MATHEMATICA[solve numerical integration and find min value]

In summary, the conversation discusses the problem of finding the minimum sum of an equation using numerical integration, specifically in Excel. The equation involves integration and the goal is to find a value for t that will make the equation reach the minimum value. The speaker tried to solve the problem step by step, but faced difficulties with taking the derivative and plotting the function. They eventually found a possible solution through iteration.
  • #1
shafieza_garl
20
0
i want to find the minimum sum of this equation and find the t like using solver in Excel,but since the integration cannot be integrate directly,i didnt know how to use the numerical integration to apply for this problem.
i want to find the value :let say when n=2, t_0=0 and t_n=1, so i need to find t_1 that will make the equation become minimum value.I hope that anyone can help me.

Sum[Integrate[4.9*^6*E^(-0.03*t + 0.03*(t + Subscript[t, j]))*
((1 - 0.015*(6 + t))/E^(0.015*Subscript[t, j]) +
(-0.91 + 0.015*Subscript[t, j])/E^(0.015*t))^2,
{t, Subscript[t, -1 + j], Subscript[t, j]}], {j, 1, n}]
 
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  • #2
write the equation you want to solve in normal way (like what written in your book) so i can help
 
  • #3
i had attached the equation.how can we use numerical integration in mathematica to solve the problem?or any other method can be used?
 

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  • #4
Let us try to solve it step by step
first I'll open the sum and get the following equation
http://www.3mints.info/upload/uploads/bbb65e0e17.jpg
and x is t1

press enter and get this answer
http://www.3mints.info/upload/uploads/f53bc6e9d8.jpg

now to find minimum you can take derivative and make it equal zero but i faced problem when trying solve function to zero i got the following massege

olve::tdep: The equations appear to involve the variables to be \
solved for in an essentially non-algebraic way

so i have to find another way . I plot the function to see the shape
http://www.3mints.info/upload/uploads/65a285154e.jpg

as you notice the minimum value of x is around 0.5 corresponding to arround 1 as a value for cos t.
I evaluated the function x at numbers between 0.4 to 0.6 with 0.00002 step and found that cos t = 0.894953
see figure
http://www.3mints.info/upload/uploads/116575a4af.jpg

and by iteration you can find value of x

x = 0.51219

please this is not for sure the right answer but it could give an idea
 
  • #5


Thank you for reaching out regarding your question on Mathematica and numerical integration. The equation you have provided appears to be a complex one with multiple variables and nested functions. In order to find the minimum value of this equation, you can use Mathematica's built-in function "NMinimize". This function will numerically search for the minimum value of a given expression by varying the specified variables. In your case, you can specify the variables as n and t_1 and set the appropriate constraints for their values. For example, you can set n=2 and t_1 to be between 0 and 1.

Here is an example code that you can try:

NMinimize[{Sum[Integrate[4.9*^6*E^(-0.03*t + 0.03*(t + Subscript[t, j]))*
((1 - 0.015*(6 + t))/E^(0.015*Subscript[t, j]) +
(-0.91 + 0.015*Subscript[t, j])/E^(0.015*t))^2,
{t, Subscript[t, -1 + j], Subscript[t, j]}], {j, 1, n}], n == 2 && 0 < t_1 < 1}, {n, t_1}]

This code will give you the minimum value of the expression and the corresponding values of n and t_1 that minimize it. You can also adjust the constraints and variables as needed for your specific problem.

I hope this helps and feel free to reach out if you have any further questions or concerns.
 

1. How can I use Mathematica to solve numerical integration problems?

To solve a numerical integration problem using Mathematica, you can use the function "NIntegrate". This function takes in the function you want to integrate, the limits of integration, and any other necessary parameters, and returns the numerical value of the integral. For example, if you want to integrate the function f(x) from 0 to 1, you can use the command NIntegrate[f[x], {x, 0, 1}].

2. What kind of functions can be integrated using Mathematica?

Mathematica can integrate a wide range of functions, including algebraic, trigonometric, exponential, and logarithmic functions. It can also handle more complex functions such as piecewise-defined functions and functions involving special functions like Bessel and Gamma functions.

3. Can Mathematica find the minimum value of a function?

Yes, Mathematica has a built-in function called "Minimize" that can find the minimum value of a function. This function takes in the function you want to minimize, the variables to be varied, and any constraints, and returns the minimum value and the values of the variables at which the minimum occurs.

4. How accurate are Mathematica's numerical integration and minimum value calculations?

Mathematica uses advanced algorithms and numerical methods to ensure high accuracy in its integration and optimization calculations. However, the accuracy may vary depending on the complexity of the function and the precision settings specified by the user. It is recommended to check the accuracy of the results using different precision settings and by comparing with known solutions.

5. Can Mathematica handle multidimensional integration and optimization?

Yes, Mathematica has functions such as "NIntegrate" and "Minimize" that can handle multidimensional integration and optimization problems. These functions can take in multiple variables and can handle functions with multiple input dimensions.

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