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We will study a maximal solution of the time-space fractional heat equation in complex domain. The fractional time is taken in the sense of the Riemann-Liouville operator, while the fractional space is assumed in the Srivastava-Owa operator. Here we employ some properties of the univalent functions in the unit disk to determine the upper bound of this solution. The maximal solution is illustrated in terms of the generalized hypergeometric functions.

Fractional calculus (integrals and derivatives) of any positive order can be considered a branch of mathematical physics which concerns with differential equations, integral equations, and integrodifferential equations, where integrals are of convolution form with weakly singular kernels of power law type. It has prevailed more and more interest in applications in several fields of applied sciences.

Fractional differential equations (real and complex) are viewed as models for nonlinear differential equations. Varieties of them play important roles and tools not only in mathematics but also in physics, dynamical systems, control systems, and engineering to create the mathematical modeling of many physical phenomena. Furthermore, they are employed in social science such as food supplement, climate, and economics. One of these equations is the heat equation of fractional order. Over the last two decades, many authors studied this equation in fractional type. Recently, Vázquez et al. analyzed a set of fractional generalized heat equations [

In this paper, we shall study a solution of the time-space fractional heat equation in the unit disk. The fractional time is taken in the sense of the Riemann-Liouville operator while the fractional space is assumed in the Srivastava-Owa operator. Here we shall apply some properties of the univalent functions to determine the upper bound of this solution. The maximal solution is illustrated in terms of the generalized hypergeometric functions.

The idea of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was found over 300 years ago. Abel in 1823 scrutinized the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem.

This section concerns with some preliminaries and notations regarding the Riemann-Liouville operators. The Riemann-Liouville fractional derivative strongly poses the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms [

The fractional (arbitrary) order integral of the function

The fractional (arbitrary) order derivative of the function

From Definitions

In [

The fractional derivative of order

The fractional integral of order

Consider,

Note that the Srivastava-Owa operators are the complex version (in a simply-connected region) of the Riemann-Liouville operators.

One of the major branches of complex analysis is univalent function theory: the study of one-to-one analytic functions. A domain

Let

A function

For

In this section, we will express the maximal solution (a solution which has no extension is called a maximal solution) for the problem

Distributions of the heat in the unit disk.

We have the following results.

Let

Since

Let

Since

We proceed to determine the maximal solution of (

Let

By applying the upper bound of the operator

Note that any extension of the solution

In the same manner of Theorem

Let

We introduced a method depending on some properties of the geometric function theory (univalent, starlike, and convex) for obtaining the maximal solution of heat equation of fractional order in a complex domain. This solution was obtained for two cases: homogeneous and nonhomogeneous distributions of the source. We realized that this solution can be considered as a positive, bounded, and stable solution in the unit disk. We may apply this new method on well-known diffusion equations of fractional order such as the fractional wave equation in complex domain. Furthermore, since these solutions are determined in terms of the hypergeometric function and its generalization, then this leads to have a convergence solution. Note that the hypergeometric function involves the Mittag-Leffler function functions. Figure

Starlike distribution (nonhomogeneous).

Convex distribution (homogeneous).

The authors would like to thank the reviewers for their comments on earlier versions of this paper. This research has been funded by university of Malaya, under Grant no. (HIR/132).