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Partial Differential Equations

A partial differential equation is a differential equation which contains unknown multivariable functions and their partial derivatives. They are used to solve problems involving functions of several variables. They can usually be solved by hand however occasionally a computer is necessary. In physics problems, partial differential equations are used to describe a variety of phenomena such as electrostatics, fluid flow, sound, heat and elasticity.

Partial differential equations are problems that involve rates of change with respect to a continuous variable. The equation for a partial differential equation function is:

F(x_1,……,x_n,u,∂u/∂x_1 ,……,∂u/(∂x_n ),(∂^2 u)/(∂x_1 ∂x_2 ),……,(∂^2 u)/(∂x_1 ∂x_n ),…)=0

Where F is a linear function of u and its derivatives.

Partial differential equations can be solved using Laplace transforms, numerical methods or on a computer. The method depends on the order of the equation. Therefore, partial differential equations are extremely useful when dealing with single order or multi-variable systems which occur very often in physics problems.

First order differential Equations, partial DE's

1. Find solutions to the given Cauchy- Euler equation (a) xy'+ y =0 (b) x2y'' + xy'+y =0 ; y(1) =1, y'(1) =0 2. Find a solution to the initial value problem x2y' + 2xy = 0; y (1) = 2 3. Find the general solution to the given problems (a) Y' + (cot x)y = 2cosx (b) (x-5)(xy'+3y) = 2 4. Solve t

Solutions to the Helmholtz Equation

Please help with the following problem. Provide step by step calculations for each problem. Consider the Helmholtz partial differential equation: u subscript (xx) + u subscript (yy) +(k^2)(u) =0 Where u(x,y) is a function of two variables, and k is a positive constant. a) By putting u(x,y)=f(x)g(y), derive ordinary diff

Characteristic curves

Find a partial differential equation whose characteristic curves are the lines x-y=a, x+2y=b where a,b are arbitrary real constants.

Solution to a Partial Differential Equation BVP

For n>0, find the solution to the boundary value problem -Ã?"u=(n/Ã?â?¬)e^{-n(xÃ?²+yÃ?²)}, xÃ?²+yÃ?²<1, u(x,y)=0, xÃ?²+yÃ?²=1. What happens in the limit as nââ? 'âË??? (Ã?"u=((âË?â??Ã?²u)/(âË?â??xÃ?²))+((âË?â??Ã?²u)/(âË?â??yÃ?²))).

Laplace problems

** Please see the attached file for the full problem description ** Hello, Please address some inverse Laplace problems as attached. Find the inverse Laplace transform of each of the following functions: (a) F(s) = s^3/s^4 + 4a^4 (b) G(s) = s^2/s^4 + 4a^4 (c) H(s) = s/s^4 + 4a^4 (d) K(s) = 1/s^4 + 4a^4

DE Spring Problem

DE: 32 pound weight is attached to the lower end of a coiled spring suspended from the ceiling, spring constant being 3 pounds per foot. The weight is pushed upward by 5 feet and given downward velocity of 3 feet per second. The medium it is in offers a resistance in pounds numerically equal to 4 times the instantaneous velocity

Cartesian Coordinates Problem Separation of Variables

Required to solve the attached Laplace Equation problems using separation of variables. Let €:= {(x,y) : 0 < x < 1, 0 < y < 1}. (1) Let u(x,y) satisfy the PDE uxx + uyy = 0, u(0, y) = 0, u(1, y) = 0, 0 < y < 1; u(x, 0) = 0, u(x,1) = f(x), 0 < x <1. (2) If f(x) = x, 0 < x < ½; an

Partial Differential Equation Boundary Conditions

Find the general solution of the wave equation U(tt) = U(xx) subject to the boundary conditions u(0,t) = u(1,t) = 0. Then find the unique solution of the wave equation subject to the initial conditions: u(x,0) = 2sin3pix and ut(x,0) = 5 sinpix Thanks

Partial Differential Equations with a Function

See the attached file. D(phi)/d(tau)+d(phi)/d(chi)+d(phi)/d(epsilon)+1=0 Boundary conditions: phi as a f(chi,epsilon,0)=0 phi as a f(0,epsilon,tau)=1 phi as a f(chi,0,tau)=1 Solve the PDE..... solution and explanation would be fine Any method to solving the solution would be fine (numerical or analytical).

1D Heat Equation with Variable Diffusivity

Please provide a detailed solution to the attached problem. Consider the solution of the heat equation for the temperature in a rod of length L=1 with variable diffusivity: u_t = A^2 d/dx (x^2 du/dx) The derivatives are partial derivatives. The boundary conditions are: u(1,t)=u(2,t)=0 And u(x,0) = f(x) sol

Differential equations

Hi, Please help working on section 1.1 problems 2,4,8,14,16 section 1.2 problems 6,10,20,24,27 thank you See attached Classify each as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ODE, ind

Solving Partial Differential Equations by Change of Variables

In solving this problem, derive the general solution of the given equation by using an appropriate change of variables. 1. ∂u/∂t - 2 ∂u/∂x = 2 Answer: u(x,t) = f(x + 2t) - x In this exercise, (a) solve the given equation by the method of characteristic curves, and (b) check you answer by plugging it back int

Financial Partial Differential Equations : Black-Scholes and Ito's Lemma

Please see the attached file for the fully formatted problems. If and we let , then , where and . Define . Suppose that the stock pays dividends continuously: ? D(S,t) => dividend ? If dividend is paid continuosly: * D(S,t)dt = D0Sdt * D0 is a constant dividend rate Derive the equation for directly by using

Partial Differential Equations : Wiener Process and Ito's Lemma

Please help with the following problems. See the following posting for the complete equations Consider a random variable r satisfying the stochastic differential equation: Where are positive constants and dX is a Wiener process. Let (see attached file) which transforms the domain for r into (-1,1) for Xi Suppose th

Partial Differential Equations : Wiener Process

Please see the attached file for the fully formatted problems. 1) Suppose: dS = a(S,t)dt + b(S,t)dX, where dX is a Wiener process. Let f be a function of S and t. Show that: (see the attached file for equations) 2) Suppose that S satisfies (see the attached file for equations) where u >=0, signa> 0, and dX is a Wi

Partial Differential Equations : Separation of Variables (6 Problems)

Please see the attached file for the fully formatted problems. 11. Separation of Variables. By usingu(x, t) = X(x)T(t) or u(x,y, t) = X(x)Y(y)T(t), separate the following PDEs into two or three ODEs for X and T or X, Y, and T. The parameters c and k are constants. You do not need to solve the equations. NOTE: one of the equa

Haase Diagrams and Partial Ordering Relations

Consider the following Hasse diagram of a partial ordering relation R on a set A: (a) List the ordered pairs that belong to the relation. (b) Find the (boolean) matrix of the relation. See attached file for full problem description.

Partial Differential Equations : Characteristic Curves

1.(a) Solve the given equation by the method of characteristic curves. (b) Check your answer by plugging it back into the equation. x &#8706;u/&#8706;x + y &#8706;u/&#8706;y = 0 See attached file for full problem description.

Cauchy Reimann condition and analytic functions

1. a) The Cauchy-Riemann equation is the name given to the following pair of equations, ∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂x which connects the partial derivatives of two functions u(x, y) and v(x, y) i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions satisfy the Cauchy-Riemann equa

Cauchy Problem: One-Parameter Family of Solution Curves

I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of us with respect to x. Furthermore, I will let u*x=p and u*y=q. Also, I will let ^ denote a power. For example, x^2 means x squared, and, I will let / denote division. Here is my problem: The PDE

Partial Differential Equations

A) Classify and find general expressions for the characteristic coordinates for the equation {see attachment} b) Use the canonical coordinates {see attachment} and transfer the above PDE into the new coordinates. Solve it in the new coordinates and show that {see attachments} where F and G are arbitrary functions of their ar

Analyticity : Cauchy-Riemann Equation and Antiderivatives

Let D be a domain in C and assume that f is analytic in D. Decide whether the statements below are true or false and give a short reason for your answer. a) If there exists an open subset U of D such that Im f&#8801;0 in U, then f is constant in D. Please see attached for all three questions.

Cauchy-Riemann Equations

6. Let u and v denote the real and imaginary components of the function f defined by the equations _ f(z) = (z^2)/z when z &#8800; 0, f (z) = 0 when z = 0. Verify that the Cauchy-Riemann equations ux = vy and uy = -vx are satisfied at the origin z = (0, 0).