We obtain solution representation formulas for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well-defined provided that certain analyticity issues are appropriately addressed.
A method for solving linear initial boundary value problems was recently reimplemented as a true spectral transform method. As part of this reformulation, the precise sense in which the spectral transforms diagonalize the underlying spatial differential operator was elucidated. That work concentrated on two point initial boundary value problems and interface problems on networks of finite intervals. In the present work, we extend these results, by means of three examples, to new classes of problems: problems on semiinfinite domains, problems with nonlocal boundary conditions, and problems in w…
This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $ψ$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.
In the present paper, we determine the estimations on Atangana-Baleanu-Caputo fractional derivative at extreme points. With the assistance of the estimations obtained, we derive the comparison results. Peano's type existence results established for nonlinear fractional differential equations involving Atangana-Baleanu-Caputo fractional derivative. The acquired comparison results are then utilized to deal with the existence of local, extremal and global solution.
In this paper we investigate fractional differential equations with Hilfer fractional derivative of order $1<γ<2$ and type $δ\in [0,1]$ in a Banach space. We introduce a family of general fractional cosine operator functions of order $1<γ<2$ and type $δ\in [0,1]$ and discuss their properties to give a suitable definition of mild solution of a semilinear evolution equation. In last section an example is presented.
In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial problems for a differential equation with the bi-ordinal Hilfer's derivative and regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting po…
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and methods in the theory of composites then extend to these equations.
