In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.

We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting w...

-Using the He’s variational iteration method, it is possible to find the exact solutions or better approximate solutions of the partial differential equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. In this paper, this method is used for solving a nonlinear partial differential equation, a three-di...

Uncertain partial differential equation is a type of partial differential equation driven by Liu processes. This paper first provides two methods for solving the first order linear uncertain partial differential equation and the inverse uncertainty distribution of solution is discussed. Moreover, on the basis of the study of uncertainty theory and the propagation process of internet public opin...

In this paper a new analytical method for solving the truly (pure) non-linear differential equation with slow-time variable parameter is developed. The method is based on the solution of the corresponding differential equation with a constant parameter, where the exact frequency of vibration is introduced. The approximate solution of the differential equation with time variable parameter is ass...

The aim of this paper is to investigate the Hyers-Ulam stability of the  linear differential equation$$y''(x)+alpha y'(x)+beta y(x)=f(x)$$in general case, where $yin C^2[a,b],$  $fin C[a,b]$ and $-infty

where u (x, t) denotes the unknown function, f (s) is the given nonlinear function, u0 (x) and u1 (x) are the given initial value functions, k is a constant, the subscript t indicates the partial derivative with respect to t, n is the dimension of space variable x, and △ denotes the Laplace operator in R. The effects of small nonlinearity and dispersion are taken into consideration in the deriv...

The Fuchsian differential equation is a linear differential equation whose singularities are all regular. It frequently appears in a range of problems in mathematics and physics. For example, the famous Gauss hypergeometric differential equation is a canonical form of the second-order Fuchsian differential equation with three singularities on the Riemann sphere C ∪ {∞}. Global properties of sol...