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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

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

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DE Spring Problem

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Cartesian Coordinates Problem Separation of Variables

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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

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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

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

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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

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Partial Differential Equations : Separation of Variables (6 Problems)

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Partial Differential Equations : Characteristic Curves

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Cauchy Reimann condition and analytic functions

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Cauchy Problem: One-Parameter Family of Solution Curves

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Partial Differential Equations

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Analyticity : Cauchy-Riemann Equation and Antiderivatives

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Cauchy-Riemann Equations

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Partial Differential Equations

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Impulse Forcing

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