# Nonlinear mathematical relationship between energy

Department of Mathematics, Faculty of Science, Urmia University, P.O. Box , One of the ways that energy transports in fluid is electron thermal conduction. we analyze relation between the Emden-Fowler equations and the nonlinear. This article is part of the theme issue 'Nonlinear energy transfer in .. differential- difference equations and Fourier analysis. J. Math. Phys. We consider the periodic defocusing cubic nonlinear Schrödinger (NLS) equation. (). {−i∂tu + ∆u .. the Γ(n) defining relation to observe the bound ( ). □.

With Calculus If you are familiar with calculus, then you probably realize that all we did in the section above was take a special case of an integral - in this case, the area of a triangle.

More generally, we want to integrate Equation 8 to arrive at the stored potential energy this will be useful in the nonlinear springs section. So, the work done by a varying force F u over a distance x is: If we substitute Equation 1 from above into Equation 11 and replace W with PE you can do this because, assuming there are no frictional losses, the work required to stretch the spring will be exactly equal to the resulting potential energy stored in the springwe determine that: The preceding section deals with linear springs, which follow a simple set of equations and behave "nicely.

For example, Figure 5 shows a force-displacement curve that we measured for a rubber band in the Science Buddies project, Launch Time: The Physics of Catapult Projectile Motion.

This curve is nonlinear, so it does not follow Equation 1. Do you see a problem? It is not a straight line. The rubber band behaves as a nonlinear spring. We can use a curve-fitting tool to find the equation for the line. In this case, the equation for the data in Figure 5 is: As in the Linear Springs section, F is force in newtons and x is displacement from the spring's neutral position in meters.

### Linear & Nonlinear Springs Tutorial

The values of This type of equation is called a power law because the variable x is raised to the power of p. The values of a and p will be different for different rubber bands. However, you cannot always assume that a nonlinear spring will follow a power law!

That just happened to be the case in the catapult experiment — there are other potential equations for the force vs. We do not have space to explain these different types of curves here — the point is that you should always perform an experiment to create a force vs.

## Linear & Nonlinear Springs Tutorial

Now, notice how Equation 14 above is different from Equation 1 in the Linear Springs section. Because we have a nonlinear spring, the slope of the force-displacement curve is not constant. The definition of k as "the slope of the force-displacement curve" is still true, but now that value can change.

In general, you can take the derivative of the force-displacement curve at any point to find the stiffness we do not call it the "spring constant" anymore, because it is not actually a constant: We can then plug Equation 13 into Equation 15, and we arrive at our equation for the stiffness of the rubber band: Figure 6 shows the stiffness vs.

Notice how, as the rubber band stretches more, the stiffness decreases instead of remaining constant. This means that the rubber band actually gets weaker as you stretch it.

### What Is a Non Linear Relationship? | Sciencing

The stiffness decreases with increasing displacement, instead of remaining constant. We also want to calculate the potential energy of this nonlinear spring. We can use the general equation: Remember that PE is potential energy and u is a dummy variable that we use for displacement, because we are already using x for the limit of the integral.

In a target which has been heated and compressed to the point of thermonuclear ignition, energy can then heat surrounding fuel to cause it to fuse as well, creating a chain reaction that burns the fuel load, potentially releasing tremendous amounts of energy.

Theoretically, if the reaction completes with perfect efficiency though this is a practically impossible feata small amount of fuel about the size of a pinhead, or around 10 milligrams, is capable of releasing the energy equivalent to burning a barrel of oil that is, chemically combining its hydrocarbon molecules with oxygen.

Fusion reactions combine lightweight atoms, such as hydrogen, together to form larger ones. Generally the reactions take place at such high temperatures that the atoms have been ionized, their electrons stripped off by the heat; thus, fusion is typically described in terms of ''nuclei'' instead of ''atoms''.

Fusion reactions on a scale useful for energy production require a very large amount of energy to initiate in order to overcome the so-called Coulomb barrier or fusion barrier energy. Since the positively-charged nuclei are naturally repelling each other, this repulsive force must be overcome by providing some form of external energy.

When this occurs, however, the reaction is a rather energetic one. Generally less energy will be needed to cause lighter nuclei to fuse, and when they do, more energy will be released.

As the mass of the nuclei increase, there is a point where the reaction no longer gives off net energy - the energy needed to overcome the energy barrier is greater than the energy released in the resulting fusion reaction. This point occurs when iron nuclei are formed and is the cause of death in some massive stars.

This phenomenon plays no role in laboratory induced fusion however, since the energy and temperature required to form iron nuclei are very large. Fusion of heavy nuclei is possible using particle accelerators and is the method used to form very heavy transuranic elements such as Roentgenium for instance, though the method of achieving this fusion of heavy elements is far removed from the methods used in large scale fusion reactions which are desired in tokamak or ICF fusion reactors.

The key to practical fusion power is to select a fuel that requires the minimum amount of energy to start, that is, the lowest barrier energy. The best fuel from this standpoint is a one to one mix of deuterium and tritium; both are heavy isotopes of hydrogen.

The D-T Deuterium and Tritium mix has a low barrier because of its high ratio of neutrons to protons. The presence of neutral neutrons in the nuclei helps pull them together via the strong force; while the presence of positively charged protons pushes the nuclei apart via Coloumbic forces the electromagnetic force. Tritium has one of the highest ratios of neutrons to protons of any element - two neutrons and one proton. Adding protons or removing neutrons increases the energy barrier.

The temperature and pressure required for any particular fuel to fuse is known as the Lawson criterion. These conditions have been known since the s when the first H-bombs were built.

ICF experiments started in earnest in the mids, when lasers of the required power were first designed. This was long after the successful design of magnetic confinement fusion systems, and even the particularly successful tokamak design that was introduced in the early s.

Nevertheless, high funding for fusion research stimulated by the multiple energy crises during the 's produced rapid gains in performance, and inertial designs were soon reaching the same sort of ''below breakeven'' conditions of the best magnetic systems.

Shiva was a ''proof of concept'' design, followed by the NOVA design with 10 times the power. Funding for fusion research was severely constrained in the 80's, but NOVA nevertheless successfully gathered enough information for a next generation machine whose goal was ignition. Although net energy can be released even without ignition the breakeven pointignition is considered necessary for a practical power system.

In [1] and [2], we solved some classes of nonlinear Volterra integral equations and Emden-Fowler equation, by introducing the Discrete Group Transformations DGT and RF-pair operations, which thus fur, could not be integrated using the classical method. Also in [3], we analyze relation between the Emden-Fowler equations and the nonlinear heat conduction problems with variable transfer coefficients by using the discrete group method.

The exact solutions of classical Emden-Fowler equation, by means of appropriate transformations of discrete group analysis is transformed into the exact solution of nonlinear applied problem of ICF. This approach is effective to find a new integrable equations which thus far, could not be integrated using the classical methods. This approach shows that, under the discrete group transformation, the solution of transformed equations can be converted into the solution of the reference equation [1,2,5,6].