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A Simple Introduction to Integral EQUATIONS

A Simple Introduction to Integral EQUATIONS. Glenn Ledder Department of Mathematics University of Nebraska-Lincoln gledder@math.unl.edu. Some Integral Equation Examples. Volterra. second kind. Volterra. second kind. Fredholm. second kind. Volterra. first kind.

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A Simple Introduction to Integral EQUATIONS

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  1. ASimpleIntroductionto Integral EQUATIONS Glenn Ledder Department of Mathematics University of Nebraska-Lincoln gledder@math.unl.edu

  2. Some Integral Equation Examples Volterra secondkind Volterra secondkind Fredholm secondkind Volterra firstkind

  3. Integral Equations from Differential Equations First-Order Initial Value Problem Volterra Integral Equation Of the Second Kind Used to prove theorems about differential equations. Used to derive numerical methods for differential equations.

  4. Volterra IEs of the second kind (VESKs) • Symbolic solution of separable VESKs • Volterra sequences and iterative solutions • Theory of linear VESKs • Brief mention of Fredholm IEs of 2nd kind • An age-structured population model

  5. Volterra Integral Equationsof the Second Kind (VESKs) Giveng : R3→Randf: R→R,findy: R→Rsuch that [equivalent to an initial value problem if g is independent of t]

  6. Volterra Integral Equationsof the Second Kind (VESKs) Giveng : R3→Randf: R→R,findy: R→Rsuch that Linear: Separable: Convolved:

  7. Symbolic Solution of Separable VESKs Solve Let Then New Problem: Solution:

  8. Solve Let and New Problem: Solution:

  9. Volterra Sequences Givenf, g, and y0, definey1, y2, … by and so on.

  10. Volterra Sequences Givenf, g, and y0, definey1, y2, …by • Does ynconverge to some y? • If so, does ysolve the VESK?

  11. Volterra Sequences Givenf, g, and y0, definey1, y2, …by • Does ynconverge to some y? • If so, does ysolve the VESK? • If so, is this a useful iterative method?

  12. A Linear Example Lety0=1. Then ↓↓ The sequence converges to the known solution.

  13. Convergence of the Sequence y0 y1 y2 y3 y4 y5 y6 y

  14. A Nonlinear Example Lety0=0. Then

  15. Convergence of the Sequence y3 y2 y1

  16. Lemma 1 Homogeneous linear Volterra sequences decay to 0 on [a, T] whenever k is bounded. Proof: (a=0) Choose constants K,Y and T such that Then

  17. Given we have In general, so

  18. Theorem 2Homogeneous linear VESKshave the unique solutiony≡0. Proof: Letφ be a solution. Choosey0=φ. Thenyn=φfor alln, soyn→φ. But the lemma requiresyn→0. Thereforeφ≡0.

  19. Theorem 3Linear VESKs have at most one solution. Proof: Suppose and Lety=φ–ψ. Then By Theorem 2, y≡0;hence,φ=ψ.

  20. Lemma 4Linear Volterra sequences with y0=f converge. Sketch of Proof: Define 1) gn→ 0 2) Together, these properties prove the Lemma.

  21. Theorem 5Linear VESKs have a unique solution. Proof: By Lemma 4, the sequence converges to some y. Taking a limit as n→∞: The solution is unique by Theorem 3.

  22. Fredholm Integral Equations of the Second Kind (linear) • Separable FESKs can be solved symbolically. • Fredholm sequences converge (more slowly than Volterra sequences) only if||k||is small enough. • The theorems hold if and onlyif ||k|| is small enough. Giveng:R3→Randf:R→R,findy:R→Rsuch that

  23. An Age-Structured Population Given An initial population of known age distribution Age-dependent birthrate Age-dependent death rate Find Total birthrate Total population Age distribution

  24. An Age-Structured Population Simplified Version Given An initial population of newborns Age-dependent birthrate Age-independent death rate Find Total birthrate Total population

  25. The Founding Mothers Assume a one-sex population. Starting population is 1 unit. Life expectancy is 1 time unit. p0′=-p0, p0(0)=1 Result: p0(t) = e-t

  26. Basic Birthrate Facts = Total Birthrate Births to Founding Mothers + Births to Native Daughters b(t)=m(t)+d(t) Letf(t) be the number of births per mother of agetper unit time. m(t) = f(t)e-t

  27. Births to Native Daughters Consider daughters of ages x to x+dx. All were born between t-x-dx and t-x. The initial number was b(t-x)dx. The number at time t is b(t-x)e-xdx. The rate of births is f(x)b(t-x)e-xdx =m(x)b(t-x)dx.

  28. The Renewal Equation or This is a linear VESK with convolved kernel. It can be solved by Laplace transform.

  29. Solution of the Renewal Equation Given the fecundity functionf, define where L is the Laplace transform. Then

  30. A Specific Example Letf(x)=axe-2x. Then Ifa=9, thenp(t) →1.5. Exponential growth fora>9. The population dies if a<9.

  31. Fecundity a=10 a=9 a=8

  32. Birth Rate a=10 a=9 a=8

  33. Population a=10 a=9 a=8

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