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ravicplk
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can anyone help me on the concept picard iteration?
thanks in advance
thanks in advance
thanks bro.HallsofIvy said:What do you want? A full course in the subject?
Here is a simple example: To solve the equation y'= y, y(0)= 1 using Picard iteration, start by approximating y by a constant. Since we know y(0)= 1, "1" is a good choice. Then the equation becomes y'= 1 and, integrating, y= x+ C. When x= 0, y(0)= C= 1 so the first "iteration" gives y= x+ 1.
Now the equation is y'= x+ 1. Integrating, y= (1/2)x2+ x+ C and, setting x= 0, y(0)= C= 1 so y= (1/2)x2+ x+ 1.
Now the equation is y'= (1/2)x2+ x+ 1. Integrating, y= (1/6)x3+ (1/2)x+ x+ C and, setting x= 0, y(0)= C= 1 so y= (1/6)x3+ (1/2)x2+ x+ 1.
Now the equation is y'= (1/6)x3+ (1/2)x2+ x+ 1. Integrating, y= (1/24)x4+ (1/6)x3+ (1/2)x2+ x+ C and, letting x= 0, y(0)= C= 1 so y= y= (1/24)x4+ (1/6)x3+ (1/2)x2+ x+ 1.
Continuing the iteration will give higher and higher powers of x. It should be clear now that we are getting terms of the form (1/n!)xn and that this is giving higher and higher order Taylor Polynomials for ex, the actual solution to y'= y, y(0)= 1.
Picard Iteration is a numerical method used to approximate the solution of a differential equation. It works by breaking down the differential equation into smaller, simpler equations and then solving them iteratively until the solution converges to the desired accuracy.
Picard Iteration can be used to solve a wide range of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and integral equations. It is particularly useful for solving non-linear problems that cannot be solved analytically.
Picard Iteration is a simple and efficient method for solving differential equations. It is easy to implement and does not require a lot of computational resources. It also provides a good approximation of the solution, even for complex non-linear problems.
Picard Iteration may not converge for all types of differential equations. In some cases, the solution may oscillate or diverge instead of converging. It also requires a good initial guess to start the iteration process, which can be difficult to determine for certain problems.
Yes, there are other methods similar to Picard Iteration, such as Newton's method and Runge-Kutta methods. These methods also use iterative processes to approximate the solution of a differential equation, but they may converge faster or be more suitable for certain types of problems.