Aem pde 1
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This document provides an overview of higher order homogeneous partial differential equations and their applications. It discusses that a partial differential equation involves a dependent variable and two or more independent variables. Homogeneous equations have solutions called complementary functions where the non-homogeneous term is equal to zero. The auxiliary equation is obtained by replacing the differential operator with the independent variable to find the complementary functions. Several examples are provided to demonstrate finding the complementary functions based on whether the roots of the auxiliary equation are real and distinct or repeated. Applications of partial differential equations discussed include shape processing, feature extraction, and economics.
Aem pde 1
- 1. Advanced Engineering Mathematics (2130002) Guided by : Asst. Prof. Jaimin Patel Prepare By : Name : Shah Jainam (160410119115) Shah Prayag (160410119116) Shah Preet (160410119117) Shah Rishabh (160410119118) Shah Saumil (160410119119) 1
- 2. Topic : Higher Order Homogeneous Partial Differential equations & its applications 2
- 3. Higher Order Homogeneous Partial Differential equations & its applications A partial differential equation is an equation involving a function of two or more variables and some of its partial derivatives. Therefore a partial differential equation contains one dependent variable and more than one independent variable. Here z will be taken as the dependent variable and x and y the independent variable so that . We will use the following standard notations to denote the partial derivatives. ,, q y z p x z t y z s yx z r x z 2 22 2 2 ,, 3
- 4. Homogeneous Equation The case where ƒ = 0 is called a homogeneous equation and its solutions are called complementary functions. It is particularly important to the solution of the general case, since any complementary function can be added to a solution of the non-homogeneous equation to give another solution where D is the differential operator d/dx (i.e. Dy = y' , D2y = y",... ), and the ai are given functions. Such an equation is said to have order n, the index of the highest derivative of y that is involved. 4
- 5. Linear Differential Operator Above equation is linear Differential operator 5
- 6. Rules To Obtain The C.F. 6
- 8. Case- 1 : Roots Are Real And Distint The n solution of the form z = fᵣ(y + mᵣx) for r = 1,2,3...,n exists. Therefore C.F. Of above equation is given by f₁(y + m₁x) + f₂(y + m₂x) + ... + fᵣ(y + mᵣx) Where f₁, f₂, ..., fᵣ are arbitrary functions. 8
- 9. Case – 2 Roots Are Repeated 9
- 10. Remark The auxiliary equation is obtained by replacing D with m and Dᶦ with 1 in the given differential equation. If f(x , y) = 0, the particular integral = 0 10
- 11. Example-1 11
- 12. Example-2 12
- 13. Example-3 13
- 14. Example-4 14
- 16. Shape Processing Using Pdes Shape processing refers to operations such as denoising, fairing, feature extraction, segmentation, simplification, classification, and editing. Such operations are the basic building blocks of many applications in computer graphics, animation, computer vision, and shape retrieval. Many shape processing operations can be achieved by means of partial differential equationsor PDEs. The desired operation is described as a (set of) PDE(s) that act on surface information, such as area, normals, curvature, and similar quantities. PDEs are a very attractive instrument: They allow complex manipulations to be described precisely, compactly, and measurably, and come with efficient and effective numerical methods for solving them. We present several applications of PDEs in shape processing. 16
- 17. Partial Derivative In Economics In economics the demand of quantity and quantity supplied are affected by several factors such as selling price, consumer buying power and taxation which means there are multi variable factors that affect the demand and supply. In economics marginal analysis is used to find out or evaluate the change in value of a function resulting from 1-unit increase in one of its variables. For example Partial derivative is used in marginal demand to obtain condition for determining whether two goods are substitute or complimentary. Two goods are said to be substitute goods if an increase in the demand for either result in the decrease or the other. While two goods are said to be complimentary goods if a decrease of either result in a decrease of the 17
- 18. Partial Derivative In Engineering In image processing edge detection algorithm is used which uses partial derivatives to improve edge detection. Grayscale digital images can be considered as 2D sampled points of a graph of a function u(x,y) where the domain of the function is the area of the image. 18
- 19. Reference Sites www.wileyplus.com www.elsevier.com www.sabre.org 19
- 20. Thank You... 20