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Confirming the Fourier Coefficient Formulas: Bsic Fourier Coefficient Formula which are given as: Assume v is a signal of the form:

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Inverse Fourier Transform-1: Remaining Part of Inverse Fourier Transform are given as: In the second integrals;

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Fourier Series: The basic idea of the Fourier series is that any periodic waveform can be represented with a sum of harmonically related sinusoids. Let’s break this statement down. First, a waveformis a function of time. A waveform is periodic if it repeats itself identically after a period of time. Let the period be denoted T. Then mathematically, a T-periodic waveform v satisfies

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Fourier Series of Functions with Arbitrary Periods: So far, we’ve worked with 2π-periodic functions for convenience. Similar results exist for functions having any period. These results can be obtained in a similarmanner. However, there is an easier way, one calculus II students are familiar with: substitution. One way to obtain a new series representation is to perform a substitution in a known series. Suppose that f is a function with period T = 2p > 0 for which we want to find a Fourier series. In other words f (x 2p) = f (x). If we let t = πx / p and we define:

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Theorem on Inverse Fourier Transform: Theorem. Let f ∈ L¹ (R) and let f be piecewise smooth on R. Then for every x ∈R,

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Integration of Fourier Series: The following is a cosequence of the fundamental theroem of calculus: Preposition 6: If f : [a ,b]→C is sectionally contineous, then

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Preposition-4: If f satiesfies a Lapschitz conditions, then S_N f → f uniformly as N → . Proof: We have

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Euler Method: Recall the Differential Equation we get:

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Half Range Fourier Series: It is often convenient to represent a given function as a Fourier series which contains only cosine terms or only sine terms, as in the initial examples. If f(x) is piecewise continuous on [0, π ], it can be extended to [–π , 0] as either an even function or as an odd function. Then f is periodically extended to the whole real lines as a 2π - periodic functions by

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Partial Differentiation: Functions of two or more variables: In many applications, we come across function involving more than one independent variable. For example, the area of rectangle is a function of two variables, the length and breadth of the rectangle. Definitions: Let u = f (x , y) be a function of two independent variables x and y. The derivatives of u with respect to x varies and y remains constant is called the partial derivatives of u with respect to x is and is denoted by

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Period of Multiple Function: If f and g are periodic of period p then so is f g Proof : Denote f g by h want h (x p) = h (x)

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Hyperbolic Equation with Constant Coefficients: The Charectric equations is;

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Even and Odd Function: If f (-x) = f (x) the function, f (x) is even. This means that f (x) is symmetric about the y-axis, f (x) = x² and f (x) = cos ( nπx/L ) are examples of even functions. If g (-x) = - g (x) the function, g (x) is odd. This means that g (x) is anti - symmetric about the y- axis. g (x) = x³ and g (x) = sin ( nπx/L ) are examples of odd functions. A function may be neither even or odd. h (x) = exp (x) and h (x) = x cos (x) are examples of function that are neither even or odd.

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Periodic Functions: Periodic functions are functions which repeat: f (t P)= f (t) for all t. For example, if f (t) is the amount of time between sunrise and sunset at a certain latitude, as a function of time t, and P is the length of the year, then f (t P) = f (t) for all t, since the Earth and Sun are in the same position after one full revolution of the Earth around the Sun. We state this explicitly as the following definition: a function f (t) is periodic with period P > 0 if, f (t P)= f (t) for all t.

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Fourier Coefficients: Given a T-periodic signal v, it is possible to compute the Fourier coefficients of (4) with the following equations:

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Orthogonal Function of Fourier Series: Basic Orthogonal Function used in Fourier Series are given as: If we review the above example you will see that for each coefficients, if we wanted a_k, the coefficients of the k^th harmonic cosine, we multiplied the signals by cos ( 2πkf₀t ). which translate the components of interest to DC where it is extrated while discarding all other terms by integrating over exactly one cycle of the fundamental frequency. Similarly if we are interested in b_k, the coefficients of the k^th harmonic sine, we multiply the signal by sin ( 2πkf₀t ) which translate the components of interest to DC, where it is extracted while discarding all other terms by integrating over exactly one cycle of the fundamental frequency. This work because cos ( 2πkf₀t ) and sin ( 2πkf₀t ) are orthogonal functions, meaning that following orthogonality conditions are satiesfied by these functions.

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Parabolic and Elliptic Equation with Constant Coefficients: Parabolic: The only solutions of main equations is;

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Trigonometric Polynomials: A Trigonometric Polynomial is a polynomial expression involving cos x and sin x: cos⁵x 6 cos³x sin²x 3 sin⁴x 2 cos²x 5

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Properties of the Fourier Series: Basic properties of Fourier Series which are given as follows: Linearity:

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General Interval: General Intervals of Fourier Series are given as: The theory of Fourier series can be expanded to include functions which have arbitrary period. Let f : R→R have period 2l > 0 . That is f (x 2l) = f (x), xεR. Also define Φ: R→R by

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Ordinary Differential Equation: Higher order differential equations with constant coefficients – Method of variation of parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations with constant coefficients. The study of a differential equation in applied mathematics consists of three phases.

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Problems on Method of Undetermined Coefficients: 3. Solve using the method of undetermined method: y'' - 5y' 6y = sin 2x.

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Problems on Differential Equation-2: 7. Solve Solution: Given

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Problems on Euler's Equation: 1. Solve Solution: Given (x²D² 4xD 2)y = x logx ...........(1)

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Method of Variation of Parameter-1: 2. Solve the method of variation of parameters Solutions: The A.E is m² 1 = 0

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Cauchy's Homogeneous Linear Equation: Consider Homogeneous Linear differential equation as:

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Problems-2: 9. Solve the initial value problem Solutions: We have (D² 4D 5)y = - 2 cos hx

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Special Form of X in Differential Equation: Type 2: P.I of the form We have D ( Sin ax) = a cos ax

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Problems on Simultaneous Differential Equations:

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Problems on Special Form of X in Differential Equation (Type-2): 1. Solve (D² 9)y = cos 4x. Solution: Given equation is (D² 9)y = cos 4x

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Linear Equation with Constant Coefficients: 1. Solve the simultanous equations Solution:

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Solution of Initial and Boundary Value Function: The differential equation in which the conditions are specified at a single value of the independent variable say x = x₀ is called an Initial Value Problem (IVP). If y = y(x), the initial conditions usually will be of the form.

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Inverse Differential Operator and Particular Integral: Consider a Differential equation: f (D) y = x Define 1 / f(D) such that,

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Properties of Bessel Function-1: Basic properties are given as:

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Legendre Linear Equation: It is the Form: Where A₁,A₂,..........,A_n are constants. It can be reduced to linear differential equation with constant coefficients.

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Method of Undetermined Coefficients: Here we consider a method of finding the particular integral of the equation f (D) y = φ(x). When φ(x) is of some special forms. In this method first we assume that the particular integral is of certain form with some coefficients. Then substituting the value of this particular integral in the given equation and comparing the coefficients, we get the value of these “undetermined” coefficients. Therefore, the particular integral can be obtained. This method is applicable only when the equation is with constant coefficients. In the following cases we give the forms of the particular integral corresponding to a special form of φ(x). Case(i): If φ(x) is polynomial of degree n

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Second Order ODE with Variable Coefficients: Let us now return to the more general 2nd order, linear ODE y’’ p(x) y’ q(x) y = g(x)

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Bessel Equation: The second order differential equation given as:

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Problems on Frobenius Series Solution: Exampel: 1 Take the first case of;

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Properties of Legendre Polynomials: The Legendre Polynomials are the everywhere regular solutions of Legendre’s Equation, (1 − x²)u′′ − 2xu′ mu = [ (1 − x²)u′ ] ′ mu = 0,

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Orthogonality of Legendre Polynomials: The differential equation and boundary condition satiesfies by the Legendre Polynomials forms a generalized system where the boundary condition amount to insisting on regularity of the solutions at the boundaries). They should therefore satiesfies the Orthogonality Relations.

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Bessel function of Second Kind: The Bessel function of the second kind, Y_ν(x) is sometimes referred to as a Weber function or a Neumann function (which can be denoted as N_ν(x)). It is related to the Bessel function of the first kind as follows:

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Method of Separation of Variables For PDE: The method of separation of variables can be applied to linear partial differential equations, especially those with constant coefficients in the equation. To understand the strategy of this method, let us start by the following simple example: Example: Consider the equations:

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Laplace's Equation: Consider Laplace equations in two variables on the upper half - plane y > 0. Then

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Wave Equation: The equation is given as;

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Heisenberg's Inequality: This is the Basic Heiseberg's Inequality Theorem:

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Problems on Application of Partial Differential Equation: Example 1. Classify the following linear second order partial differential equation and find its general solutions:

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Transmission Line Equations: In a long electrical cable or a telephone wire both the current and voltage depend upon position along the wire as well as the time.

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Parsevals Identities: By the Convolution Theorem for,

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Heat Conduction Equation: The heat equations, For u (x, t) a function of two variables, with initial conditions

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Laplace Transform: The Laplace transform method is used for solving the differential equations with initial and boundary conditions. The advantage of this method is that it solves the differential equations with initial conditions directly without the necessity of first finding the general solution and then evaluating the arbitrary constants using the initial conditions. In particular, this method is used in problems where the driving force (mechanical or electrical) has discontinuities for a short time or is periodic. In this unit we study the basic concepts of Laplace transforms and its applications to solve the differential equations arising in mechanics, electrical circuits and bending of beams. Definition:

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Laplace Transforms of Unit Step Function: Unit Step Function ( Heaviside Function ): The unit step function or Heaviside Function u (t - a) is defined as follows:

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Dirac Delta Generalized Function: Since this is an elementary presentation of few generalized functions, we do not mention the theory of distributions, which should be the most appropriate way to present a complete theory of generalized functions. Since we restrict ourselves mainly to the Dirac delta generalized function only, we have chosen to present it as a limit of a function sequence. A continuous function sequence is a sequence whose elements are continuous functions. An example is the continuous function sequence {u_n}, for n ≥ 1, is given by

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Problems on Solution of Linear Differential equations-2: 5. Solve using laplace transforms y'' 2y' 5y = 8sint 4cost given that y (0) = 1 and y (π) = e^-π

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Laplace Transforms of the Derivatives: If the Laplace Transform of f(t) is known then by using the following results we can find the Laplace transform of the derivatives;

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Laplace Transforms of Unit Impulse Function: Unit Impulse Functions: Definition: The unit impulse function denoted by is defined as follows:

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Problems on Convolution Theorem: 1. Using Convolution Theorem find the inverse Laplace transform of the following functions:

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Laplace Transforms of Standard Function: 1. Laplace Transform of a Constant: Let f(t) = a, where a is constant. Then from the definition of Laplace transform, we get;

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Problems on Laplace Transformation-1: (5).

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Laplace Transformation on Integral Function: 1. Laplace Transforms of the form e^at f (t): If the Laplace transform of f (t) is known, then the Laplace transform of e^at f (t) where a is a constant can be determined by using the shifting property.

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Laplace Transform of Periodic Function: A function f (t) is said to be periodic function with periods α > 0, if f (t a) = f (t) Theorem: if f (t) is a periodic function of period α > 0, then

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Convolution Theorem: Convolution Theorem are given as: If Proof :

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Problems Laplace Transformation on Integral Function: 1. Find the Laplace transforms of the following:

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Inverse Laplace Transforms: Introduction: If L { f (t) } = F (s), then f(t) is called the Inverse Laplace Transforms of F (s) and symbolically, we write f(t) = L ^-1 { F(s) }. Here L^-1 is called Inverse Laplace Transform Operators.

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Problems on Solution of Simultanoeous Differential Equation-1: 10. Solve by using laplace transforms

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Solution of Linear Differential equations: One of the important applications of Laplace transforms is to solve linear differential equations with constant coefficients with initial conditions. For example, consider a second order linear differential equations:

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Problems on Inverse Laplace Transform-1: (8)

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Properties of Inverse Laplace Transform: 1. Linear Property If a and b are two constants then

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Problems on Laplace Transforms of Unit Step Function-1: 2. Express the following functions in terms of the Heaviside's unit step function and hence find their Laplace Transform:

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Problems on Laplace Transformation on Periodic Function-1: 4. A periodic function of periods is defined by