MATLAB solves equations and integrals

Let's use the numerical solution.

function hahaha

t,n] = ode23t(@myfun,[0 1000],[100 100])

plot(t,n)

function dn=myfun(t,n)r1=,r2=,k1=1000,k2=1000,m=9e-6,n=4e-5;

dn1=(r1*(1-n(1)/k1)-m*n(2))*n(1);

dn2=(r2*(1-n(2)/k2)-n*n(1))*n(2);

dn=[dn1;dn2]

Result: t = *n =

You just look at the symbolic operation part, and it is not difficult to solve your equation with MATLAB, just three statements, as follows:

a='2/(sqrt(2*pi)*x))*exp(-y^2/(2*x^2))';

b='y*int(a,''x'',0,+inf)';

solve('b=2*x/sqrt(2*pi)','x'The result of the run is as follows, that is, the result of solving the equation:

ans =1/2*b*pi^(1/2)*2^(1/2)

k=[1 2 3 4 5 6 7];

xk=[ ;

fxk=[ ;

s1=sum(fxk(1:end-1).*diff(xk))) Rectangle.

s2=trapz(xk,fxk);Trapezium.

The rectangle formula treats the differential area element as a rectangle δs(i)=fxk(i)*δx(i).

The trapezoidal formula is to treat the area element of the differential as a trapezoidal δs(i)=(fxk(i)+fxk(i+1)) 2*δx(i).

Rectangle. <>

Trapezium. <>

In general, the area calculation of the trapezoid is more accurate.

1. First of all, the function to find the integral in MATLAB is the int function, you can help int, take a look at the function usage, int(fx, x, m, n) in fx is the function, x is the variable, m and n are the upper and lower limits, as shown in the figure below.

2. Let's take a look at the example of finding the integral, enter syms x a in the command line window, and define the symbol variables x and a, as shown in the figure below.

3. Enter fx = a*x 2 and press the enter key to define the function fx, as shown in the following figure.

4. Enter int(fx,x,1,10) to find the integral, as shown in the figure below.

5. Finally, after pressing the enter key, you can see the integration result of the function in the range of 1-10, if a is a constant, the integration result is multiplied by the constant, as shown in the figure below.

trapz, the rectangular formula is a definition of calculus, assuming that the value of each integral interval is the same, and the integral interval and the resulting value form a rectangle.

The solution of the system of equations with integral you gave is a bit complicated, think about it. This can be solved with the fsolve() function. Ideas for the solution:

1. Create two custom functions, one is a custom equation function, and the other is an integral equation function 2. Use the fsolve() function to solve the values of v, sigma, d1, d2.

x0=[1,,10,;Initial.

x=fsolve(@func,x0);

3. Solve the result.

v=，σ=，p=

If you have any questions, you can discuss them further or discuss them in private messages.

1. Use the int function, which is abbreviated from integrate, int function expression, variable, integration upper limit, integration lower limit.

2. For example, to find an fx = a*x 2, to integrate x in the interval (m, n), first define the four variables m, x, a, b as symbolic variables.

syms m x a b;

fx = a*x^2;

int(fx,x,m,n)

3. Through the above method, you can find the integral of any function in a given interval, if you want to see the writing format, you can use the pretty command, so that the display is closer to the usual representation.

1. In MATLAB, there are many ways to integrate operations, in order to facilitate the processing of similarities and differences in different ways, take the following integration as an example:

2. Trapezoidal integral method.

First, in the simplest way, take the function trapz as an example, z = trapz(x,y) where x is the discretization vector of the integral interval, y is the vector of the same dimension as x, which represents the integrand, and z is the return integral approximation.

clc，clear。

Trapezoidal integral method.

x = :1，y = exp(-x.^2)，s = trapz(x,y)

Result: s =

3. High-precision numerical integration (1).

In order to overcome the problem of low accuracy of the trapezoidal integration method, the high-precision integration method can be adopted, the first method can be z = quad(fun,a,b) This method is the adaptive step Simpson scoring method to obtain the function fun on the interval [a,b] definite integral, as follows:

clc;clear;

Trapezoidal integral method.

s = quad(inline('exp(-x.^2)')1,1)

Result: s =

4. High-precision data integration (2).

The high-precision Lobatto integral method is used in the format: z = quadl(fun, a, b).

clc;clear;

Trapezoidal integral method.

s = quadl(inline('exp(-x.^2)')1,1)

Result: s =

In life or research, you may encounter situations that require integration operations, such as calculating the area of an irregular graph, etc. MATLAB has strong data processing capabilities, as long as any integrable function and integral upper and lower lines are given, it can be used for integration operations. Here's how:

Tools Raw materials.

MATLAB software.

Method steps.

Take f(x)=e 2x+sin(x+ 3), the lower limit of integration: a=0, and the upper limit of integration: b= 4 as an example.

First, create the integrand m file.

Click New and select Function

Enter the command in the device:

function f=f(x)

f=exp(2*x).*sin(x+pi/3);

Note that with*

Save the function file to your own work path, e.g. G: Matlab Work.

Enter the command: CD G: MATLAB WORK

It is the path that becomes the current path.

There are two types of instructions for calculating integrals in MATLAB:

1.f=quad('fname', a, b, tol, trace) Simpson numerical integration method.

fname', a, b, tol, trace) Newton-Cotes numerical integration method.

Wherein: fname is the expression or function name of the integrand, a and b are the upper and lower bounds respectively, and tol can control the integration precision, and omitted is taken; trace=1 is a graph for the integration process, trace=0 for no graph.

Compared with the two, the quad8 has higher accuracy.

Call the integral function squad to perform the calculation. Enter the command:

f=quad('f(x)',0,pi/4)

For other functions, you only need to modify the expression in the function file.

The specific steps for MATLAB to calculate definite integrals are as follows:

1. Take f(x)=e 2x+sin(x+ 3), the lower limit of integration: a=0, and the upper limit of integration: b= 4 as an example.

First, create the integrand m file. Click New and select Function

2. Enter the command in the device: function f=f(x); f=exp(2*x).*sin(x+pi/3);Note that with*

3. Save the function file to your own work path, such as G: Matlab Work.

Enter the command: cd g: matlab work, so that the path becomes the current path.

4. Two kinds of instructions for calculating integrals in MATLAB:

fname', a, b, tol, trace) Simpson numerical integration method.

fname', a, b, tol, trace) Newton-Cotes numerical integration method.

Wherein: fname is the expression or function name of the integrand, a and b are the upper and lower bounds respectively, and tol can control the integration precision, and omitted is taken; trace=1 is a graph for the integration process, trace=0 for no graph.

Compared with the two, the quad8 has higher accuracy.

5. Call the integral function squad for calculation. Enter the command: f=quad('f(x)', 0, pi 4), as shown in the figure, the calculation result can be obtained after entering. For other functions, you only need to modify the expression in the function file.

int is the analytic solution, using Newton's Leibniz formula to find the definite integral, that is, first find the indefinite integral, and then use the upper and lower version limits to substitute, the obtained solution weight is the exact solution, of course, the premise formula has the ability to integrate, and some formulas do not have indefinite integrals.

quadl is a numerical solution, and the basic idea is to solve it according to the original definition of integration, that is, to find the area of each microelement (quadl to find the area of each small trapezoid) in the infinite division of the integration region (between the upper and lower limits) (the division into many segments in the program).

eps is a positive number that is very close to 0 because 0 is substituted into t-3*t^2+2*t.^3).1 3) will make an error and replace it with a very small number.

Numerical solutions are like many methods and many functions.

syms x f1 f2

f1= ((sin(x)).bai3 - sin(x)).5).^1/2);

f2=int(f1,0,pi);

simplify(f2)

The name is formed by the combination of the first three letters of the words matrix and dulaboratory.

The meaning of MATLAB is DAO

The matrix laboratory is mainly used for the access of convenient matrices, and its basic non-prime matrix is a matrix that does not need to define dimensions.

Find the definite integral of f to t over the interval [a,b].

Symbolic solution: e.g. f=t 2*exp(-t); a=0,b=1>> clear

syms t% defines symbolic variables.

f=t^2*exp(-t);

int(f,t,0,1)

ans =2 - 5*exp(-1)

Numerical solution: e.g. f=t 2*exp(-t 3); a=0,b=1>> clear

t=:1;% in the middle is the step size.

f=t.^2.*exp(-t.

3);% to find the value of the function at the node. This one"^"with"*"Front"."Represents the power and multiplication of the corresponding elements.

Since t is a vector, only the square matrix can be multiplied, and the matrix multiplication must also satisfy the corresponding dimensionality relation.

The value of the function at the sum(f* node is equivalent to the height of the rectangle, the step size is equal to the width of the rectangle, and the product is then summed to obtain an approximation of the total area.

ans =

The specific steps for MATLAB to calculate definite integrals are as follows:

1. Take f(x)=e 2x+sin(x+ 3), the lower limit of integration: a=0, and the upper limit of integration: b= 4 as an example. First, create the integrand m file. Click New and select Function

3. Save the function file to your own work path, such as G: Matlab Work.

Enter the command: cd g: matlab work, so that the path becomes the current path.

4. Two kinds of instructions for calculating integrals in MATLAB:

fname', a, b, tol, trace) Simpson numerical integration method.

fname', a, b, tol, trace) Newton-Cotes numerical integration method.

Wherein: fname is the expression or function name of the integrand, a and b are the upper and lower bounds respectively, and tol can control the integration precision, and omitted is taken; trace=1 is a graph for the integration process, trace=0 for no graph.

Compared with the two, the quad8 has higher accuracy.

5. Call the integral function squad for calculation. Enter the command: f=quad('f(x)', 0, pi 4), as shown in the figure, the calculation result can be obtained after entering. For other functions, you only need to modify the expression in the function file.