Friday, April 3, 2020

HP Prime: Solving Integral Equations

HP Prime:  Solving Integral Equations

Introduction

The program INTEGRALSOLVE solves the following integral equation for x:

x
∫ f(X) dX  - a = 0
0

using Newton's method.

Big X represents the variable of f(X) to be integrated while small x is the x that needs to be solved for.

Taking the derivative of the above integral using the Second Fundamental Theorem of Calculus:

d/dx [ ∫( f(X) dX from X=0 to X=x ) - a ]
= d/dx [  F(x) - F(0) - a ]
= d/dx [ F(x) ] - d/dx [ F(0) ] - d/dx [ a ]
= d/dx [ F(x) ]
= f(x)

F(X) is the anti-derivative of f(X).  F(0) and a are numerical constants, hence the derivative of each evaluates to 0.

Newton's Method to solve for any function g(x) is:

x_n+1 = x_n - g(x_n) / g'(x_n)

Applying this to the equation, Newton's Method gives:

x_n+1 = x_n - [ ∫( f(X) dX from X=0 to X=x_n ) - a ] / f(x_n)

HP Prime Program INTEGRALSOLVE

Note:  Enter f(X) as a string and use capital X.  This program is designed to be use in Home mode.

EXPORT INTEGRALSOLVE(f,a,x)
BEGIN
// f(X) as a string, area, guess
// ∫(f(X) dX,0,x) = a
// EWS 2019-07-26
// uses Function app
LOCAL x1,x2,s,i,w;
F0:=f;
s:=0;
x1:=x;
WHILE s==0 DO
i:=AREA(F0,0,x1)-a;
w:=F0(x1);
x2:=x1-i/w;
IF ABS(x1-x2)<1 font="" then="">
s:=1;
ELSE
x1:=x2;
END;
END;

RETURN approx(x2);
END;

Examples



Radians angle mode is set.

Example 1:

Solve for x:

x
∫ sin(X) dX = 0.75
0

Initial guess:  1

Result:  x ≈ 1.31811607165

Example 2:

Solve for x:

x
∫ e^(X^2) dX = 0.95
0

Initial guess:  2

Result:  x ≈ 0.768032819934

Pretty powerful.

April 2020:  New Posts Every Friday, Saturday, and Sunday! 

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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