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On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner (2006)

Colloquium Mathematicae

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function f : ℝ s → ℝ can be approximated with arbitrary accuracy by an infinite sum ∑ r = 1 ∞ H r ( x ₁ , . . . , x s ) ∈ C ∞ ( ℝ s ) of analytic functions H r , each solving the same system of universal partial differential equations, namely P ( x σ ; H r , ∂ H r / ∂ x σ , . . . , ∂ ⁵ H r / ∂ x ⁵ σ ⁵ ) = 0 (σ = 1,..., s).

On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method

Haci Mehmet Baskonus, Hasan Bulut (2015)

Open Mathematics

In this paper, we apply the Fractional Adams-Bashforth-Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Then, we construct a table including numerical results for both fractional differential equations. Then, we draw two dimensional surfaces of numerical solutions and analytical solutions by considering the suitable values of parameters. Finally, we use the L2 nodal norm and L∞ maximum nodal norm to evaluate the accuracy...

On the solutions of Knizhnik-Zamolodchikov system

Lev Sakhnovich (2009)

Open Mathematics

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.

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