analytical solution Definition and Topics - 11 Discussions

In mathematics, a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions admitted in a closed-form expression may vary with author and context.

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1. Poisson Ratio -- Finding a corresponding analytical solution for the strain

Hi, I ran into problems using the poisson ratio. For a FE simulation I created a simple 2D 1mm x 1mm block, and prescribed a 0.1 mm displacement at the top edge. Furthermore, the bottom edge is constraint in the y-dir, and the left edge in the x-dir. The material parameters are E = 100, and v...
2. I Analytical Solution: Open Channel Rectangular Fluid Flow

Hi All, I'm looking for an analytical solution to the open channel rectangular fluid flow profile. The flow is bounded by three walls but the top is open to atmosphere. Assume steady state flow that is parallel and incompressible. I've already found information involving a rectangular flow...
3. Discrepancies between numerical and analytical solutions

The analytical solutions are: $$\psi(x) = \begin{cases} Ce^{\alpha x}, \text{if } x < -\frac{L}{2}\\ Asin(kx) + Bcos(kx), \text{if } -\frac{L}{2} \leq x \leq \frac{L}{2}\\ Fe^{-\alpha x} , \text{if } x > \frac{L}{2} \end{cases}$$...
4. A Solving two simultaneous integro-differential equations

I am trying to find a closed-form (analytical) solution for the two following simultaneous integro-differential equations : du[x]/dx= - a v[x] +b ∫〖[1-(y-x)^4 〗].(v[y]-v[x])dy And (dv[x])/dx= - f u[x] -g ∫〖[1-(y-x)^4 〗].u[y]dy With the initial conditions: v[0]=e and u[1]=0 a,b,f,g...
5. A Need help with a differential equation

Hi all, I have derived a differential equation, which I don't know how to solve. I can do some numerical simulations, but would really be interested in, at least, knowing if an analytical solution exists, so would appreciate any help with it: (I have removed argument from y)...
6. Preparing a Solution using ppm and ppb -

I would like to prepare 50.0 g of 50 ppb Pb standard solution in 1% HNO3 gravimetrically, in a 50.0mL plastic tube from a stock solution of 50 ppm Pb. From what I understand, this is a dilution...so C1V1 = C2V2 I like working in mg/L more than ppb and ppm so I converted 50 ppb = 0.05 mg/L 50...
7. Analytical solution to mx"(t)+b(x'(t))x'(t)+k(p)x(t)=0

Hello guys! I have following differential equation mx"(t)+b(x'(t))x'(t)+k(p)x(t)=0. As can be seen, "attenuation term" is dependent of velocity x'(t). Also stiffness term k(p) is dependent of term p, which is p=k(p)x(t)/A. In this equation A is constant and k(p) means, of course, same term as...
8. Is there an analytic solution to this system of equations?

I have the following system of equations with variables ##a,m##, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 \frac{\ln{\frac{1}{2}}}{26.8} \right) }{\left(\dfrac{\ln{\frac{1}{2}}}{26.8} + a \right)} \\ \\ 50 &= me^{-a\left(...
9. Optimal Control of Linear-Affine System w/ Constraints

Although this could fall under engineering, I thought the Diff Eq forum was the most relevant. Let me know if I should post elsewhere. I have a fairly basic system for which I'm trying to find a minimum-time optimal control policy. I know there are many ways to do this numerically, but since...
10. Analytical solution for bending stiffness of tapered tube

How to derive the formula to find the bending stiffness of an isotropic tapered tube which is cantilevered with a point load applied at the free end?
11. Analytical solution for bound state energies of infinite well

Hi there I am trying to find bound state energies assuming infinite potential. I have been told it can be done by analytically solving Right Hand Side and Left Hand Side of an equation such as: E^1/2 tan(2ma^2E/4hbar)^1/2 = (V0-E)^1/2 If solved properly, it should give one curve (RHS), crossed...