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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

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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

Undetermined Coefficients for Differential Equations

Please show all steps to solution. (see attached for equations) A) Apply the undetermined coefficients to find a particular solution to the second-order linear nonhomogenous differential equation Write the general solution B) Use the solution to question

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

Partial Differential Equations - Convolution of Functions

See attached problem (convolution of two functions).

Partial Differential Equations Functions

1. Consider the first order PDE, ∂u/∂t = ct(∂u/∂x) -&#8734; < x < &#8734; c does not equal 0 a) Find the fundamental solution b) Use the fundamental solution and convolution to find a formula for the solution to: ∂u/∂t = ct(∂u/∂x) -&#8734; < x < &#8734; c does not equal 0 u(x,0) =

Partial Differential Equations and the Lagrange Method

1. solve X*U_x + Y*U_y =0 answer is suppose to be U(X,Y)=f(Y/X) 2. solve U_x + U_y =1 answer unknown

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

Quasi-Linear Partial Differential Equation

Show that the solution u of the quasi-linear partial differential equation u_y + a(u)u_x = 0. With initial condition u(x,0) = h(x) is given implicitly by u(x,y) = h(x-a(u)y). Show that the solution becomes singular for some positive y, unless a(h(s)) is a nondecreasing function of s.

Nonhomogeneous Differential Equations : Particular and General

14. Consider the nonhomogeneous linear equation dy/dt = λy + cos(2t) To find its general solution, we add the general solution of the associated homogeneous equation and a particular solution yp(t) of the nonhomogeneous equation. Briefly explain why it does not matter which solution of the nonhomogeneous equation we use

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 and Probability Density Functions

Please see the attached file for the fully formatted problems. 1) Suppose that S is a random variable that is defined on [0,&#8734;) and whose probability density function is: (see the attached file) a and b being positive numbers. Show: where (see the attached file) 2) We know that the solution of the final valu

Partial Differential Equations : Wiener Process

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Partial Differential Equations: Show that the Drag Force is Zero

10. Show that the drag force is zero for a uniform flow past a cylinder with circulation. See attached file.

Calculating Partial Differential Equations

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Partial Differential Equations : Separation of Variables (6 Problems)

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Partial differential equation

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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.

Equation of a Tangent Plane and Area of a Surface

Write an equation of the plane tangent to the surface S given by x = u^2 + v, y = u + v^2, z = uv, ((u,v)?R^2) at the point P with u = 2, v = 1. Find the area of the surface z = 3sqrt(x^2 + y^2), y >= 0, 0 <= z <= 6.

Partial Differential Equations : Harmonic Function and Constant Function

Suppose U is a positive harmonic function which is defined everywhere in the plane. Show that U must be a constant function. This is a question regarding Poisson Integral Formula in PDE. I need to find it using the Harnak's inequalities.

Partial Differential Equations : Neumann and Dirichlet Problems

Please see the attached file for the fully formatted problems.