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Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

Mohammed H. Aqlan, Ahmed Alsaedi, Bashir Ahmad, Juan J. Nieto (2016)

Open Mathematics

We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration...

Explicit rational solutions of Knizhnik-Zamolodchikov equation

Lev Sakhnovich (2008)

Open Mathematics

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group 𝒮 n n. We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.

Explicit solution for Lamé and other PDE systems

Alexei Rodionov (2006)

Applications of Mathematics

We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem. The equations of three dimensional elastic equilibrium are solved as an example. Another convergence theorem is proved for this particular system. We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables.

Explicit solutions for boundary value problems related to the operator equations X ( 2 ) - A X = 0

Lucas Jódar, Enrique A. Navarro (1991)

Applications of Mathematics

Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation X ( 2 ) - A X = 0 are studied for the general case, even when the algebraic equation X 2 - A = 0 is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.

Explicit two-step Runge-Kutta methods

Zdzisław Jackiewicz, Rosemary Anne Renaut, Marino Zennaro (1995)

Applications of Mathematics

The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order p ≤ 5 the minimal number of stages for explicit TSRK method of order p is equal to the minimal number of stages for explicit Runge-Kutta method of order p - 1 . Numerical results are presented which...

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