Higher accurate approximate solutions for the simple pendulum in terms of elementary functions

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied. The coefficients of the Fourier series are found, and it turns out that they are rapidly decreasing. The solution is found to be very close to the exact solution. In the small-angle regime the solution formula for the displacement becomes equivalent to the solution describing the linear pendulum. During the analysis a new formula for the period of the simple pendulum is found, which is more accurate as compared to most previously published results. The formula gives the period with an accuracy better than 0.0004% for angles up to , and within 0.025% for angles up to radians. In the small-angle regime the formula becomes equivalent to the result for the period of the linear pendulum. The present derivation of an anharmonic solution to the equation of motion describing a simple pendulum, as well as the derivation of a new expression for the pendulum period, is obtained in terms of elementary functions. It is believed to give valuable insight into a better understanding of the behaviour of the simple pendulum in undergraduate courses on classical mechanics.

The shooting method consists of guessing unknown initial values, transforming a second-order nonlinear boundary value problem (BVP) to an initial value problem and integrating it to obtain the values at the right end to match the specified boundary condition, which acts as a target equation. In the shooting method, the key issue is accurately solving the target equation to obtain highly precise initial values. Due to the implicit and nonlinear property, we develop a generalized derivative-free Newton method (GDFNM) to solve the target equation, which offers very accurate initial values. Numerical examples are examined to show that the shooting method together with the GDFNM can generate a very accurate solution. Additionally, the GDFNM can successfully solve the three-point nonlinear BVPs with high accuracy. A new splitting-linearizing method is developed to express the approximate analytic solutions of nonlinear BVPs in terms of elementary functions, which adopts the Lyapunov technique by inserting a dummy parameter into the governing equation and the power series solution. Then, linearized differential equations are sequentially solved to derive the analytic solution.

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

The bifilar pendulum is often used for indirect measurements of mass moments of inertia of bodies that possess complex geometries. The exact equation of motion of the bifilar pendulum is highly nonlinear, and has not been solved in terms of elementary functions. Extensive use has been made, however, of the linearized approximation to the exact equation, and it has been assumed that the simple harmonic oscillator adequately describes the motion of the bifilar pendulum. It is shown here that such is generally not the case. Numerical solutions to the exact nonlinear differential equations of motion are obtained for a range of values of initial angular displacement, filament length, and radius of gyration. The filament length and the radius of gyration are normalized with respect to the half-spacing between the filaments. It is shown that the approximate solution gives good results only for small ranges of the system parameters.

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.

General solution and Green’s function for fluid-saturated infinite and semi-infinite orthotropic poro-thermoelastic materials

General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application

The one-dimensional potential well V(x)=−(ℏ2ν(ν+1)∕2ma2)sech2(x∕a) does not reflect waves of any energy when ν is a positive integer. We show that in this reflectionless case the solutions of Schrödinger’s equation can be expressed in terms of elementary functions. Wave packets can be constructed from these energy eigenstates, and the propagation of such wave packets through the potential region can be studied analytically. We find that the group velocity of a particular packet can substantially exceed the group velocity of a free-space Gaussian packet. The bound states of the potential can also be expressed in terms of elementary functions when ν is an integer. The special properties of the integer ν potentials are associated with critical binding.

Although it is known theoretically that certain infinite integrals of Bessel functions can be expressed in terms of elementary functions, the practical evaluation of such integrals was quite difficult due to the algebraic complexity of the expressions involved. A simple and elegant algebra is introduced here which allows these integrals to be calculated in an elementary way in terms of elementary functions. Some relationships are shown between the integrals involving Bessel functions and two-dimensional integrals over a circle of elementary functions involving distances between points. A comparison is made with existing results, and some of them were found in error (or were misprints).

In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrodinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors. Tens of papers studying such models have been published (see [1] and the references therein), but representations of solutions to the arising Schrodinger- and heat-type equations, which go back to Feynman, have not been considered so far. One of the possible reasons is that the traditional application of Feynman’s approach involves integrals with respect to diffusion processes whose transition probabilities have no explicit representation (in terms of elementary functions) in the situation under consideration. In this paper, instead of these transition probabilities, we use their approximations, which can be expressed in terms of elementary functions. Apparently, similar approximations were first applied in [4, 5] to study the diffusion and the quantum evolution of particles of constant mass on Riemannian manifolds. It turns out that the central idea of the approach developed in [4, 5] can also be applied (after appropriate modifications) to the situation considered in this paper. In what follows, we assume that solutions to the Cauchy problems for the equations under examination exist and are unique; thus, we can and shall consider not only solutions to equations but also the corresponding semigroups. Somewhat changing the terminology of [2, 6], we define a real (complex) Schrodinger semigroup as e – tH (respectively, e ith ), where H is a self-adjoint positive operator on a Hilbert space or the generator of a diffusion process.

The results of this paper are due to some previous studies of the application problem of motion of a rigid body in a resisting medium. During these studies, a transcendental integral expressed in terms of elementary functions was obtained for a particular case. This fact allowed one to completely analyze all phase trajectories and to indicate those properties that were “rough” and were retained for some more general systems. The integrability of such a system is related to latent symmetries. Therefore, the study of sufficiently wide classes of dynamic systems with similar symmetries is of interest.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

An exact closed form solution for fragmentation of Weibull fibers in a single filament composite with applications to fiber-reinforced ceramics

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators