# What is Analytical solution: Definition and 64 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. ### A Is the following sum a part of any known generalized function?

I have a sum that looks like the following: ## \sum_{k = 0}^{\infty} \left( \frac{A}{A + k} \right)^{\eta} \frac{z^k}{k!} ## Here, A is positive real. If \eta is an integer, this can be written as: ## \sum_{k = 0}^{\infty} \left( \frac{A(A +1)(A+2) \cdots (A + k - 1)}{(A + 1)(A+2)(A+3)...
2. ### A Analytical solution for an integral in polar coordinates?

Hi, I am trying to find open-form solutions to the integrals attached below. Lambda and Eta are positive, known constants, smaller than 10 (if it helps). I would appreciate any help! Thank you!

The freefall wiki entry wiki Freefall has an analytic solution for freefall distance in a gravitational field, but ... it doesn't seem to work ... at least i can't get it to work ... here is my MATLAB program to test it ... clear G=6.7e-11; % gravitational constant m^3/(kg*s^2) mEarth =...
4. ### 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...
5. ### Fully analytical solution for the "Leaky Integrate and Fire" Neuron model

Does anyone know if it is possible to develop a fully analytical solution for a leaky integrate and fire neuron driven by arbitrary time-varying current? Here's what I have so far (setting as many possible constants to 0 and 1): The equations: ## \dot{V} = - V + I(t) ## and if ##V(t) = 1##...
6. ### I Mixed topic : from FEM to analytical solution via limits?

Is this anyhow possible ? The system would be a wave equation modelized by a finite elements basis in space and time. Is there any method to do the limit discretization->continuum with paper and pencil ?
7. ### B Is there an analytical solution to this equation?

So I've derived the rocket equation in empty space and with constant gravity. Now I am interested in adding air resistance. I'm aware that there are 2 different models as if 0<Re<1 then F_drag=k*v and if 1000<Re<30000 then F_drag=1/2*A*rho*CD*v^2. And for my purpose the second model is most...
8. ### Analytical solution to mono-group kinetics with a ramp of reactivity

Consider the mono-group model and a ramp in reactivity like ##\rho = -\gamma \beta t## The system is $$\frac {dP}{dt} = \frac {\rho - \beta}{\Lambda} P + \lambda C$$ $$\frac {dC}{dt} = \frac {\beta}{\Lambda} P - \lambda C$$ 1st method: assume that the concentration of precursor doesn't...
9. ### I Analytical 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...
10. ### Analytical solution of the Photon Diffusion Equation

Homework Statement Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique. Homework Equations Starting equation integrated over t: int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1) Result...
11. ### A Determine PDE Boundary Condition via Analytical solution

I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$\frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t}$$...
12. ### Discrepancies between numerical and analytical solutions

The analytical solutions are: $$\psi(x) = \begin{cases} Ce^{\alpha x}, \text{if } x < -\frac{L}{2}\\ Asin(kx) + Bcos(kx), \text{if } -\frac{L}{2} \leq x \leq \frac{L}{2}\\ Fe^{-\alpha x} , \text{if } x > \frac{L}{2} \end{cases}$$...
13. ### Modeling analytical solution of 1D heat equation

I am trying to write code for analytical solution of 1D heat conduction equation in semi-infinite rod. The analytical solution is given by Carslaw and Jaeger 1959 (p305) as $$h(x,t) = \Delta H .erfc( \frac{x}{2 \sqrt[]{vt} } )$$ where x is distance, v is diffusivity (material property) and t...
14. ### Basic electrostatics problem; analytical solution?

Homework Statement For the system given, both objects have the same charge and same mass (both given). I'm also given string length, L. I need to solve for θ. Homework Equations Coulomb's Law, W=mg The Attempt at a Solution Using simple equilibrium force analysis (with weight, tension, and...
15. ### 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...
16. ### 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)...
17. ### Analytical solution of photon diffusion in finite media

Homework Statement I'm trying to derive an analytical expression for the photon backscatter flux in finite turbid media using the diffusion equation and the method of images. What I want to write is: for a given volume (x,y,z), where a coherent light source is incident on the x-y plane and z is...
18. ### I Analytical solution to b=a*x^-theta - x?

I wonder if there is an analytical solution to: b=a*x-θ - x with a>0, b>0, x>0, θ>1
19. ### A Is there an analytical solution to the following problem?

Differential equation \begin{align} \frac{{\partial}^2 T (x,t)}{\partial x^2} = \frac{1}{\alpha}\frac{\partial T(x,t)}{\partial t}, \end{align} boundary conditions \begin{aligned} T(x, 0) = T_\infty \\ -k\frac{\partial T(0,t)}{\partial x}= h(T_{max} - T(0,t)) \\...
20. ### Analytical Exp[((-i Phi)/2)*h3]

the closed analytical solution of exp [((- i phi)/2) * h3] someone could help!
21. ### 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...
22. ### 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...
23. ### 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(...
24. ### 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...
25. ### How can I find the analytical solution for the system?

Homework Statement dx(t)/dt = N0*sin(omega*t) * x(t) - ( N0*x^2 / k ) Omega,N0 and k are positive . Homework EquationsThe Attempt at a Solution I tried to solve it using the Bernoulli equations but I could not get the last result.
26. ### 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?
27. ### Can you prove a differential equation has no analytical solution?

Can you prove a differential equation has no analytical solution? Teach me ,please! Thank you a lot!
28. ### 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...
29. ### Analytical solution of Discrete-time Algebraic Riccati Equation

Homework Statement Solve for X in the DARE (Discrete-time Algebraic Riccati Equation) analytically. A is diagonal A = [-a\;0; 0 \;a], and B = [b; 0] (in MATLAB notation). Any help is very much appreciated! Homework Equations The DARE is given as A'XA - X - (A'PB+S)(B'XB+R)^{-1}(A'XB+S)' + Q =...
30. ### Problem in understanding analytical solution of LCR circuit

Hi all, I in my text they first did a phasor-diagram solution to a series LCR circuit and brought Z= under root of (R^2 +(Xc^2-XL^2)). After this they use a differential equation for series LCR circuit and actually did not solve such hard two degree differential equation, rather they...
31. M

### Solving a PDE: Analytical Solution for $$\alpha \frac{\partial f}{\partial r}$$

hey pf! i was wondering if you could help me out with a pde, namely \alpha ( \frac{z}{r} \frac{\partial f}{\partial r} + \frac{\partial f}{\partial z} ) = \frac{2}{r} \frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + 2 \frac{z}{r}...
32. ### A system of differential equations not having a analytical solution

I am going to quote an article below and there is a part I would like clarification.I do not understand why chaotic differential systems do not have an analytical solutions. In the simplest form,an equation such as this one [sin(x) + x - 0.5 = 0] does not have an analytical solution i.e it can...
33. ### Analytical solution of nonlinear ordinary differential equation

Dear All, I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution. \frac{dx}{dt} = 2Wx + 2xy - 4x^{3}\frac{dy}{dt} = \gamma \, (x^{2} -...
34. ### Analytical solution of an infinite series

How to find the value of an infinite series. for e.g.Ʃ_{n=1}^{\infty} (β^{n-1}y^{R^{n}}e^{A(1-R^{2n})}) where β<1, R<1, y>1, and A>0? Note that this series is covergent by Ratio test. I already have the numerical solution of the above. However, I am interested in analytical solution...
35. ### Analytical solution of the Euler equations in 1D?

Hi. Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can...
36. ### Analytical solution for thermal stresses in a rectangular plate

Hello everybody, I am solving a 2D problem of thermal stresses in a rectangular plate in which temperature is changing only y direction. Plate has fixed displacement conditions. Could anyone help me to find out analytical solution of thermal stresses for my problem? Does anyone suggest me...
37. ### Solving System of 2 ODEs: Analytical Solution

hallo I ams earching analytical solution for system of two ODE in next form x*(dy/dr) - (y*y/r) = constant1*r*r*x*x*x x*(dy/dr)+(x*y/r)=constant2*r*r*x*x*(r*constant3-y) where x(r) and y(r). conditions are x=constant4 at r=constant5 and y=0 at r=constant5 thnx r.
38. ### Analytical solution for coupled partial differential equations

Hello, In my study i came across to solve the analytical solution for coupled equation y(x,t) and z(x,t).The equations contains" f " function which is a function of the first variable exponentially. The first equation is : ∂y/∂t=∂^2(y)/∂x^2- 2*f(y)*z; The second equation ...
39. ### Analytical solution to integro-differential equation

Hi all, I'm trying to find an analytical solution to the following integro-differential equation: a f'(x)\int_0^x f(x)dx + b f'(x) + a [f(x)]^2 - a f(x) = 0 with initial condition: f(0)=1 This is a simplified problem for which I know the solution: f(x)=1. I'm trying to find...
40. ### Step by step analytical solution of temperature distribution

Homework Statement Lets say we have a 2D rectangular plate with a point heat source and some boundary conditions. I would like to solve and understand step by step the solution to this second order differential equation. Let's say dimensions of rectangular are a and b. 2 opposite sides are at...
41. ### Analytical solution to the diffusion equation with variable diffusivity

Hi, I'm trying to find an analytical solution (if one exists) to the 1d diffusion equation with variable diffusivity κ(x); \partial_t u(x,t) = \partial_x[\kappa(x) \partial_x u(x,t)] Could someone point me in the right direction to solve this if its possible to do so analytically. I've...
42. ### Accuracy of Runge-Kutta method when no analytical solution

Homework Statement I have to find eigenvalues to \frac{d^2y}{dx^2} + p^2 e^x y = 0,\, y(0)=0,y(1)=0 using the Runge-Kutta single step method to solve the ODE (splitting it up), with step length h and then another numerical method. This is not a problem. However, I need to be able to find...
43. ### Analytical Solution to a Magnetic Field

Hello all, I'm currently working on some magnetic shielding, and my supervisor wants me to try and find an analytical solution to the magnetic field inside the shielding. The shielding is basically bucket shaped (a hollow cylinder with the top cut out), and its meant to shield its interior...
44. ### Analytical solution of these coupled differential equations

Homework Statement I don't know how to type math equations of I have included a image file. Take initial conditiona as [0 1] Homework Equations The Attempt at a Solution No idea
45. ### Possible analytical solution of transcendental equation by an assumption.

Is it possible to solve the next transcendental equation analytically (obviously for k): sinh(k)=(b/2)(1+(e^(-2kl))) making the assumption that (bl >>1). I think that is not possible, but in an article that i found, they solve it by making that assumption, and they reach to the solution...
46. ### Any approximate analytical solution to this ode?

I'm working on this differential equation this few days... Could you give some guidance on approximate solutions to it? i(t) is the only function while all others are parameters. \frac{di(t)}{dt} = -\lambda(\sigma\phi\sqrt{i(t)(1-i(t))} + N\mu i(t)(1-i(t)) Thank you a lot!
47. ### Analytical solution for heat equation with simple boundary conditions

I am trying to solve the following heat equation ODE: d^2T/dr^2+1/r*dT/dr=0 (steady state) or dT/dt=d^2T/dr^2+1/r*dT/dr (transient state) The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2. I have searched the analytical solution for this kind of ODEs in polar coordinate...
48. ### Contact problem Analytical solution

Hi, I am trying to do a vibration analysis of a cantilever beam in contact with Earth (in ABAQUS). Now I need to match these results with the analytiocal solution. Could any of you guide me any paper or book where this klind of a problem is solved. So basically I am looking for the...
49. ### Multiple Energy Sources - Analytical Solution

Guys, I was wondering if any of you has encountered a problem with multiple energy sources and the analytical solution for that problem. Is there any way to formulate this problem "without" superposition? Thanks, Frank
50. ### Is there an analytical solution to the integral of a PDF?

Hey all, I have an engineering background with a pretty terrible grounding in statistics and just about all maths that I can't immediately visualize as some kind of dynamic system, so forgive me if this is an obvious question! I am working through a set of derivations for a conditional...