In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fra...

where we understand that y is a function of an independent variable t . (We use t because in many examples the independent variable happens to be time, but of course any other variable could be used. In current versions of Maple, the dependence of y on t must be explicit, i.e., one must write y(t).) It is sometimes convenient to use informal notation and refer to this example as y′ = 0 (a physi...

We prove that degrees of rational solutions of an algebraic differential equation F (dw/dz,w, z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations. Consider the differential equation F (w′, w, z) = 0, (w′ = dw/dz) (1) where F is a polynomial in three variables. Theorem 1 For ...

The generalized Maxwell equations including an additional scalar field are considered in the first order formalism. The gauge invariance of the Lagrangian and equations is broken resulting the appearance of a scalar field. We find the canonical and symmetric energy momentum tensors. It is shown that the trace of the symmetric energymomentum tensor is not equal to zero in the theory considered. ...

The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local HamiltonJacobi equations. The idea is to combine an integral representation of the operator and Duhamel’s formula to prove, on the one side, the key a priori estimates for the scalar conservation law and t...

(4) v′ r(s) = c(xr(s), yr(s), vr(s)). Note that here the subscript r denotes a parameter, not a derivative! We may equally well write xr(s) = x(r, s), yr(s) = y(r, s), and we will do so; we adopted the subscript notation to emphasize that along each integral curve r is fixed, i.e. is a constant. Which parameterization should we use? We are normally also given initial conditions along a curve Γ ...

(1) a(x, y, u)ux + b(x, y, u)uy = c(x, y, u), with a, b, c at least C, given real valued functions. There is an immediate difference between semilinear and quasilinear equations at this point: since a and b depend on u, we cannot associate a vector field on R to the equation: we need to work on R at least to account for the (x, y, u) dependence. To achieve this, we proceed as follows. We consid...