This document discusses finite difference and interpolation methods. It defines finite differences of various orders (first, second, etc.) and describes forward, backward, and central difference tables. It also covers Newton's forward and backward interpolation formulas for unequal intervals using forward and backward differences. An example is provided to illustrate calculating interpolated values using these formulas.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
This document explains Simpson's 1/3rd rule for numerical integration. Simpson's 1/3rd rule approximates the integral of a function over an interval by breaking the interval into equal subintervals and approximating the function within each subinterval as a quadratic polynomial. The approximation takes the function values at the endpoints and midpoint of each subinterval. The approximations over all subintervals are then summed to give an approximation of the full integral. Important considerations for applying Simpson's 1/3rd rule include using an even number of equal subintervals and having a minimum of 3 points defined in each subinterval.
B.tech ii unit-5 material vector integrationRai University
This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
Stirling's formula provides an approximation of factorials and is derived as the average of the Gauss forward and backward interpolation formulae. It is most accurate when -1/4 < p < 1/4. The formula is f(x) = f(x0) + f'(x0)(x - x0) + (f"(x0)/2!)(x - x0)^2 + ... + (f^((n))(x0)/n!)(x - x0)^n, where f^((n))(x0) is the nth derivative of f evaluated at x0. Stirling's formula is obtained by taking the average of the Gauss forward and backward difference formulae.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
This document discusses the Cauchy-Euler differential equation, which is a linear homogeneous ordinary differential equation with variable coefficients. It has a particularly simple structure that allows it to be solved explicitly. The key steps to solve this type of equation are to substitute x=et to convert it into a linear equation, then use operators related to differentiation with respect to t. This transforms the equation into one that can be solved using standard techniques for linear differential equations.
This document defines and provides examples of linear differential equations. It discusses:
1) Linear differential equations can be written in the form P(x)y'=Q(x) or P(y)x'=Q(y), where multiplying both sides by an integrating factor μ results in a total derivative.
2) First order linear differential equations of the form P(x)y'=Q(x) have an integrating factor of e∫P(x)dx. The general solution is y(IF)=C.
3) Bernoulli's equation is a differential equation of the form P(x)y'+Q(x)y^n=R(x), where the general solution depends
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
This document explains Simpson's 1/3rd rule for numerical integration. Simpson's 1/3rd rule approximates the integral of a function over an interval by breaking the interval into equal subintervals and approximating the function within each subinterval as a quadratic polynomial. The approximation takes the function values at the endpoints and midpoint of each subinterval. The approximations over all subintervals are then summed to give an approximation of the full integral. Important considerations for applying Simpson's 1/3rd rule include using an even number of equal subintervals and having a minimum of 3 points defined in each subinterval.
B.tech ii unit-5 material vector integrationRai University
This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
Stirling's formula provides an approximation of factorials and is derived as the average of the Gauss forward and backward interpolation formulae. It is most accurate when -1/4 < p < 1/4. The formula is f(x) = f(x0) + f'(x0)(x - x0) + (f"(x0)/2!)(x - x0)^2 + ... + (f^((n))(x0)/n!)(x - x0)^n, where f^((n))(x0) is the nth derivative of f evaluated at x0. Stirling's formula is obtained by taking the average of the Gauss forward and backward difference formulae.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
This document discusses the Cauchy-Euler differential equation, which is a linear homogeneous ordinary differential equation with variable coefficients. It has a particularly simple structure that allows it to be solved explicitly. The key steps to solve this type of equation are to substitute x=et to convert it into a linear equation, then use operators related to differentiation with respect to t. This transforms the equation into one that can be solved using standard techniques for linear differential equations.
This document defines and provides examples of linear differential equations. It discusses:
1) Linear differential equations can be written in the form P(x)y'=Q(x) or P(y)x'=Q(y), where multiplying both sides by an integrating factor μ results in a total derivative.
2) First order linear differential equations of the form P(x)y'=Q(x) have an integrating factor of e∫P(x)dx. The general solution is y(IF)=C.
3) Bernoulli's equation is a differential equation of the form P(x)y'+Q(x)y^n=R(x), where the general solution depends
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
- A differential equation involves an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the exponent of the highest order derivative.
- Linear differential equations involve the dependent variable and its derivatives only to the first power. Non-linear equations do not meet this criterion.
- The general solution of a differential equation contains as many arbitrary constants as the order of the equation. A particular solution results from assigning values to the arbitrary constants.
- Differential equations can be solved through methods like variable separation, inspection of reducible forms, and finding homogeneous or linear representations.
1) A differential equation contains an independent variable (x), a dependent variable (y), and the derivative of the dependent variable with respect to the independent variable (dy/dx).
2) The order of a differential equation refers to the highest order derivative present. For example, an equation containing dy/dx would be first order, while one containing d2y/dx2 would be second order.
3) The degree of a differential equation refers to the highest power of the highest order derivative. For example, an equation containing (d2y/dx) would be degree 1, while one containing (d2y/dx)2 would be degree 2.
4) There are several methods for solving first
1. The document defines ordinary and partial differential equations and discusses the order and degree of differential equations.
2. Examples of common second order linear differential equations with constant coefficients are given, including equations for free fall, spring displacement, and RLC circuits.
3. The document also discusses homogeneous linear equations and Newton's law of cooling as examples of differential equations.
This document contains the syllabus for a course on Mathematical Methods taught according to the JNTU-Hyderabad new syllabus. It covers topics like matrices and linear systems, eigenvalues and eigenvectors, linear transformations, solution of nonlinear systems, curve fitting, numerical integration, Fourier series, and partial differential equations. The specific section summarized discusses numerical differentiation using forward, backward, and central differences. It also covers numerical integration techniques like the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule.
This document discusses numerical integration and interpolation formulas. It begins by explaining the general formula for numerical integration using equidistant values of a function f(x) between bounds a and b. It then derives Trapezoidal, Simpson's, and Weddle's rules by putting different values for n in the general formula. The document also discusses Newton's forward and backward interpolation formulas, Lagrange interpolation formula, and provides examples of their application. It concludes by comparing Lagrange and Newton interpolation and discussing uses of interpolation in computer science and engineering fields.
1) First order ordinary linear differential equations can be expressed in the form dy/dx = p(x)y + q(x), where p and q are functions of x.
2) There are several types of first order linear differential equations, including separable, homogeneous, exact, and linear equations.
3) Separable equations can be solved by separating the variables and integrating both sides. Homogeneous equations involve functions that are homogeneous of the same degree in x and y.
This document discusses techniques for solving eigen problems, including the power method, inverse power method, QR decomposition, and QR algorithm. It provides details on implementing these techniques, such as the steps of the QR algorithm and ways to accelerate its convergence like deflation and ad hoc shifts. References are also included.
This document discusses sequences and series. It provides definitions of key terms like sequence, finite sequence, infinite sequence, convergent sequence, divergent sequence, monotonic sequence, and geometric progression. It then goes on to solve 4 example problems:
1) It shows that the sequence 2n^2+n/n^2+1 is convergent by taking the limit as n approaches infinity.
2) It uses the ratio test to show that the sequence n!/n^n is convergent.
3) It proves that the sequence 1/1! + 1/2! +...+ 1/n! is convergent by showing it is increasing and bounded.
4) It shows that the sequence
applications of second order differential equationsly infinitryx
1) Second-order differential equations are used to model vibrating springs and electric circuits. They describe oscillations, vibrations, and resonance.
2) Springs obey Hooke's law, resulting in a second-order differential equation relating position to time. The solutions describe simple harmonic motion.
3) Damping forces can be added, resulting in overdamped, critically damped, or underdamped systems with different behavior.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
This document provides an overview of numerical differentiation and integration methods. It discusses Newton's forward and backward difference formulas for computing derivatives, as well as Newton-Cote's formula, the trapezoidal rule, and Simpson's one-third and three-eighths rules for numerical integration. Examples of applying these methods to real-world problems are provided. The document also compares Simpson's one-third and three-eighths rules, noting their different assumptions about the polynomial order of the integrated function and requirements for the number of intervals.
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
- A differential equation involves an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the exponent of the highest order derivative.
- Linear differential equations involve the dependent variable and its derivatives only to the first power. Non-linear equations do not meet this criterion.
- The general solution of a differential equation contains as many arbitrary constants as the order of the equation. A particular solution results from assigning values to the arbitrary constants.
- Differential equations can be solved through methods like variable separation, inspection of reducible forms, and finding homogeneous or linear representations.
1) A differential equation contains an independent variable (x), a dependent variable (y), and the derivative of the dependent variable with respect to the independent variable (dy/dx).
2) The order of a differential equation refers to the highest order derivative present. For example, an equation containing dy/dx would be first order, while one containing d2y/dx2 would be second order.
3) The degree of a differential equation refers to the highest power of the highest order derivative. For example, an equation containing (d2y/dx) would be degree 1, while one containing (d2y/dx)2 would be degree 2.
4) There are several methods for solving first
1. The document defines ordinary and partial differential equations and discusses the order and degree of differential equations.
2. Examples of common second order linear differential equations with constant coefficients are given, including equations for free fall, spring displacement, and RLC circuits.
3. The document also discusses homogeneous linear equations and Newton's law of cooling as examples of differential equations.
This document contains the syllabus for a course on Mathematical Methods taught according to the JNTU-Hyderabad new syllabus. It covers topics like matrices and linear systems, eigenvalues and eigenvectors, linear transformations, solution of nonlinear systems, curve fitting, numerical integration, Fourier series, and partial differential equations. The specific section summarized discusses numerical differentiation using forward, backward, and central differences. It also covers numerical integration techniques like the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule.
This document discusses numerical integration and interpolation formulas. It begins by explaining the general formula for numerical integration using equidistant values of a function f(x) between bounds a and b. It then derives Trapezoidal, Simpson's, and Weddle's rules by putting different values for n in the general formula. The document also discusses Newton's forward and backward interpolation formulas, Lagrange interpolation formula, and provides examples of their application. It concludes by comparing Lagrange and Newton interpolation and discussing uses of interpolation in computer science and engineering fields.
1) First order ordinary linear differential equations can be expressed in the form dy/dx = p(x)y + q(x), where p and q are functions of x.
2) There are several types of first order linear differential equations, including separable, homogeneous, exact, and linear equations.
3) Separable equations can be solved by separating the variables and integrating both sides. Homogeneous equations involve functions that are homogeneous of the same degree in x and y.
This document discusses techniques for solving eigen problems, including the power method, inverse power method, QR decomposition, and QR algorithm. It provides details on implementing these techniques, such as the steps of the QR algorithm and ways to accelerate its convergence like deflation and ad hoc shifts. References are also included.
This document discusses sequences and series. It provides definitions of key terms like sequence, finite sequence, infinite sequence, convergent sequence, divergent sequence, monotonic sequence, and geometric progression. It then goes on to solve 4 example problems:
1) It shows that the sequence 2n^2+n/n^2+1 is convergent by taking the limit as n approaches infinity.
2) It uses the ratio test to show that the sequence n!/n^n is convergent.
3) It proves that the sequence 1/1! + 1/2! +...+ 1/n! is convergent by showing it is increasing and bounded.
4) It shows that the sequence
applications of second order differential equationsly infinitryx
1) Second-order differential equations are used to model vibrating springs and electric circuits. They describe oscillations, vibrations, and resonance.
2) Springs obey Hooke's law, resulting in a second-order differential equation relating position to time. The solutions describe simple harmonic motion.
3) Damping forces can be added, resulting in overdamped, critically damped, or underdamped systems with different behavior.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
This document provides an overview of numerical differentiation and integration methods. It discusses Newton's forward and backward difference formulas for computing derivatives, as well as Newton-Cote's formula, the trapezoidal rule, and Simpson's one-third and three-eighths rules for numerical integration. Examples of applying these methods to real-world problems are provided. The document also compares Simpson's one-third and three-eighths rules, noting their different assumptions about the polynomial order of the integrated function and requirements for the number of intervals.
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
This document provides an overview of topics in vector integration, including line integrals, surface integrals, and volume integrals. It includes examples of calculating each type of integral. The key theorems covered are Green's theorem, Stokes' theorem, and Gauss's theorem of divergence. Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region. Stokes' theorem relates a line integral around a closed curve to a surface integral over the enclosed surface. Gauss's theorem relates the surface integral of the normal component of a vector field over a closed surface to the volume integral of the divergence of the vector field over the enclosed volume.
This document is an internship project report submitted by Siddharth Pujari to the Indian Institute of Space Science and Technology. The report focuses on advanced control system design for aircraft and simulating aircraft trajectory. It includes modeling an aircraft's state space model in MATLAB to test controllability. The report also covers theoretical aspects of stability of linear systems, linearizing nonlinear models, controllability of linear systems using the Kalman criterion and transition matrix, and applying these concepts to simulate aircraft controllability in MATLAB.
This document describes numerical integration and differentiation techniques taught in a B.Tech Engineering Mathematics course. It covers the Trapezoidal, Simpson's 1/3 and 3/8 rules for numerical integration of functions. For numerical differentiation, it discusses Euler's method, Picard's method, and Taylor series for solving ordinary differential equations. Examples are provided to illustrate the application of these numerical methods to evaluate integrals and solve initial value problems.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-VRai University
This document describes numerical integration and differentiation techniques taught in a B.Tech Engineering Mathematics course. It covers the Trapezoidal, Simpson's 1/3 and 3/8 rules for numerical integration of functions. For numerical differentiation, it discusses Euler's method, Picard's method, and Taylor series for solving ordinary differential equations. Examples are provided to illustrate the application of these numerical methods to evaluate integrals and solve initial value problems.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
The document provides an overview of complex integration, including:
1) Definitions of complex line integrals and applications in engineering.
2) Cauchy's integral theorem and formula, used to evaluate line integrals.
3) Taylor and Laurent series expansions of analytic functions.
4) The residue theorem, used to evaluate integrals along closed paths via residues of singularities enclosed by the path.
Diploma_Semester-II_Advanced Mathematics_Indefinite integrationRai University
This document is from a unit on indefinite integration. It defines indefinite integration and antiderivatives. It provides rules for integration, including algebra rules and standard integrals involving trigonometric, exponential, logarithmic and algebraic functions. It describes methods for integration like integration by substitution, integration by parts, and integrating multiple trigonometric functions. It provides examples demonstrating these integration techniques and exercises for students to practice indefinite integration.
The document discusses various mathematical methods for interpolation and solving equations including:
1) Bisection method, iteration method, and Newton-Raphson method for finding roots of equations.
2) Finite difference methods for numerical differentiation and interpolation using forward, backward, and central difference operators.
3) Newton's forward and backward interpolation formulas for equally spaced data using finite differences.
4) Gauss interpolation and Lagrange interpolation for unequally spaced data points.
This document provides an introduction to integration, which is the inverse process of differentiation. It defines indefinite and definite integrals, and discusses techniques for evaluating integrals such as basic integral formulas, integration by parts, integration by substitution, and integrals of trigonometric functions. Examples are provided to illustrate each technique, with practice exercises included at the end. The document serves as a tutorial on basic concepts and methods in integral calculus.
This document provides an overview of analytic functions in engineering mathematics. It defines analytic functions as functions whose derivatives exist in some neighborhood of a point, making them continuously differentiable. The Cauchy-Riemann equations are derived as necessary conditions for a function to be analytic. It also defines entire functions as analytic functions over the entire finite plane. Examples of entire functions include exponential, sine, cosine, and hyperbolic functions. The document discusses analyticity in both Cartesian and polar coordinates.
This document discusses numerical solutions of partial differential equations. It contains an introduction and four chapters:
1. Preliminaries - Defines basic concepts like differential equations, partial derivatives, order of a differential equation.
2. Partial Differential Equations of Second Order - Classifies second order PDEs and provides examples.
3. Parabolic Equations - Discusses explicit and implicit finite difference methods like Schmidt's method and Crank-Nicolson method to solve heat equation.
4. Hyperbolic Equations - Will discuss numerical methods to solve hyperbolic PDEs like the wave equation.
The document provides information about algebraic expressions and polynomials. It begins by explaining that algebra uses a language of numbers, variables, and symbols to represent quantitative relationships. The document then covers topics like equivalent expressions, simplifying expressions, evaluating expressions, and different types of polynomials. It aims to help students acquire skills in performing operations with polynomials and using those skills to simplify, evaluate, and solve algebraic expressions and problems.
This document describes a numerical analysis lab assignment to create M-files to calculate forward and backward difference tables. It includes the objective, theory on calculating difference tables, sample code to generate the tables for given data sets, output plots of the data, analysis of the time complexity, and notes on error propagation in difference tables.
The document discusses the concept of derivatives in calculus. It defines a derivative as the limit of the ratio of change in y over change in x as the change in x approaches zero. This represents the instantaneous rate of change of a function with respect to its variable. The document provides examples of finding the derivatives of various functions by applying this definition and using algebraic manipulation of limits. It also explains how derivatives can be used to find the slope of a curve and solve problems involving rates of change.
This document discusses finite-difference calculus techniques used to approximate values of functions and derivatives at discrete points in reservoir simulation models. It introduces common finite-difference operators - including forward, backward, central, shift, and average operators - and examines their relationships to derivative operators in Taylor series expansions. Examples are provided to demonstrate calculating finite-difference approximations of first and second derivatives in 1D and 2D. The document also covers solving the Poisson equation and time-independent partial differential equations using finite-difference methods.
Numerical Methods and Analysis discusses various root-finding methods including bisection, false position, and Newton-Raphson. Bisection uses interval halving to find a root between two values with opposite signs. False position uses the slope of a line between two points to estimate the next root. Newton-Raphson approximates the root using Taylor series expansion neglecting higher order terms. Interpolation uses forward difference tables to construct a polynomial approximation of a function.
This document provides information about coordinate geometry and various geometric concepts that can be represented using a Cartesian coordinate system. It includes:
1) An introduction to coordinate geometry and Cartesian coordinate systems.
2) Equations and properties of lines, including finding slopes, angles between lines, parallel/perpendicular lines, and intersections.
3) Equations and properties of circles, including center-radius form, diameter form, tangents, and normals.
4) Worked examples and exercises on finding equations of lines and circles given information about points, slopes, radii, etc.
This document discusses different numerical methods for finding the roots or zeros of equations, including the bisection method, Newton-Raphson method, and regula-falsi method. It provides definitions, steps, examples, and analyses of the convergence properties and advantages/disadvantages of each method. The bisection method uses interval halving to bracket the root, while Newton-Raphson and regula-falsi are iterative methods that successively approximate the root using derivative information or the chord between initial points.
This document provides an overview of functions of complex variables. It discusses key topics including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known. Specific methods covered are direct, Milne-Thomson's, exact differential equations, and a shortcut method. Examples are provided to illustrate determining the analytic function given properties of its real or imaginary part. The document also briefly outlines applications of complex variables and standard conformal transformations.
This document describes topics related to multiple integrals, including double and triple integrals. It provides examples of calculating double integrals using different methods, such as changing the order of integration or changing variables. It also discusses evaluating double integrals over regions defined implicitly or bounded by curves. Several practice problems are included for calculating double integrals over given regions.
This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.
The document provides information about curve tracing including important definitions, the method of tracing a curve, and examples of tracing specific curves. It defines singular points, multiple points, nodes, cusps, and points of inflection. The method of tracing involves analyzing the curve for symmetry, points of intersection with the axes, regions where the curve does not exist, asymptotes, and tangents. Examples analyze the curves y=(x-a)^2, (x+y)^2=(x-a)^2, y=(2-x)^2, and y=x^2 for these properties and sketch the curves.
The document is an acceptance letter from the International Journal of Scientific and Engineering Research notifying Kundan Kumar that his paper titled "A Parabola Symmetrical to y=x Line" has been accepted for publication. The paper is categorized as a research paper and was found to have high significance, originality, technical quality, and clarity based on the reviewer's evaluation. The paper will be published in Volume 6, Issue 1 of the January 2015 edition of the journal.
This certificate acknowledges that Kundan Kumar successfully published a research paper titled "A Parabola Symmetrical to y=x Line" in January 2015. The paper was reviewed and published by the International Journal of Scientific & Engineering Research.
2. Unit-IV Finite Difference and Interpolation
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Content
Finite differences, difference tables, Newton’s forward interpolation & it’s problems, Newton’s
backward interpolation & it’s problems, stirling's interpolation formula & Problems based on it,
Newton’s divided difference formula for unequal intervals & it’s problems ,Lagrange’s divided
difference formula for unequal intervals & it’s problems
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1.1 Finite differences— Let = ( ) be a function and ∆ = ℎ denote the increment in the
independent variable . Assume that∆ , increment in the argument (also known as the
interval or spacing) is fixed. i.e. ℎ =constant . Then the first finite difference y is defined as—
∆ = ∆ ( ) = ( + ∆ ) − ( )
Similarly finite differences of higher orders are denoted as follows—
∆ = ∆(∆ ) = ∆ ( + ∆ ) − ( )
= ∆ ( + ∆ ) − ∆ ( )
= [ ( + 2∆ ) − ( + ∆ )] − [ ( + ∆ ) − ( )]
∆ = ( + 2∆ ) − 2 ( + ∆ ) + ( )
In general ∆ = ∆(∆ ), for = 2,3,4 …
Now consider the function = ( )specified by the tabulated series = ( ) for a set of
equivalent points where = 0,1,2, … , and ∆ = ∆ − = ℎ =constant. Thus the
tabulated function consists of ordered pairs ( , ), ( , ), … , ( , ), …. Here entries
are known as entries.
1.2 Forward Difference—
The first forward difference is denoted by ∆ and defined as
∆ = − .
The symbol ∆ is the forward difference operator.
Properties—
1. ∆ = 0 (Differences of constant function are zero)
2. ∆( ) = ∆( ), where is a constant .
3. ∆( + ) = ∆ + ∆
4. ∆( ) = ∆ + ∆
5. ∆ (∆ ) = ∆
6. Where and are non-negative integers and ∆ = (By definition).
7. The higher order forward difference are defined as:
8. The second order forward difference of is
9. ∆ = ∆(∆ ) = ∆ − ∆
In general,
∆ = ∆(∆ ) = ∆ − ∆
It defines the nth
order forward differences.
Any higher order forward differences can be expressed in terms of the successive values of
the function.
Example:
1.
∆ = − 2 +
2.
∆ = − 3 + 3 −
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Forward Difference Table
Value of Value of 1st
diff 2nd
diff 3rd
diff 4th
diff 5th
diff
∆ ∆ ∆ ∆ ∆
+ ℎ ∆ ∆ ∆ ∆
+ 2ℎ ∆ ∆ ∆
+ 3ℎ ∆ ∆
+ 4ℎ ∆
+ 5ℎ
1.3 Backward Difference—
The first backward difference is denoted by ∇ and defined as—
∇ = − .
The symbol ∇ is the backward difference operator.
Second order backward difference
∇ = ∇(∇ ) = ∇ − ∇
In general
∇ = ∇(∇ ) = ∇ − ∇
Now
∇ = ∇ − ∇ = (∇ − ∇ ) − (∇ − ∇ )
= ∇ − ∇ − ∇ + ∇
= ∇ − 2∇ + ∇
= ( − ) − 2( − ) − ( − )
= − −2 + 2 − +
= − 3 + 3 +
In general,
∇ = ∑ (−1)
Backward Difference Table
Value of Value of 1st
diff 2nd
diff 3rd
diff 4th
diff 5th
diff
= + ℎ ∇
= + 2ℎ ∇ ∇
= + 3ℎ ∇ ∇ ∇
= + 4ℎ ∇ ∇ ∇ ∇
= + 5ℎ ∇ ∇ ∇ ∇ ∇
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1.4 Central Difference— Sometimes it is convenient to employ another system of differences
known as central differences. The central difference operator is denoted by and defined by the
relation—
− = δ /
− = δ
.
.
.
− = δ ( )/
Similarly, the higher order central differences are defined as:
δ − δ = δ
δ / − δ = δ
δ − δ = δ / and so on.
So the table will be:
Central Difference Table
Value of Value of 1st
diff 2nd
diff 3rd
diff 4th
diff 5th
diff
δ /
+ ℎ δ
δ / δ /
+ 2ℎ δ δ
δ / δ / δ /
+ 3ℎ δ δ
δ / δ /
+ 4ℎ δ
δ /
+ 5ℎ
Example —Evaluate the followings—
(i) ∆ (ii) ∆ log 2 (iii) ∆ (iv)∆ ( )
Solution—
(i) ∆ = ( + ℎ) −
= ( )
= tan
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Similarly
∆ ( ) = ∆[ ( + ℎ) − ( )] = ∆ ( + ℎ) − ∆ ( )
= ℎ[( + ℎ) − ] + [( + ℎ) − ] + ⋯ + ℎ
= ( − 1)ℎ + + + ⋯ + ′′
∴ the second differences represent a polynomial of degree ( − 2).
Continuing in this process for the nth differences we get a polynomial of degree zero.
i.e. ∆ ( ) = ( − 1)( − 2) … 1
ℎ = ! ℎ , which is constant.
Hence the ( + 1)th
and higher order differences of a polynomial of nth
degree will be zero.
Example— Evaluate ∆ [(1 − )(1 − )(1 − )(1 − )].
Solution— ∆ [(1 − )(1 − )(1 − )(1 − )]
= ∆ [ + (. . . ) + (… ) + ⋯ + 1]
= (10!) [∵ ∆ ( ) = 0 < 10]
Example— Find the missing value of the following table:
: 45 50 55 60 65
: 3.0 _______ 2.0 ________ -2.4
Solution—The difference is –
∆ ∆ ∆
45 = 3.0 − 3 5 − 2 3 + − 9
50 2 − + − 4 3.6 − − 3
55 = 2.0 − 2 −0.4 − 2
60 −2.4 −
65 = −2.4
Solving the two equations 3 + − 9 = 0 and 3.6 − − 3 = 0.
we can find the value of and .
3 + = 9 … ( )
+ 3 = 3.6 … ( )
From ( )
= 9 − 3 .
Substituting the value of ( )
+ 3(9 − 3 ) = 3.6
⟹ −8 = 3.6 − 27
⟹ =
−23.4
−8
= 2.935
= 9 − 3(2.935) = 9 − 8.775 = 0.225.
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Other Difference Operators—
1. Shift Operator(E)—
Shift operator is the operation of increasing the argument by ℎ so that
( ) = ( + ℎ) ( ) = ( + 2ℎ) ( ) = ( + 3ℎ) …
The inverse of shift operator is defined as ( ) = ( − ℎ).
If is the function of ( ), then = , = , = , where may
be any real number.
2. Averaging operator( )—
Averaging operator =
Relation between the operators—
1. ∆= − 1
2. ∇= 1 −
3. = −
4. = +
5. ∆= ∇= ∇ =
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2.1 Interpolation—Let = ( ) is tabulated for the equally spaced values of
= , , , … , , where = + ℎ, = 0,1,2, … ,
⟹ = + ℎ,
= + 2ℎ
= + 3ℎ
= + ℎ
It gives— = , , ,… ,
…
…
The process of finding the values of corresponding to any value of = between and
is called interpolation.
The study of interpolation is based on the concept of difference of a function.
To determine the values of ( ) of ′( ) for some intermediate values of various types of
difference are very much useful.
2.2 Newton’s forward difference interpolation— Let the function = ( ) takes the values
, , ,… corresponding to the values , + ℎ, + 2ℎ, … of . Suppose it is required to
evaluate ( ) for = + ℎ, where is any real number.
For any real number , we have defined such that—
( ) = ( + ℎ)
= ( + ℎ) = ( ) = (1 + ∆) [∵ = 1 +△]
= 1 + △ +
( )
!
△ +
( )( )
!
△ + ⋯ [Binomial theorem]
= + △ +
( )
!
△ +
( )( )
!
△ + ⋯ … (1)
It is called Newton’s forward difference interpolation formula as (1) contains and the
forward differences of .
2.3 Newton’s backward difference interpolation— Let the function = ( ) takes the
values , , ,… corresponding to the values , + ℎ, + 2ℎ, … of . Suppose it is
required to evaluate ( ) for
= + ℎ, where is any real number.
Then we have
= ( + ℎ) = ( ) = (1 − ∇) [∵ = 1 − ∇]
= 1 + ∇ +
( )
!
∇ +
( )( )
!
∇ + ⋯ [ ℎ ]
= + ∇ +
( )
!
∇ +
( )( )
!
∇ + ⋯ … (2)
It is called Newton’s forward difference interpolation formula as eq (1) contains and the
forward differences of .
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Example—The table gives the distances in the nautical miles of the visible horizon for the
given heights in feet above the earth’s surface:
=height : 100 150 200 250 300 350 400
=distance: 10.63 13.03 15.04 16.81 18.42 19.90 21.27
Find the value of , when ( ) = 218 ( ) 410 .
Solution—The forward difference table is—
∆ ∆ ∆ ∆
100 10.63
150 13.03 2.40
200 15.04 2.01 -0.39
250 16.81 1.77 -0.24 0.15
300 18.42 1.61 -0.16 0.08 -0.07
350 19.90 1.48 -0.13 0.03 -0.05
400 21.27 1.37 -0.11 0.02 -0.01
(i) For = 200, = 15.04
∆ = 1.77, ∆ = −0.16, ∆ = 0.03etc.
Since = 218 and ℎ = 50
∴ = = = 0.36
Using Newton’s forward difference interpolation formula we get
= + △ +
( )
!
△ +
( )( )
!
△ + ⋯
= 15.04 + 0.36(1.77) +
. ( . )
(−0.16) +
. ( . )( . )
(0.03) + ⋯
= 15.04 + 0.637 + 0.018 + 0.001 + ⋯ = 15.696 i.e. 15.7 nautical miles.
(ii) Since = 410 is near the end of the table, we use Newton’s backward difference
interpolation formula.
∇ ∇ ∇ ∇
100 10.63 2.40 -0.39 0.15 -0.07
150 13.03 2.01 -0.24 0.08 -0.05
200 15.04 1.77 -0.16 0.03 -0.01
250 16.81 1.61 -0.13 0.02
300 18.42 1.48 -0.11
350 19.90 1.37
400 21.27
∴ Taking = 400, = = = 0.2
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Using the line of backward difference:
= 21.27, ∇ = 1.37, ∇ = −0.11, ∇ = 0.02 etc
Newton’s backward difference interpolation formula gives
= + ∇ +
( + 1)
2
∇ +
( + 1)( + 2)
6
∇ + ⋯
= 21.27 + 0.2(1.37) +
. ( . )
(−0.11) +
. ( . )( . )
(0.02)
+
0.2(1.2)(2.2)(3.2)
24
(−0.01) + ⋯
= 21.27 + 0.274 − 0.0132 + 0.00176 − 0.000704 + ⋯
= 21.531856
. . 21.5 nautical miles.
Example— From the following table, estimates the number of students who obtained marks
between 40 to 45.
Marks 30-40 40-50 50-60 60-70 70-80
No of Students 31 42 51 35 31
Solution— First we have to prepare the cumulative frequency table:
Marks less than
( )
40 50 60 70 80
No of Students
( )
31 73 124 159 190
Now the difference table is
∆ ∆ ∆ ∆
40 31 42 9 -25 37
50 73 51 -16 12
60 124 35 -4
70 159 31
80 190
We shall find i.e. the number of mark sheets less than 45.Using Newton’s forward
difference interpolation formula we get—
Taking = 40, = 45, we have = = = 0.5
= + △ +
( − 1)
2!
△ +
( − 1)( − 2)
3!
△ + ⋯
= 31 + 0.5 × 42 +
. ( . )
× 9 +
. ( . )( . )
× (−25) +
. ( . )( . )( . )
× 37
= 31 + 21 − 1.125 + 1.5625 − 1.4453125 = 50.9921875
The number of the students with marks less than 45 is 50.9921875 i.e. 51.
But the number student with marks less than 40 is 31.
Hence the number of students getting marks between 40 and 45 = 51 − 31 = 20.
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2.6 Divided difference—
TheLagrange’s formula has the drawback that if another interpolation areinterested then the
interpolation coefficient are required to recalculate.
This problem is solved in Newton’s divided difference interpolation formula.
If ( , ), ( , ), ( , ) … be given points, then the first divided difference for the argument
is , is defined by the relation [ , ] = .
Similarly,
[ , ] = and[ , ] = etc.
The second divided difference for the argument is , , is defined as [ , , ] =
[ ] [ , ]
.
The third divided difference for the argument is , , , is defined as [ , , , ] =
[ , , ] [ , , ]
and so on.
2.7 Newton’s divided difference interpolation formula— Let , , , … be the values of
= ( ) corresponding to the arguments , , , … , . Then from the definition of divided
differences, we have [ , ] =
= +( − )[ − ] … ( )
Again, [ , , ] =
[ , ] [ , ]
Which give—
[ , ] = [ , ] +( − ) + ( − ) [ , , ]
Substituting this values in the eq(i), we get
= +( − )[ − ] + ( − )( − ) [ , , ] … ( )
Also, [ , , , ] =
[ , , ] [ , , ]
Which gives [ , , ] = [ , , ] −( , )[ , , , ]
Substituting this values of [ , , ] in the eq(ii), we obtain
= +( − )[ − ] + ( − )( − ) [ , , ]
+( − )( − ) ( − )[ , , , ]
Proceeding in this way, we get
= +( − )[ − ] + ( − )( − ) [ , , ]
+( − )( − )( − )[ , , , ] + ⋯
+( − )( − ) … ( − )[ , , , … , ] … ( )
This is called as Newton’s divided difference interpolation formula.