On the Numerical Solution of Initial-Boundary Value Problems for the Convection-Diffusion Equation with a Fractional Caputo Derivative and a Nonlocal Linear Source

In a rectangular domain the first and third initial-boundary value problems are studied for the one-dimensional with respect to the spatial variable diffusion convection equation with a fractional Caputo derivative and a nonlocal linear source of integral form. Using the method of energy inequalities, under the assumption of the existence of a regular solution, a priori estimates are obtained in differential form, which implies the uniqueness and continuous dependence of the solution on the input data of the problem. On a uniform grid, two difference schemes are constructed that approximate the first and third initial-boundary value problems, respectively. For the solution of the difference problems, a priori estimates are obtained in the difference interpretation. The obtained estimates in difference form imply uniqueness and stability, as well as convergence at a rate equal to the order of the approximation error. An algorithm for the approximate solution of the third boundary value problem is constructed, numerical calculations of test examples are carried out, illustrating the theoretical results obtained in this work.

On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

We show herein the existence and uniqueness of solutions for coupled fractional order partial differential equations modeling a thermoelastic fractional Kirchhoff plate model associated with initial, Dirichlet, and nonlocal boundary conditions involving fractional Caputo derivative. Some efficient results of existence and uniqueness are obtained by employing the energy inequality method.

Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise

<p style='text-indent:20px;'>Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.</p>

The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

Approximate solutions of one-dimensional systems with fractional derivative

The fractional calculus is useful to model nonlocal phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the nonlocality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

A PRIORI ESTIMATION OF A GENERALIZED NONLOCAL BOUNDARY VALUE PROBLEM FOR A THRID ORDER EQUATION WITH A FRACTIONAL TIME CAPUTO DERIVATIVE

Рассматривается краевая задача для уравнения третьего порядка параболического типа с дробной производной Капуто. Методом энергетических неравенств получена априорная оценка решения обобщенной нелокальной краевой задачи для уравнения с кратными характеристиками с дробной производной Капуто по времени. A boundary value problem for a third-order parabolic equation with a fractional Caputo derivative is considered. A priori estimation of the solution of a generalized nonlocal boundary value problem for an equation with multiple characteristics with a fractional Caputo derivative in time is obtained by the method of energy inequalities.

Cauchy formula for the time-varying linear systems with caputo derivative

In the paper, existence, uniqueness and a Cauchy formula for the solution to a time-varying linear system containing fractional Caputo derivative is obtained. This formula shows that nonnegativity of the data of the system implies nonnegativity of the solution. In the context of a strenghthening of this result, an example illustrating the absence (in the case of Caputo derivative) of the standard relation “monotonicity of function - sign of derivative”.

Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives

In this research, we applied the variational homotopic perturbation method and q-homotopic analysis method to find a solution of the advection partial differential equation featuring time-fractional Caputo derivative and time-fractional Caputo–Fabrizio derivative. A detailed comparison of the obtained results was reported. All computations were done using Mathematica.