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NUMERICAL INTEGRATION

NUMERICAL INTEGRATION. Motivation: Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than evaluating them exactly using a complicated antiderivative of f(x) Example:

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NUMERICAL INTEGRATION

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  1. NUMERICAL INTEGRATION • Motivation: Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than evaluating them exactly using a complicated antiderivative of f(x) • Example: The solution of this integral equation with Matlab is 1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x) we cannot find this solution analytically by techniques in calculus.

  2. Course content • Methods of Numerical Integration • Trapezoidal Rule’s • 1/3 Simpson’s method • 3/8 Simpson’s method • Applied in two dimensional domain

  3. Trapezoidal Rule’s f fp

  4. Function f approximately by function fp. Then, where fp is a linear polynomial interpolation, that is By substitution u=x-x0we have where

  5. Trapezoidal Rule’s f fp

  6. For two interval, we can use summation operation to derive the formula of two interval trapezoidal that is where

  7. Trapezoidal Rule’s f fp

  8. Similar to two interval trapezoidal, we can derive three interval trapezoidal formula that is where • Thus, for n interval we have where and for

  9. 1/3 Simpson’s f fp

  10. Function f approximately by function fp. Then, where fp is a quadratic polynomial interpolation, that is By substitution u=x-x0we have where

  11. 1/3 Simpson’s f fp

  12. For 4 subinterval we have where • Thus, for n subinterval we have where and

  13. 3/8 Simpson’s fp f

  14. Similar to 1/3 Simpson’s method, f approximately by function fp where fp is a cubic polynomial interpolation, that is By substitution u=x-x0we have where and

  15. Numerical Integration in a Two Dimensional Domain c(x) d(x) b= =a

  16. A double integration in the domain is written as • The numerical integration of above equation is to reduce to a combination of one-dimensional problems

  17. Procedure: • Step 1: Define So, the solution is • Step 2: Divided the range of integration [a,b] into N equispaced intervals with the interval size So, the grid points will be denoted by and then we have

  18. Step 3: Divided the domain of integration into N equispaced intervals with the interval size So, the grid points denoted by • Step 4: By Applying numerical integration for one-dimensional (for example the trapezoidal rule) we have for

  19. Step 5: By applying numerical integration (for example trapezoidal rule) in one-dimensional domain we have the solution of double integration is

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