Explore BrainMass

Explore BrainMass

    Partial Differential Equation

    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 April 25, 2024, 6:28 am ad1c9bdddf

    BrainMass Solutions Available for Instant Download

    Equation of Motion and Integration Constants

    Consider a particle moving under the influence of a driving force F(t) with damping proportional to velocity. The equation describing the system is, m dv/dt + αv = F (t) The boundary condition in time are v → 0 as t → ±∞. Find v(t) if the input F(t) is a force that is a constant F0 between times t = 0 and t = T and ze

    Legendre equation parity

    A differential equation that occurs frequently in physics (as part of the solution of Laplace's equation, which occurs in such areas as electrodynamics and quantum mechanics, among others) is Legendre's equation. In this post, we'll have a look at the equations and some of the properties of its simplest solutions: the Legendre p

    Frictionless Train Through the Earth

    I need some help answering these questions: a) The gravitational potential energy U(r) of a mass m due to a mass density ρ(r) satisfies ∇^2 U = 4 π Gmρ, where G is the gravitational constant. If the earth is considered to be a uniform sphere of mass M, radius R, show that the gravitational potential energy of a mass m ins

    Prove The Solution of a Rectangular Reactor Core

    A reactor core is in the form of a rectangular prism of height h with a rectangular base having sides of length a and b. Assuming that the boundary condition is psi = 0 at the sides of the reactor show, by direct substitution or otherwise, that the solution has the form psi = A cos(pi*x / a) cos(pi*x / b) cos(pi*z / h)

    Heat Flow Equation

    The heat flow in one dimension is governed by the partial differential equation [see the attachment for equation] where [attached] is the temperature in space as a function of time. Using the Fourier transform in x solve this equation given the condition [attached]; C is a constant.

    Derivatives in Damped Wave Equation

    I need to substitute a solution to the damped wave equation back into the original differential equation to get an identity. I am unsure of how to solve the partial derivatives and the steps I take mathematically going through the differential equation. A more detailed explanation is attached. Thanks!

    Thermodynamics: Maxwell's Relations

    Derive Maxwell's Relations from First and Second Laws of Thermodynamics and Thermodynamic Functions like Internal Energy, Helmholtz's Function, Enthalpy and Gibbs Free Energy. And also explain how they are satisfied by an ideal monatomic gas?

    Solving the PDE in physics

    Consider a flexible string held stationary at both ends and free to vibrate transversely subject only to the restoring forces due to tension in the string. Deduce the PDE for such systems and define all parameters that distinguish the systems. Do this with a string of normal length l and using Cartesian coordinates. Using a

    Change in specific entropy of a copper block

    The temperature of a block of copper is increased from T0 to T without any appreciable change in its volume. Show that the change in its specific entropy is Δs = cp ln (T/T0) - v0β2/κ (T-T0) (equation is attached in a better form) Δs = change in specific entropy cp = specific heat at constant pres

    A Grounded Conducting Sphere

    A point charge +q is situated a distance a (a>R)from the center of a grounded conducting sphere of radius R. Find the potential outside the sphere.

    Find the entropy change of an ideal gas and of a van der Waals gas in case of isothermal expansion in which the volume increases by a factor alpha. Also, find the entropy change in case of a free expansion in which the volume increases by a factor alpha. In case of the free expansion of the van der Waals gas, find the temperature change.

    Find the entropy change of an ideal gas and of a van der Waals gas in case of isothermal expansion in which the volume increases by a factor alpha. Also, find the entropy change in case of a free expansion in which the volume increases by a factor alpha. In case of the free expansion of the van der Waals gas, find the tempera

    Electric Force and Fields

    Find the force on the charge +q in the attached figure. Note: the XY plane is a grounded conductor.

    entropy, heat, and temperature during compression of lead

    Please see attached. The change in volume is small enough to be ignored (.2% of the initial volume). 1.) 10^(-3) m^3 of lead is compressed reversibly and isothermally at room temperature from 1 to 1000 atm pressure. Using one of Maxwell's thermodynamic relations to find the following: a) the change in entropy b) The hea

    Characteristic frequencies and modes of a system of springs

    Consider a system of 3 springs (A, B and C) connected in a line. The left hand end of spring A is anchored to a wall as is the right hand end of spring C. Between springs A and B is mass M and between springs B and C is mass N. Consider the following 2 situations: 1. mass M = N and the spring constants are spring A = k. B

    Expansions of a Perfect Gas

    Please see attached file for full problem description. I started doing this problem but I could way off here. I think that for part C it is the Joule-Kelvin expansion. So the final temperature would be T1 plus a hairy integral. For part B) it seem like it is just the Joule expansion. However, I am not sure if I manipulated th

    Solve Problems Using Laplace Transforms

    Calculate and graph the output voltage in the RL circuit of the above circuit if: E(t) = k when t greater than or equal to 0 and less than 5 E(t) = 0 when t greater than or equal to 5 The current is initially 0 See attached file for full problem description.

    Lagrangian Dynamics and Orbital Mechanics problem

    1. Two blocks each of mass M are connected by an extension less, uniform string of length l. One block is placed on a smooth horizontal surface and the other block hangs over the side of the table with the string passing over a frictionless pulley. Describe the motion of the system: a) when the mass of the string