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Integral Equation :Integral Equation: 1. Definition Integral Equation: An integral equation is an equation is which an unknown function appears under one or more integral sign. 2. Linear and Non-Linear Integral equation: An integral equation is called linear if only linear operation are performed in it upon the unknown function. An integral equation which is not linear is known as a non-linear integral equation. Most general type of integral equation is of the form ( ) ( ) ( ) ( ) ( ) . In this case upper limit may be variable or constant. And = + ∫ , lower limit is always constant. 3. If upper limit is constant then integral equation is known as Fredholm integral equation. If upper limit is variable ( ) then integral equation is known as Volterra integral equation. 4. In above expression if () ≠ then integral equation is third kind. And if () = 0 then integral equation is first kind. Again when () = 1 then integral equation is second kind. If () = 0 then integral equation is homogeneous. 5. Singular Integral equation: When one or both limits of integration become infinite or when the kernel become infinite at one or more points within the range of integration, the integral equation is known as singular integral equation. 6. Symmetric kernel: A kernel (, ) is symmetric (hermitian) if (, ) = (, ) in case of real values = . 7. Seprable of Degenerate kernel: (, ) = ∑ () ℎ() note that () can be regarded as linearly independent. 8. Integral equation of the Convolution type: Kernel must be of type (, ) = ( − ) 9. Iterated Kernels: Iterated kernels (, ), = 1,2,3, … are defined as: (, ) = (, ) and ( ) ( ) ( ) , = ∫ , , = 2,3, …. 10. Resolvent Kernel or Reciprocal Kernel: It written in place of (, ) to find solution in form (, ; ) Γ(x, t; λ) 11. Eigen Values and Eigen functions: The values of parameter λ for which integral equation has a non-zero solution. λ = 0 is not an eigen value. If () is an eigen function then () is also eigen function on same eigen value. 12. Note that a homogeneous fredholm integral equation of the second kind may, generally, have no eigen value and eigenfunction, or it may not have nay real eigen value or eigen functions. () 13. Important integral: ∫ () = ∫ () Note that LHS is multiple integral of order ()! n and RHS is ordinary integral of order one. 14. Inner product: ( ) ( ) ( ) , = ∫ / ‖ ‖ | ( )| 15. () = ∫ 16. |(, )| ≤ ‖‖‖‖ (Schwarz inequality) ‖ + ‖ ≤ ‖‖ + ‖‖ (Minkowski inequality) :Conversion of Ordinary differential equation into integral equation: 1. Initial value problem: When an ordinary differential equation is to be solved under conditions involving dependent variable and its derivative at the same vale of the independent variable. 19 2. Boundary value problem: When an ordinary differentiable equation is to be solved under conditions involving dependent variable and its derivative at two different values of independent variable. Then the problem consideration is said to be boundary value problem. :Homogeneous Fredholm integral equations of the second kind with separable or degenerate kernels: 1. Characteristic values: The number = 0 is not eigenvalue since for = 0 yields () = 0 which is zero solution. 2. If the kernel (, ) is continuous in the rectangle ∶ ≤ ≤ , ≤ ≤ , and the number are finite, then to every eigen value there exist a finite number of linearly independent eigenfunctions. The number of such functions is known as index of eigen value. Different eigen value has different indices. 3. A homogeneous Fredholm integral equation may, generally, have no eigen values and eigen functions or it may not have any real eigen value and eigen function. 4. Solution of homogeneous fredholm integral equation of the second kind with separable kernel: ( ) ( ) ( ) ( ) ∑ ( ) ( ) Given = ∫ , and , = () and = ∫ () then 1 − − − ⎡ ⋯ ⎤ − 1 − eigen value is given by () = ⎢ ⎥ = 0 and = ⎢ ⋮ ⋱ ⋮ ⎥ − − ⎣ ⋯ 1 − ⎦ ( ) ( ) and solution is ( ) ∑ ( ) ∫ = :Method of Successive Approximations: 1. Iterated Kernels or Functions: Let fredholm integral equation of the second kind () = () + ( ) ( ) then Iterated kernels ( ) are defined as follows ( ) ∫ , , , = 1,2,3, … , = ( ) ( ) ( ) ( ) , and , = ∫ , ( , ) , ∫ , ( , ) = 2,3, ….. this can be done for Volterra integral equation of the second kind by putting = . ( ) ( ) ( ) ( ) 2. ℎ interated kernel , satisfies the relation , = ∫ , , where is any positive integer less than . ( ) ( ) ( ) ( ) 3. Let = + ∫ , be given Fredholm integral equation of the second kind. Suppose that (a) Kernel (, ) ≢ 0, is real and continuous in the rectangle R, for which ≤ ≤ , ≤ ≤ also let |(, )| ≤ , (b) () ≢ 0, is real and continuous in the interval I, for which ≤ ≤ . Also let |()| ≤ (c) is constant such that || ≤ () Then equation has a unique solution in I and this solution is given by the absolutely and uniformly ( ) ( ) ( ) ( ) ( ) ( ) ( ) convergent series = + ∫ , + ∫ , ∫ , + ⋯ 20 4. Let ( ) ( ) ( ) ( ) be given Volterra integral equation of the second kind. = + ∫ , Suppose that (d) Kernel (, ) ≢ 0, is real and continuous in the rectangle R, for which ≤ ≤ , ≤ ≤ also let |(, )| ≤ , (e) () ≢ 0, is real and continuous in the interval I, for which ≤ ≤ . Also let |()| ≤ (f) is constant such that || ≤ () Then equation has a unique solution in I and this solution is given by the absolutely and uniformly convergent series ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + ∫ , + ∫ , ∫ , + ⋯ 5. Solution of Fredholm integral equation of the second kind by successive approximations, Iterative methods: Consider Fredholm integral equation of the second kind ( ) ( ) ( ) ( ) then ( ) ∑ and solution is ( ) = + ∫ , , ; = (, ) = ( ) ( ) ( ) + ∫ , ; 6. Let (, ; ) be the resolvent kernel of a Fredholm integral equation ( ) ( ) ( ) ( ) = + ∫ , then the resolvent kernel satisfies the integral equation ( ) ( ) ( ) ( ) . , ; = , + ∫ , , ; 7. The series of resolvent kernel (, ; ) , (, ; ) = ∑ (, ) is absolutely and uniformly convergent for all values of and in the circle || < . (,;) 8. The resolvent kernel satisfies the integro-differential equation = ∫ (, ; ) (, ; ) 9. (, ), = 1,2,3, … are iterated kernels and – (, ) = ∑ (, ), then (, ) and (, ) are known as reciprocal if they are both real and continuous in R as satisfy (, ) + (, ) = ( ) ( ) ( ) ( ) ( ) ( ) ∫ , , and , + , = ∫ , , 10. Volterra’s solution of Fredholm integral equation of the second kind given () = () + ( ) ( ) if (1) ( ) is real and continuous in ( ) ∫ , , ≤ ≤ ≤ ≤ , ≠ 0 (2) () is real and continuous in I and () ≠ 0 (3) If a function (, ) reciprocal to (, ) exists, then the given integral equation has unique solution given by ( ) ( ) ( ) ( ) = − ∫ , 11. Solution of Volterra integral equation of the second kind by successive Approximations: Volterra integral equation of the second kind ( ) ( ) ( ) ( ) , as a zero order = + ∫ , approximation to the required solution (), take () = () and find further approximation by relation ( ) ( ) ( ) ( ) = + ∫ , 12. Important Result: (, ; ) be the resolvent kernel of a Volterra integral equation. () = () + ( ) ( ) then resolvent kernel satisfies the integral equation ( ) ( ) ∫ , , ; = , + ( ) ( ) ∫ , , ; ( ) ( ) ( ) ( ) 13. Solution of Voterra integral equation of second kind : = + ∫ , when its ( ) kernel (, ) is of some particular form. (1) (, ) = () + ()( − ) + () + ! 21 ( ) ( ) ⋯ + () then (, ; ) = ,; where (, ; ) is a solution of the differential ()! equation − () + () + ⋯ + () = 0 satisfying the condition = = = ⋯ = = 0 ℎ = and = 1 ℎ = then solution is given by () = () () + ∫ (, ; )(). (2) (, ) = () + ()( − ) + () + ⋯ + ! ( ) ( ) () then (, ; ) = ,; where ℎ(, ; ) is a solution of the differential ()! equation − () + () + ⋯ + ()ℎ = 0 satisfying the condition ℎ = = = ⋯ = = 0 ℎ = and = 1 ℎ = then solution is given by () = ( ) ( ) ( ) + ∫ , ; . 14. Solution of Voterra integral equation of the second kind by reducing to differential equation (by earlier method given above). 15. Voterra integral equation of the first kind can be converted to a Voterra integral equation of the ( ) ( ) ( ) () second kind. As given ∫ , = () then second kind is given as = − (,) (,) ( ) ∫ here p is depend on condition (, ) ≠ 0 (,) :Classical Fredholm Theory: 1. Fredholm’s First Fundamenta theorem: The non-homogeneous Fredholdm integral equation of the second kind ( ) ( ) ( ) ( ) where the functions ( ) and ( ) are = + ∫ , ( ) ( ) ( ) ( ) integrable, has a unique solution = + ∫ , ; where the resolvent kernel ( ) (, ; ) is given by (, ; ) = ,; with D() ≠ 0, is a meromorphic function of the complex () variable , being the ration of two entire functions definied by the series (, ; ) = (, ) + , … , () ∑ ∫ ∫ … ∫ … and ! , … , , … , () () = 1 + ∑ ∫ ∫ … ∫ … both which converge for all values of . ! , … , , … , (, ) ⋯ (, ) = ⋮ ⋱ ⋮ which is known as Fredholm determinant. , … , (, ) ⋯ (, ) ( ) ( ) ( ) 2. The solution of the Fredholm homogeneous equation = ∫ , is identically zero. (,;) 3. () = () + ∫ (, )() resolvent kernel is (, ; ) = where (, ; ) = () ( ) (, ) + ∑ (, ) and () = 1 + ∑ () where ! !
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