501 PHYS ِProf. Awatif Ahmad Hindi ُEnter

Reference 1- W.E boyce and R.C Diprema , "elementary differential equations " 3rd edition (1975), johnwily 2- E.A coddington , “ an introduction to ordinary differential equation “ , prentice –hali (1961) 3- E.Kreyszig “advanced engineering mathematics “ 7th edition , johnwily (1993) 4- L.S. ross , “introduction to ordinary differential equations” 4th edition , john wily (1989) 5- Abramowitz , M. stegun , I.A. hand book of mathematical function . dover, new York (1962)

Reference 6- hochstadt , H. “special function of mathematical physics “ hold , rineheart , winstone , new york (1961) 7- Lebedev,N.N.Special Functions and their Applications,Prentice-Hall,Englewood Cliffs,N.J.(1965) 8-Rainvile,E.D.”Special Functions,Macmillan,New York(1960).

contents Special Functions of mathematics Integral Equation Differential equation

Special Functions of mathematics Gamma and Beta functions Definition Properties of the Beta and Gamma functions: some examples

Definition We define the Gamma and Beta functions respectively by

Properties of the Beta and Gamma functions:

Definition of the Gamma function for negative values of the argument

simplify where possible Some examples Express each of the following integrals in terms of Gamma or Beta functions and simplify where possible

Bessel Functions Bessel’s equation of order n is We shall solve (1)by using Frobinous method the solution of (1) is given by :

The explicit relation ship between and for integral n is shown in the following theorems Theorem 4 Theorem 1 Theorem 5 Theorem 2 Theorem 6 Theorem 3 Theorem 7

Theorem 1 When n is an integer (positive or negative)

The two independent solutions of Bessel’s equation may be taken to be Theorem 2 The two independent solutions of Bessel’s equation may be taken to be For all values of n.

Generating function for the Bessel functions Theorem 3 Generating function for the Bessel functions

Integral representations for Bessel functions: Theorem 4 Integral representations for Bessel functions:

Theorem 5

Recurrence Relations

graphs of the Bessel functions

graphs of the Bessel functions

graphs of the Bessel functions

Orthogonally of the Bessel functions Theorem 6 Orthogonally of the Bessel functions If are roots of the equation

Theorem 7 Bessel Series If f(x) is defined in the region and can be expanded in the form Where the are the roots of the equation then

problems 1-Use the generating function to prove that 2- Show that

Integral equation Definition(1) Defintion (2) Defintion (3)

Integral equation Definition(1)) An integral equation is an equation in which an unknown function appears under one or more integral signs. Naturally, in such an equation there can occur other terms as well .

For example The equation for for

Where the function is the unknown function which all the functions are known are integral equation . These functions may be complex –valued functions of the variables S and t for

Integral equation Definition(2)) An integral equation is called linear if only linear operations are performed function on it upon the unknown function . The equations (1) and (2) are Linear while (3) is nonlinear.

Integral equation Definition(3)) The most general type of linear integral equation is of the form: Where the upper limit be either variable or fixed. The functions are known functions,

While is to be determined ; is a nonzero,real or complex, parameter for is called the kernel . Is called the Kernel

The following special cases of equation (4) are of main interest : I) Fredholm integral equation II) Volterra Equations

I) Fredholm integral equation: In all Fredholm integral equation of the first kind the upper limit of integration b,say,is fixed. i) In the Fredholm integral equation of the first kind Thus, =0 (5)

I) Fredholm integral equation: ii) In the Fredholm integral equation of the second kind,

I) Fredholm integral equation: iii) The homogeneous Fredholm integral equation of the second kind is a special case of(ii) above . In this case

II) Volterra Equations Volterra Equations of the first, homogeneous , and second kinds are precisely as above except that is the variable upper limit of integration. Equation (4) itself is called an integral equation of the third kind

Singular Integral equation: Definition (4) When one or both limits of integration become infinite or when the kernel becomes infinite at one or more points within the range of integration ,the integral equation is called Singular .

Are singular integral equations. For example ,the Integral equations Are singular integral equations.

Special Kinds of kernel separable or degenerate kernels Symmetric kernel

I) separable or degenerate kernels A kernel to k(s,t) is called separable or degenerate if it can be expressed as the sum of a finite number of terms each of which is the product of a function s only and a function of only ; that is,

II) Symmetric kernel A complex-valued function K(s,t) is called symmetric (or Hermitian) if where the asterisk denotes the complex conjugate. For a real kernel, this coincides with definition

Eigen values and eigen functions If we write the homogeneous Fredholm equation as We have the classical eigen value or characteristic value problem; is the eigen value and is the corresponding eigen function or characteristic function.

Relationship between linear differential equations and Volterra integral equation: The solution of the linear differential equation With continuous coefficients given the initial conditions may be reduced to a solution of some Volterra integral equation of the second kind

From this hypothesis and some mathematical treatment we reach to where

We shall explain some methods for solving linear integral equations ; Methods of solution We shall explain some methods for solving linear integral equations ; These methods are : 1- Analytical methods 2- Numerical methods

Analytical methods for solving Volterra integral equation: Resolvent kernel of Volterra integral equation. The method of successive approximation. using Laplace Transform. Solution of integro- differential equations with the aid of the Laplace transformation. in in

Resolvent kernel of Volterra integral equation If the kernel has the general form k(x,t). If the kernel is a polynomial of degree (n-1) in x or (n-1) in t. iii) the kernel is dependent on the difference of the arguments. If the kernel is a polynomial of degree in in

And after some manipulation we shall have In the three cases above we shall begin with Volterra integral equation of the form in And after some manipulation we shall have in Where is called the resolvent kernel .

The method of successive approximation Suppose we have a Volterra type integral equation (14).Take some function Suppose we have a Volterra type integral equation (14). Take some function continuous in [0,a] into the right side of (14 ) in place of we got Continuing the process, we obtain a sequence of Functions where, we got , where

to the solution of the integral equation (14) Where the sequence converges as in to the solution of the integral equation (14) in

Using Laplace transform The Laplace transformation may be employed in the solution of systems of Volterra integral equations of the type we got Where are known continuous functions having Laplace transforms . , where

Taking the Laplace transform of both sides of (15) we get : we got This is asymptotic of linear algebraic equations in Solving it ,we find

Analytical methods for solving Fredholm integral equation: If the kernel is a polynomial of degree The method of Fredholm Determinants Integral Equation with degenerate kernels in

The method of Fredholm Determinants The solution of the Fredholm equation of the second kind we got is given by the formula

the Fredholm resolvent kernel of equation (17) Where the function is called the Fredholm resolvent kernel of equation (17) and defined by the equation in Provided the Here, are power series in : in

in in

Integral Equation with degenerate kernels The kernel The integral equation (17) with degenerate kernel (20) has the form we got

After some manipulation ,it has the form Where in in

solving Volterra integral equation: Numerical methods for solving Volterra integral equation: using the trapezoidal rule

the trapezoidal rule Consider the nonhomogeneous Volterra integral equation of the second kind we got To apply the trapezoidal rule , let and Define applying the trapezoidal rule to the integral of (23) ,we obtain:

the integration in (23) is over Thus for we take the equation (24) can be written in the form : we got

The system of equation in (25) can be written in a more compact form as After some manipulation , we obtain

By solving the system (27) we find Which is an approximatetion of the solution of (23)

solving Fredholm integral equation: Numerical methods for solving Fredholm integral equation: of the second kind The approximate method that we will discuss here for solving Fredholm equation of the second kind:

Are based on approximating the solution of (28) by a partial sum: Of N linearly independent functions On the internal (a,b).If we substitute from (29) into (28) for there will be an error

Involved which depends on x and on the way the coefficients are chosen Our main goal is how we can find or impose N conditions of the approximate solution (30).

The Galerkin approximate method In this method the N conditions are established for the determination of the N coefficients in (29) By making of (30) we got orthogonal to N given linearly independent Functions on the interval (a,b).

We will use the definition of orthogonality on in (30) , Then these N conditions become

After some manipulation ,we obtain :

Differential Equations Series solutions of linear differential equations 1- Power series solutions about an ordinary point this will not be discussed here because it has been taken in the past {Bs.C} 2-Solutions for singular points; the method of Frobineous 3- Bessel’s equation and Bessel Functions

The method of Frobineous We consider the homogeneous linear differential equation we got And we assume that is singular point of (1) under certain conditions we are justified in assuming a solution of the form

Where r is a certain (real or complex ) constsnt. Again,we write the differential equation (1) in the equivalent normalized form where Outline of the method of Frobenius: 1- Let a regular singular point of the differential Equation (1), seek solutions valid in some interval

and assume a solution Where we write the solution in the 2- Assuming term by term differentiation of (4) Is valid ,we obtain

Where K is a certain integer and the coefficients Now we substitute the series (4),(5) and (6) for y and its derivatives, respectively ,into the differential equation (1) are functions of 3- Now we proceed to simplify the resulting expression So that it takes the form Where K is a certain integer and the coefficients are functions of r and certain of the coefficients of the solution of (4)

5- Upon equating to zero the coefficient of the 4- In order that (7) be valid for all X in the deleted interval we must set 5- Upon equating to zero the coefficient of the lowest power of , we obtain a quadratic equation in r called the indicial equation of the differential equation (1) .The two roots of this quadratic equation in r , called the indicial equation of the differential equation (1) we must set

The two roots of this quadratic equation in r , called the indicial equation of the differential equation (1).The two roots of this quadratic equation are often called the exponents of the differential equation (1) and are the only possible values for the constant r in the assumed solution (4) .Thus at this stage the unknown constant is determined .We denote the roots of the indicial equation by where Here denotes the real part of and of course if is real ,then is simplify it self.

6- Now we equateto zero the remaining coefficients in (7) ,we are thus led to a set of conditions, involving the constants r which must be satisfied by the various coefficients in the series (4). 7- We now substitute the root into the conditions obtained is step 6 , and then choose the to satisfy these conditions . If the are so chosen , the resulting series (4) with is a solution of the desired form. Note that if and are real and unequal , then is the larger root. and are real and unequal , then is the larger root.

8- if ,we may repeat the procedure of step (7) using the root instead of In this way a second solution of the desired form (4) may be obtained . Note that if and are real and unequal , then is the smaller root . However , in the case in which and real and unequal , the second solution of the desired form (4) obtained in this step may not be linearly independent of the solution obtained in step (7) . also , in the case which are real and equal , the solution obtained in the step is clearly identical with the one obtained in step (7) and are real and unequal , then is the larger root.

I hope that the presentation is useful 