An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear nonpolynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.


1-Introduction
Fractional calculus is the study of fractional order derivatives and integrals. It gained extensive attention form the researchers in the last few decades. It has exceptional applications in diverse fields of science and engineering. Spline functions are piecewise polynomials of degree that are joined together at the break points with continuous derivatives. A piecewise non-polynomial spline function is a blend of trigonometric, as well as polynomial basis functions, which form a complete extension. In [1], Saeah solved linear Volterra integral equations using non-polynomial spline function. In [2], Batool used the variational iteration method to solve partial integro differential equations of fractional order. In [3], Ghulam et al. studied the application of Caputo K-fractional derivatives. In [4], a method based the fractional shifted Legendre polynomials was applied to solve non-homogeneous space and time fractional partial differential equations (FPDEs), in which space and time fractional derivatives are described in the Caputo sense. In [5], some applications of the nonpolynomial spline approach to the solution of the Burgers' equation were studied. In [6], the non-

ISSN: 0067-2904
Hasan and Salim Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2327-2333 2328 polynomial spline methods were used for the solution of a system of obstacle problems. The main objective of the present paper lies on introducing a new approximate solution of time-space fractional heat equations by using the linear non-polynomial spline method.

2-Tensor Product of Two-dimensional Problems
The treatment of high-dimensional problems, such as heat equation, can be approached by concepts of tensor product approximation. Let be a regain such that { }. The method of finding two dimensional functions in a tensor product space , such that are two spline spaces may be considered as the basis for the generalized spline functions, so that are integer numbers. The interpolation condition is hence ∑ ∑ ( ) In the matrix form, we solve the system We find the coefficients which are unknown of the function g, which is given in eq (1), where is tensor product of two matrices .

3-Non-Polynomial Spline Function for Caputo derivative Definition (3.1) [7]
The Caputo fractional derivative of the fractional order is defined as ∫ Now consider the partition of . Let denotes the set of piecewise polynomials on subinterval of partition ∆. Let be the exact solution. This new method provides an approximation, not only for at the knots, but also for , , at every point in the range of integration . Also, the differentiability of the trigonometric part of non-polynomial splines compensates for the loss of smoothness inherited by the polynomial [1] . The non-polynomial spline function is obtained by the segment . Each non-polynomial spline of the n order has the form where are constants and is the frequency of the trigonometric functions which will be used to raise the accuracy of the method.
In this paper, the linear non-polynomial spline function will be used for solving fractional partial differential equations.

Definition (3.1) [1]: Linear Non-Polynomial Spline Function
The form of the linear non-polynomial spline function is where are constants to be determined.

Definition (3.2) [8]
Let and be two Hilbert spaces. If have the orthonormal bases and , respectively , then is an orthonormal basis for . In particular , the Hilbert dimension of the tensor product is the product of the Hilbert dimensions.

Definition (3.3) [9]
A set of functions is an orthogonal set of functions on the interval if any two functions in the set are orthogonal to each other, so that ∫ In the following theorem, the convergence of the method to the exact solution using a tensor product is given.

Theorem (3.4)
Non-polynomial solutions of the two variables given in equation (1)

4-Solution of Fractional Heat Equation Using Tensor Product of Linear Non-Polynomial Spline
The non-polynomial spline method will be used here to approximate the solution of the failure probability density function (FPDF): …(3) with boundary condition: …(4) and initial condition: is a given function. By equation (2), the function is replaced by which is used to approximate the solution of the FPDE. Hence, ∑ ∑ …(6) where , are the basis of the linear non-polynomial spline function and , are the basis of the linear non-polynomial spline function. From the initial condition given by equation (5), one my get:

∑ ∑
We substitute the knot points for the -axis to get an equation for each knot point, and form the boundary condition given in equation (6), we have ∑ ∑ and ∑ ∑ Similarly , we substitute the knot points for the t-axis to get an equation for each knot point at and Form the boundary condition given in equation (4), we have We substitute the mesh points ,for , , to get an equation for each pair , for all , Then, from equations (7-10), a system with unknown coefficients must be determined to compute equation (6).

5-Illustrative Example
To demonstrate the effectiveness of the proposed method, we consider here two test examples of one dimensional fractional heat equation problem. The software MathCad 15 is used to get the numerical results. Example(1.1): Consider the homogeneous one-dimensional fractional heat equation subject to the boundary condition : , and the initial condition : . The exact solution for is given by: . Let be a partition for the x-axis, such that: then , the mesh points for the -axis. Let be a partition for the t-axis such that: then the mesh points for -axis. By solving the linear system 16 unknown coefficients, we get , Then, the approximate solution is:  Figure-(1.1) (a), while the exact solution is illustrated in Figure (1.1) (b). The exact solution [10] for is given by : Let be a partition for the x-axis, such that: then , the mesh points for the -axis. Let 2  be a partition for the t-axis such that then the mesh points for -axis. By solving the linear system 16 unknown coefficients, we get , , Then, the approximate solution is: z u  The approximation solution is illustrated in Figure-

6-Conclusions and future work
In this paper, the approximate solutions of the fractional-order heat equation are determined using the method of a linear non-polynomial spline. The results revealed the highest agreement with the exact solutions for the problems. The solutions for the numerical examples showed the validity of the u z proposed method. In addition, it is observed that the solutions of the fractional-order equation are convergent to the exact solution for the problem. In the future, the method of linear non-polynomial spline can be used to find the solution of other FPDEs that are frequently used in science and engineering.