how can i solve eps y''+ mu a(x) y'-b(x)y=f(x) with boundary condition y(0)=y(1)=0

i want to know the function code for reaction convection diffusion equation in ode with two paremeter epsilon and mu

[Sam Chak]

Hi @Dhayalan G

Are you interested in learning how to code and then writing the function code yourself?

Or, would you prefer BVP experts in this forum to provide the full code with arbitrarily assigned values for the parameters eps, mu, a(x), b(x), and f(x)?

Clarifying this will help us determine the appropriate level of guidance to match your skill level.

Answer 1

[charan]

Hello Dhayalan,

The equations of the above form can be solved by using "diff" and "dsolve" functions in MATLAB. The "diff" function is used to define the first and seconder order derivatives of "y". Then the equation and the initial conditions are defined and solved using "dsolve". Here is an example:

syms y(x)
ode = 2*x^2*diff(y,x,2)+3*x*diff(y,x)-y-x== 0;
cond1 = y(0) == 0;
cond2 = y(1) == 0;
conds = [cond1 cond2];
ySol(x) = dsolve(ode,conds)

ySol(x) = 

[Torsten]

And if a(x), b(x) and c(x) get too complicated, use "bvp4c".

Answer 2

[Areeba]

import numpy as np

from scipy.sparse import diags

from scipy.sparse.linalg import spsolve

def solve_rcd_equation(epsilon, mu, a, b, f, x_min, x_max, N):

# Discretization

dx = (x_max - x_min) / (N - 1)

x = np.linspace(x_min, x_max, N)

# Constructing the differentiation matrix

diagonals = [[epsilon/dx**2], [-(epsilon/dx**2 + mu*a(x)/2*dx)], [epsilon/dx**2 + mu*a(x)/2*dx - b(x)]]

D = diags(diagonals, [-1, 0, 1], shape=(N, N)).toarray()

# Boundary conditions

D[0, 0] = 1

D[0, 1] = 0

D[-1, -1] = 1

D[-1, -2] = 0

# Right-hand side

rhs = f(x)

rhs[0] = 0

rhs[-1] = 0

# Solve the system

y = spsolve(D, rhs)

return x, y

# Example functions for a, b, and f

def a(x):

return 1

def b(x):

return 1

def f(x):

return np.sin(np.pi * x)

# Parameters

epsilon = 0.1

mu = 0.5

x_min = 0

x_max = 1

N = 100

# Solve the equation

x, y = solve_rcd_equation(epsilon, mu, a, b, f, x_min, x_max, N)

# Plot the solution

import matplotlib.pyplot as plt

plt.plot(x, y)

plt.xlabel('x')

plt.ylabel('y')

plt.title('Solution of Reaction-Convection-Diffusion Equation')

plt.grid(True)

plt.show()

[Torsten]

Looks like python, not MATLAB, doesn't it ?