Explore BrainMass

Explore BrainMass

    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.

    © BrainMass Inc. brainmass.com August 12, 2022, 5:15 pm ad1c9bdddf

    BrainMass Solutions Available for Instant Download

    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

    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 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) =

    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

    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.

    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.