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Register a new userA uniqueness result on ordinary differential equations with singular coefficients
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We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the results.
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Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H > frac{1}{2} and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.
We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $varepsilon xpartial_x psi (x, varepsilon) + A (x, varepsilon) psi (x, varepsilon) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $varepsilon$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $varepsilon$ in the same sector. We use this result to construct a family in $varepsilon$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $varepsilon$; this family of filtrations recovers in the $varepsilon to 0$ limit the eigen-decomposition for the $varepsilon$-leading-order of the matrix $A (x, varepsilon)$, and also recovers in the $x to 0$ limit the eigen-decomposition of the residue matrix $A (0, varepsilon)$.
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. If the solution is a special function that is studied in the literature, then the result is more manageable using the properties of that function. It is straightforward to find the regular and irregular singular points of such an equation by a computer algebra system. However, one needs the corresponding indices for a full analysis of the singularity structure. It is shown that the $theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure which can be visualized through a Riemann P-symbol leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
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