Solving a class of generalized lyapunov operator differential equations without the exponential operator function

In this paper a method for solving operator differential equations of the type X' = A + BX + XD; X(0) = Co, avoiding the operator exponential function is given . Results are applied to solve initial value problems related to Riccati type operator differential equations whose associated algebraic equation is solvable

Analytical approximate solutions and error bounds for nonsymmetric riccati matrix differential equations

AbstractIn this paper we construct analytical approximate solutions for the nonsymmetric Riccati matrix differential equation. Given an admissible error ∈ > 0, we determine an interval where we construct an approximate solution whose error is smaller than ϵ for all the points of the interval. The approximate solution is constructed in terms of matrices related to the data

Explicit solutions for non homogeneous sturm mouville operator problems

In this paper we study existence and uniqueness conditions for the solutions of Sturm-Liouville operator problems related to the operator differential equation X" - QX = F(t). Explicit solutions of the problem in terms of a square root of the operator Q are given

An algebraic approach for solving boundary value matrix problems: existence, uniqueness and closed form solution

Sin resume

Exact, approximate solutions and error bounds for coupled implicit systems of partial differential equations

In this paper coupled implicit initial-boundary value systems of second order partial differential equations are considered. Given a finite domain and an admissible error ϵ an analytic approximate solution whose error is upper bounded by ϵ in the given domain is constructed in terms of the data

Continuous numerical solutions of coupled mixed partial differential systems using Fer's factorization

AbstractIn this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time-dependent systems of the type ut − A(t)uxx − B(t)u = 0, 0 < x < p, t > 0, u(0,t) = u(p,t) = 0, u(x,0) = f(x), 0⩽x⩽p. After truncation of an exact series solution, the numerical solution is constructed using Fer's factorization. Given ε > 0 and t0,t1, with 0< t0 < t1 and D(t0,t1) = {s(x,t); 0⩽x⩽p, t0⩽t⩽t1} the error of the approximated solution with respect to the exact series solution is less than ε uniformly in D(t0,t1). An algorithm is also included

The sign of the Green function of an n-th order linear boundary value problem

[EN] This paper provides results on the sign of the Green function (and its partial derivatives) of ann-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Almenar, P.; Jódar Sánchez, LA. (2020). The sign of the Green function of an n-th order linear boundary value problem. Mathematics. 8(5):1-22. https://doi.org/10.3390/math8050673S12285Butler, G. ., & Erbe, L. . (1983). Integral comparison theorems and extremal points for linear differential equations. Journal of Differential Equations, 47(2), 214-226. doi:10.1016/0022-0396(83)90034-7Peterson, A. C. (1979). Green’s functions for focal type boundary value problems. Rocky Mountain Journal of Mathematics, 9(4). doi:10.1216/rmj-1979-9-4-721Peterson, A. C. (1980). Focal Green’s functions for fourth-order differential equations. Journal of Mathematical Analysis and Applications, 75(2), 602-610. doi:10.1016/0022-247x(80)90104-3Elias, U. (1980). Green’s functions for a non-disconjugate differential operator. Journal of Differential Equations, 37(3), 318-350. doi:10.1016/0022-0396(80)90103-5Nehari, Z. (1967). Disconjugate linear differential operators. Transactions of the American Mathematical Society, 129(3), 500-500. doi:10.1090/s0002-9947-1967-0219781-0Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Schmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Almenar, P., & Jódar, L. (2016). Improving Results on Solvability of a Class ofnth-Order Linear Boundary Value Problems. International Journal of Differential Equations, 2016, 1-10. doi:10.1155/2016/3750530Almenar, P., & Jodar, L. (2017). SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS. Mathematical Modelling and Analysis, 22(4), 528-547. doi:10.3846/13926292.2017.1329757Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Webb, J. R. L. (2017). New fixed point index results and nonlinear boundary value problems. Bulletin of the London Mathematical Society, 49(3), 534-547. doi:10.1112/blms.12055Jiang, D., & Yuan, C. (2010). The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Analysis: Theory, Methods & Applications, 72(2), 710-719. doi:10.1016/j.na.2009.07.012Wang, Y., & Liu, L. (2017). Positive properties of the Green function for two-term fractional differential equations and its application. The Journal of Nonlinear Sciences and Applications, 10(04), 2094-2102. doi:10.22436/jnsa.010.04.63Zhang, L., & Tian, H. (2017). Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations. Advances in Difference Equations, 2017(1). doi:10.1186/s13662-017-1157-7Wang, Y. (2020). The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications. Advances in Difference Equations, 2020(1). doi:10.1186/s13662-020-02549-

The principal eigenvalue of some nth order linear boundary value problems

[EN] The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz-Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.Almenar, P.; Jódar Sánchez, LA. (2021). The principal eigenvalue of some nth order linear boundary value problems. Boundary Value Problems. 2021:1-16. https://doi.org/10.1186/s13661-021-01561-2S116202

Accurate estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

[EN] This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz-Wielandt numbers of the resulting functions.Almenar, P.; Jódar Sánchez, LA. (2021). Accurate estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems. Mathematics. 9(21):1-22. https://doi.org/10.3390/math921266312292

Convergent discrete numerical solutions of strongly coupled mixed parabolic systems

This work has been partially supported by the Spanish D.G.I.C.Y.T. grant BMF 2000-0206-C04-04Jódar Sánchez, LA.; Casabán, M. (2003). Convergent discrete numerical solutions of strongly coupled mixed parabolic systems. UTILITAS MATHEMATICA. 63:151-172. http://hdl.handle.net/10251/161860S1511726