Chebyshev Polynomial Approximation to Solutions of Ordinary Differential Equations

In this thesis, we develop a method for finding approximate particular solutions for second order ordinary differential equations. We use Chebyshev polynomials to approximate the source function and the particular solution of an ordinary differential equation. The derivatives of each Chebyshev polynomial will be represented by linear combinations of Chebyshev polynomials, and hence the derivatives will be reduced and differential equations will become algebraic equations. Another advantage of the method is that it does not need the expansion of Chebyshev polynomials. This method is also compared with an alternative approach for particular solutions. Examples including approximation, particular solution, a class of variable coefficient equation, and initial value problem are given to demonstrate the use and effectiveness of these methods

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.Comment: 19 pages, 1 figure. To appear in Nonlinearity. Minor changes. The present version includes additional historical remarks and bibliograph

A mean square chain rule and its application in solving the random Chebyshev differential equation

[EN] In this paper a new version of the chain rule for calculat- ing the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the cor- responding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.This work was completed with the support of our TEX-pert.Cortés, J.; Villafuerte, L.; Burgos-Simon, C. (2017). A mean square chain rule and its application in solving the random Chebyshev differential equation. Mediterranean Journal of Mathematics. 14(1):14-35. https://doi.org/10.1007/s00009-017-0853-6S1435141Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Analytic stochastic process solutions of second-order random differential equations. Appl. Math. Lett. 23(12), 1421–1424 (2010). doi: 10.1016/j.aml.2010.07.011El-Tawil, M.A., El-Sohaly, M.: Mean square numerical methods for initial value random differential equations. Open J. Discret. Math. 1(1), 164–171 (2011). doi: 10.4236/ojdm.2011.12009Khodabin, M., Maleknejad, K., Rostami, K., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge Kutta methods. Math. Comp. Model. 59(9–10), 1910–1920 (2010). doi: 10.1016/j.mcm.2011.01.018Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216(5), 1524–1530 (2010). doi: 10.1016/j.amc.2010.03.001González Parra, G., Chen-Charpentier, B.M., Arenas, A.J.: Polynomial Chaos for random fractional order differential equations. Appl. Math. Comput. 226(1), 123–130 (2014). doi: 10.1016/j.amc.2013.10.51El-Beltagy, M.A., El-Tawil, M.A.: Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm. Appl. Math. Model. 37(12–13), 7174–7192 (2013). doi: 10.1016/j.apm.2013.01.038Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterran. J. Math. 12(3), 1123–1140 (2015). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comp. Math. Appl. 59(1), 115–125 (2010). doi: 10.1016/j.camwa.2009.08.061Øksendal, B.: Stochastic differential equations: an introduction with applications, 6th edn. Springer, Berlin (2007)Soong, T.T.: Random differential equations in science and engineering. Academic Press, New York (1973)Wong, B., Hajek, B.: Stochastic processes in engineering systems. Springer Verlag, New York (1985)Arnold, L.: Stochastic differential equations. Theory and applications. John Wiley, New York (1974)Cortés, J.C., Jódar, L., Camacho, J., Villafuerte, L.: Random Airy type differential equations: mean square exact and numerical solutions. Comput. Math. Appl. 60(5), 1237–1244 (2010). doi: 10.1016/j.camwa.2010.05.046Calbo, G., Cortés, J.C., Jódar, L.: Random Hermite differential equations: mean square power series solutions and statistical properties. Appl. Math. Comp. 218(7), 3654–3666 (2011). doi: 10.1016/j.amc.2011.09.008Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Comp. Math. Appl. 61(9), 2782–2792 (2010). doi: 10.1016/j.camwa.2011.03.045Cortés, J.C., Jódar, L., Company, R., Villafuerte, L.: Laguerre random polynomials: definition, differential and statistical properties. Utilit. Math. 98, 283–293 (2015)Cortés, J.C., Jódar, L., Villafuerte, L.: Mean square solution of Bessel differential equation with uncertainties. J. Comp. Appl. Math. 309, 383–395 (2017). doi: 10.1016/j.cam.2016.01.034Golmankhaneh, A.K., Porghoveh, N.A., Baleanu, D.: Mean square solutions of second-order random differential equations by using homotopy analysis method. Romanian Reports Physics 65(2), 1237–1244 (2013)Khalaf, S.L.: Mean square solutions of second-order random differential equations by using homotopy perturbation method. Int. Math. Forum 6(48), 2361–2370 (2011)Khudair, A.R., Ameen, A.A., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci. 5(49), 2521–2535 (2011)Agarwal, R.P., O’Regan, D.: Ordinary and partial differential equations. Springer, New York (2009

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad

Polynomial Stability of Abstract Wave Equations

Abstract wave equations model a large class of linear dynamical systems, i.e., systems that evolve linearly over time. For example, this vast class of abstract wave equations encloses many important second-order partial differential equations that are frequent in applications. As with any partial differential equation, an important question arises whether a solution to an abstract wave equation exists for all possible initial conditions. By a solution we simply mean a function that satisfies both the abstract wave equation and the existing boundary and initial conditions. Answering this question leads us to the widely known criteria for well-posed problems by Hadamard. It turns out that all abstract wave equations are well-posed, which is implied by the theory of so-called strongly continuous semigroups. Having established the existence of solutions for abstract wave equations, it is natural to ask how the solutions behave over time. In particular, we are interested in the limiting, that is, asymptotic behaviour of the solutions as time elapses. If the solutions corresponding to all initial conditions eventually converge to some equilibria, then we call the associated abstract wave equation asymptotically stable. In case the rate of convergence is also uniform for all solutions, the solutions actually converge to their equilibria at an exponential rate, yielding exponential stability. In general, a solution to an asymptotically stable abstract wave equation can approach its equilibrium arbitrarily slowly and thus preclude any uniform rate of convergence. However, with certain assumptions we obtain results for strongly continuous semigroups that guarantee both asymptotic stability and a uniform rate of convergence for a particular subset of solutions called classical solutions. In this thesis we examine the polynomial stability of abstract wave equations. Put simply, all classical solutions to an abstract wave equation should converge to their equilibria at a polynomial rate. A polynomially stable system is always asymptotically stable but not necessarily exponentially stable. Although polynomial stability is a special case of a more general semi-uniform stability, for the time being counterparts to important results implying exponential stability only exist for polynomial stability. The key idea in these results is to investigate how the norm of a resolvent associated with the abstract wave equation grows on the imaginary axis. The slower this norm grows, the faster the classical solutions converge. At the end of this thesis we analyze a system from the literature and its two variants in great detail. We recast these systems as abstract wave equations and study their stability with the theory and tools we obtain along the way

Physical applications of second-order linear differential equations that admit polynomial solutions

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.Comment: 13 pages, no figure

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= \sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {\bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $\chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $\chi^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $2+s+s^2=0$ oustside the unit circle ($| s | > 1$).Comment: 34 pages, 1 figur