Question

4. Using inbuilt function in MATLAB, solve the differential equations: dx --t2 dt subject to the condition (01 integrated from0 tot 2. Compare the obtained numerical solution with exact solution 5. Lotka-Volterra predator prey model in the form of system of differential equations is as follows: dry dt dy dt where r denotes the number of prey, y refer to the number of predators, a defines the growth rate of prey population, B defines the efficiency rate at which predators find prey and kill them μ represents the growth rate of predator population due to prey η is the decay rate of predators. Taking 1, 0.05, H 0.02 and 0.5 (a) Solve the model system using inbuilt function ode45 with (0)10, (010. (b) Plot the solutions as r vs t and y vs t in a single graph eUingsnaphikalilc it parat

Matlab Code for these please.

Answer 1

The matlab codes for both questions are given below.

Q 4. ******************script1.m*************************

Analytical_Sol=@(t)t.^2-exp(-t)-2*t+2;

dxdt=@(t,x)-x+t^2;
x0=1;
%Numerical Solution
[t,sol] = ode45(dxdt, [0, 2], x0);

%Analytical solution
y=Analytical_Sol(t);
err=abs(sol-y);
fprintf('Numerical\t\tAnanlytical\n');
for i=1:length(t)
fprintf('%f\t\t%f\n',sol(i),y(i));
end

Q5. *******************************script2.m**********************************

clc;
clear all

alpha=1;
beta=0.05;
u=0.02;
n=0.5;
x0=10;
y0=10;
%f(1) represent dx/dt and f(2) represent dy/dt
func = @(t,f) [alpha*f(1)-beta*f(1)*f(2);u*f(1)*f(2) - n*f(2)];

[t,sol] = ode45(func, [0, 20], [x0,y0]);
subplot(211)
%sol(:,1) contain prey population
plot(t,sol(:,1))
hold on

%sol(:,2) contain predator
plot(t,sol(:,2))
legend('prey','predator')
hold off

%Now vary mu and see the change
y=0.01:0.01:0.1;
subplot(212)
for i=1:length(y)
u=y(i);
func = @(t,f) [alpha*f(1)-beta*f(1)*f(2);u*f(1)*f(2) - n*f(2)];
[t,sol] = ode45(func, [0, 20], [x0,y0]);
plot(t,sol(:,1))
hold on
plot(t,sol(:,2))
end
title('changing mu')
hold off