A New Mixed Nonpolynomial Spline Method for the Numerical Solutions of Time Fractional Bioheat Equation

In this paper, a numerical approximation for a time fractional one-dimensional bioheat equation (transfer paradigm) of temperature distribution in tissues is introduced. It deals with the Caputo fractional derivative with order for time fractional derivative and new mixed nonpolynomial spline for second order of space derivative. We also analyzed the convergence and stability by employing Von Neumann method for the present scheme.


Introduction
In the human body, the skin is considered as the largest organ. The study of skin and thermal behavior of living tissues is very fundamental, and it can be mathematically described by Pennes' bioheat transport equation: (1) Mathematical resolve of the complex thermal interaction between the vasculature and tissues is a topic of interest for numerous physiologists, physicians, and engineers [1]. Temperature distribution in skin tissues is important for medical applications such as skin cancer, skin burns, etc. [2]. At most, the ISSN: 0067-2904 Science, 2020, Vol. 61, No. 7, pp: 1724-1732 Abdullhussein and Al-Humedi Iraqi Journal of accurate solution of Pennes' equation does not exist and, therefore, approximations and numerical techniques must be used to solve this equation.
In recent years, fractional calculus has been adopted by scientists and engineers and applied in many fields, namely in the fields of materials and mechanics, medical science, fluid mechanics, viscoelasticity, physics, signal processing, anomalous diffusion, biological systems, finance, hydrology and many others [3,4] presented the solution of fractional bioheat equation by adopting the shifted Grünwald finite difference approximation for Riemann-Louville space fractional derivative method, the HPM of the fractional derivative of space, and the Caputo fractional for the fractional time. It has been spotted that the time possessed to achieve hyperthermia in a location is reduced as the order fractional derivative decreases. [5] discussed the two cases of 1D and 2D Pennes bioheat model for the implementation of triangular and quadrilateral elements method. In the 2D case, both quadrilateral and triangular elements were investigated. Through test problems, the discretization error generated from this method was reported [6] discussed the approximate solution of fractional Pennes bioheat equation with constant and sinusoidal heat flux conditions on skin, using the implicit finite difference method where the fractional time derivative is of the Caputo form. [7]. showed a numerical solution for the time-fractional Pennes bioheat transfer equation on skin tissues and solved it by Fourier Sine transform of second order derivative and the Caputo for the fractional time. [8], discussed the 2-D fractional bioheat equation by Laplace transforms of second order derivative and performed the numerical solutions to search the temperature transfer in skin exposed to immediate surface heating. Some differentiations were shown to estimate the impact of the fractional order parameter on the temperature wave. [9], presented a numerical solution of the fractional bioheat equation by finite difference of second order derivative and the fractional derivative by Grünwald Letnikov for the fractional time. They discussed and analyzed the stability and convergence. [10], studied the fractional bioheat transfer equation and solved it using an approximate solution (numerically) by finite difference of second order derivative and the fractional time derivative by Caputo derivative. They also discussed the stability and convergence by this scheme. [11], discussed the 2D fractional bioheat equation using Galerkin FEM. He found the solution method in the cylindrical living tissue and noted the effects of thermal conductivities that have significant and more remarkable effects on temperature variation in living tissue. [12], studied the fractional bioheat equation when the time-space fractional derivative in the form and solve it by Caputo fractional derivative of order and Riesz-Feller fractional derivative of order respectively. They obtained the results in terms of Fox's H-function with some specific cases using Fourier-Laplace transforms. [13], studied the fractional bioheat equation and solved the space-time fractional bioheat equation using fractional order Legendre functions of fractional space order derivative and the fractional time derivative by Caputo derivative. They observed that the quantity of the temperature at the skin surface is a strong function of the space fractional order and, conversely, the impact of the time fractional order is almost negligible.
The time fractional for Pennes' bioheat transfer equation and all the constants within it are introduced in section 2. We present the mathematical background concerned with the fractional definitions in section 3. In section 4, a new mixed spline form for the second space derivative is derived. In sections 5 and 6, we derive and apply the time fractional derivative by Caputo fractional derivative and a new mixed nonpolynomial spline form for space derivative of Pennes' bioheat transfer equation. Section 7 contains a stability analysis for Pennes bioheat transfer equation. In section 8 we apply and find the numerical solutions for time fractional Pennes bioheat equation by a new mixed nonpolynomial spline method.

Pennes Bioheat Transfer Equation with Time Fractional Derivative
The problem of the time fractional Pennes bioheat transfer equation for the modeling of skin tissue heat transfer is expressed in previous works [6,14,15,16], as follows: Science, 2020, Vol. 61, No. 7, pp: 1724-1732 Abdullhussein and Al-Humedi Iraqi Journal of 2.1 Nomenclature is the fractional order of time, is the distance from the skin surface, is a constant representing the density tissue , is a constant representing the specific heat of tissue , is the tissue thermal conductivity , is the mass flow rate of blood per unit volume of tissue , is the specific heat of blood , = 420 is the metabolic heat generation per unit volume , represents the arterial blood temperature, is the temperature of tissue, is the source of metabolic heat, represents the blood perfusion. It is worth mentioning that the constant was obtained experimentally by Pennes for a human forearm.

New Mixed Nonpolynomial Spline Form for the Second Derivative
Now we introduce the new nonpolynomial spline method which depends on a mixed spline , which can be written in the form: , (6) where and are unknown coefficients with respect to time and is the frequency of the trigonometric part of the spline functions. To find the coefficients of (6) in terms of and , at first we define , , , where Then by using (6) and (7), we obtain . From solving equation (8), we get the following expressions Therefore, by (9) and the continuity condition at knots , such that (10) From equations (6), (9) and (10), we yield the following relation (11)

Caputo fractional derivative for the time fractional derivative
The discrete approximation of time-fractional derivative at time point , can be achieved as follows [9,10] ∫ ∑ ∫ ∑ ∫ By using the forward Euler scheme to discretize the Caputo time fractional derivative, let in which is the time step size.
where , for all and 5. Derivation of a Caputo fractional and new mixed nonpolynomial spline forms for time fractional Pennes bioheat transfer equation From (7)and (12) and by putting in (2), equation (2) can be rewritten as the following system of algebraic equation: ∑ (13) where and Eq. (13) can lead to ∑ (14) By multiplying (13) and (14) by and respectively, then adding these equations, we get ( ) ∑ ( ) ( ) (15) From (11) and by substituting the value in (15), the last equation can be rewritten in the following form    where, by choosing the source function , the exact solution is given as follows: ,

Conclusions
The objective of this article is to compare the achievement of the model approach based on our new mixed nonpolynomial spline method, which have been considered for finding the numerical solutions of time fractional Pennes' bioheat equation by using Caputo fractional derivative for the time fractional derivative and a new scheme for the derivative of second order in this equation. In general, it can be concluded from and errors of the numerical approximations that the proposed method is powerful, effective, highly accurate and needed a small recurrence, as compared to the accurate solution. Furthermore, the present algorithm is simply applicable and the results clarified the activity of the suggested method. We discussed the stability of the fractional bioheat equation by the new mixed nonpolynomial spline method to clarify that the scheme is stable.