomes random and in case of mathematical language, it is known Stochastic i.e. we can predict that system with a certain probability. This kind of situation can be represented through Stochastic Differential equation (SDE). 

In our regular graduation level course we studied a lots of method in order to solve the differential equation but as i mentioned above if we can't express the underlying mechanism of any system through our known methods then it can be solved with some numerical procedure like Runga-Kutta 4th order method (RK4), Euler method, Picard method, Adams- Baseforth method, Adams-Moulton method, etc. These methods are famous due to their order of convergence i.e. how quick they approach towards their solution by minimizing the error. 

All of the aforementioned methods are available in the R software. Moreover for each class of differential equation, there exist several packages. I have listed all of the packages in the following:

• Initial value problems (IVP) of ordinary differential equations (ODE) : deSolve

• Initial value differential algebraic equations (DAE) : deSolve

• Initial value partial differential equations (PDE) : deSolve or Reactran

• Boundary value problems (BVP) of ordinary differential equations: bvpSolve or Reactran or rootSolve

• Initial value delay differential equations (DDE): deSolve or PBSddesolve 

• Stochastic differential equations (SDE) : pomp or sde

Beside all of these methods there exist another method, the discretisation method i.e. if we consider the unit step length difference between the independent variable. The following codes illustrates how to solve differential equations in R software. 

First, i will describe the easiest method i.e. the discrete approximation method 

### Let us first consider the following 1st order differential equation

### dx/dt = rx(1-x/K)

## We can easily solve this problem with the help of integration method

### But we will try to solve numerically 

## First, Create a function demonstrating the discrete approximation method

## Basically, we are using the iteration scheme here

diff.eqn = function(x0 = 0.2, K = 100, r = 0.3, t = 200)  

            {

              x = c(x0, numeric(t))

               for(i in 1:t)

                 {

                   x[i+1] = x[i] + r*x[i]*(1-x[i]/K)

                 }

              return(x)

             }

solution = diff.eqn(); solution ### Numerical solution of the method

plot(time(solution), solution, type = "l", xlab = expression(bold(time)), ylab =    expression(bold(x(t))) #### Graphical representation of the solution

#### Now, we use any standard numerical procedure as mentioned above 

diff.eqn.continuous.set.up = function(t, state, param){

  with(as.list(c(state,param)),{

    dX = r*X*(1-X/K);

    return(list(c(dX)));

  })

}

state = c(X = 0.2)

parameters = c(r=0.3, K = 17.3)

library(deSolve)

times = seq(0, 30, by = 0.01)

out = ode(y = state, times = times, func = diff.eqn.continuous.set.up, parms = parameters)

head(out)

plot(out, ylim=c(0, 100), main = "Solution of the differential equation", method = "rk4", xlab = "time", ylab = "X(t)", col = 2, lwd = 2)

##### Solving simultaneous system of equation