Solution of Fractional Derivative Homogeneous and Non-Homogeneous Parabolic Problems using the Fourier Method

M32-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye Solution of Fractional Derivative Homogeneous and Non-Homogeneous Parabolic Problems using the Fourier Method Şöhret Turhan1, İrem Bağlan1 1Department of Mathematics, Kocaeli University, Kocaeli, TR-41380, Turkey Corresponding author: [email protected] ORCID IDs: First Author: 0009-0008-5438-2709 Second Author: 0000-0003-4900-6027 DOI : 10.5281/zenodo.18033179 Abstract In this paper, the solutions of homogeneous and non-homogeneous parabolic equations with periodic boundary conditions are investigated using the Fourier method. In the solution process, the Sturm– Liouville equation is solved under periodic boundary conditions to determine the eigenvalues and corresponding eigenfunctions. Caputo fractional derivatives are employed, and the analytical solutions involve the Gamma and Beta functions, while the fractional-order solutions are expressed in terms of the Mittag–Leffler function. The results demonstrate that the Fourier method provides an effective analytical framework for solving fractional derivative semilinear pseudo-parabolic equations with periodic boundary conditions. Keywords: Fourier method, fractional derivative, periodic boundary conditions. 1. Introduction Fractional differential equations are equations in which derivatives and integrals are not limited to integer orders but can also take real or complex values [1–10]. The origins of this field date back to ancient times, and the concept of fractional derivatives has long attracted the attention of mathematicians. Many prominent mathematicians have contributed to its development. Early studies were conducted by Leibniz and L’Hôpital (1695), Bernoulli (1697), Euler (1730), and Lagrange (1772). In the following years, numerous researchers such as Laplace (1812), Fourier (1822), Abel (1823), Liouville (1832), Riemann (1847), Grünwald (1867), Letnikov (1868), Hadamard (1892), Heaviside (1892), Weyl (1917), Zygmund (1945), J. L. Lions (1959), and Liverman (1964) further advanced the theory of fractional derivatives and integrals. Fractional-order partial differential equations (PDEs) are used in many branches of science, including applied mathematics, physics, chemistry, and engineering. They have become a powerful mathematical tool for modeling complex processes and systems in these disciplines. These equations can provide more realistic models in cases where classical derivatives are insufficient. Fractional derivatives are history-dependent because they take into account not only the current state but also past M32-2 2nd Kocaeli Science Congress, November 19-21, 2025 events. They reflect effects accumulated over time, making them particularly suitable for systems with memory. These properties have led to increasing interest in fractional derivatives in recent years. Various definitions have been proposed for fractional derivatives. The Caputo and Riemann–Liouville definitions, which include integral-based expressions involving the Gamma function, are particularly prominent. Among these definitions, the Caputo derivative has become one of the most widely used in the literature due to its advantages. Because it allows initial conditions to be defined in the classical sense, the Caputo derivative yields more meaningful and applicable solutions when modeling physical processes. In this context, models incorporating the Caputo derivative produce results that are more consistent, applicable, and physically realistic compared to other fractional derivative definitions. Numerous studies supporting this view can be found in the literature [11–27]. 2. General Information 2.1 The Gamma Function (Г) The gamma function is a special function used in fractional-order integral and derivative calculations. Definition 2.1 The first definition was given by Gauss: Г(𝑧)=lim→ ! ()()…() ,𝑅𝑒(𝑧)>0. (1) With this limit definition, Gamma (Г) can be calculated for all values of 𝑧except for zero and negative integers [28]. Definition 2.2 The gamma function was also defined by Leonhard Euler in 1729, based on an integral transform. Euler’s definition was developed to generalize the factorial of natural numbers. The definition commonly used today is: Г(𝑧)=∫𝑒𝑡   ,𝑅𝑒𝑧>0. (2) With this definition, the gamma function can be calculated for all values of 𝑧[28]. Eq. (2) is valid for integer values of 𝑛. Since 𝑛!= 𝑒𝑡𝑑𝑡   the gamma function is also referred to as the factorial function. Some important properties of the gamma function include: 1) Г(1)=1 Here, if we use the limit definition made by Gauss, Г(1)=lim→ ! ..…()= lim→ ! ()! = lim→   =1. 2) Г󰇡 󰇢=√𝑛 [28]. 3) Г(𝑧+1)=𝑧.Г(𝑧) ,𝑧𝜖𝑍 M32-3 2nd Kocaeli Science Congress, November 19-21, 2025 Г(2)=Г(1)=1! Г(3)=2Г(2)=2! Г(4)=3Г(3)=3! Г(𝑛+1)=𝑛.Г(𝑛)=𝑛.(𝑛−1)!=𝑛! 2.2 The Beta Function The Gamma function, defined by Euler, forms the basis of many special functions. One of these is the Beta function. The Beta function is directly related to the Gamma function and is often used to simplify certain combinations of Gamma functions. It is also called the Euler integral of the first kind and is defined as follows: 𝐵(𝑧,𝜔)=∫𝜏(1−𝜏)𝑑𝜏   ,𝑅𝑒(𝑧)>0,𝑅𝑒(𝜔)>0 (3) The Beta function can also be expressed in terms of the Gamma function as 𝐵(𝑧,𝜔)=Г(𝑧)Г(𝜔) Г(𝑧+𝜔) and it satisfies the symmetry property [28] 𝐵(𝑧,𝜔)=𝐵(𝜔,𝑧). 2.3 Mittag-Leffler Function (E) Fractional differential equations are generally much more complex to solve than classical equations. A special function frequently used in the solution of such equations is the Mittag–Leffler function, named after Magnus Gustaf (Gösta) Mittag–Leffler. This function is a generalization of the exponential function for fractional differential equations and is particularly important in expressing solutions to initial value problems 𝐸,(𝑧)=∑ Г()   ,𝛼>0 ,𝛽>0. (4) When β=1 it is called a single parameter Mittag-Leffler function. It is shown as 𝐸(𝑧)=𝐸,(𝑧), 𝐸(𝑡)= 𝑡 Г(1+𝛼𝑘)   single parameter Mittag-Leffler function 𝐸,(𝑡)= 𝑧 Г(𝛼𝑘+𝛽)   ,𝛼>0 Two-parameter Mittag-Leffler function. When α=1,β=1 is accepted, it reduces to the classical exponential function𝑒. Therefore, Mittag-Leffler functions assume the role of the exponential function in fractional differential equations. This series converges for all values of z when 𝛼,𝛽 are real and positive. M32-4 2nd Kocaeli Science Congress, November 19-21, 2025 𝐸,(0)= Г(), 𝐸,(𝑧)=∑ Г()   =∑ !   =𝑒, 𝐸,(𝑧)=∑ Г()   =∑ ()!   = ∑ ()!   = , 𝐸,(𝑧)=∑ Г()   =∑ ()!   = ∑ ()!   = , 𝐸,(𝑧)=∑ Г()   =∑ ()!   =cosh(𝑧), 𝐸,(𝑧)= 𝑧 Г(2𝑘+2)   =1 𝑧𝑧 (2𝑘+1)!   =sinh(𝑧) 𝑧 Mittag–Lefler is a special expansion of the function [28]. 2.4 Riemann-Liouville Fractional Derivative Let f be a continuous and integrable function on the interval (𝑎,𝑡), the Reiman-Liouville fractional derivative of f of order 𝛼 for 𝑡>𝑎 such that 𝑛∈𝑁,𝑛 − 1≤𝛼<𝑛 R-L Fractional derivative 𝐷𝑓(𝑡)= (). ∫(𝑡−𝜏)𝑓(𝜏)𝑑𝜏   (5) 2.5Riemenn-Liouville Fractional Integral 𝐷𝑓(𝑡)= ()∫(𝑡−𝜏)𝑓(𝜏)𝑑𝜏   (6) 2.6 Caputo Fractional Derivative The Caputo fractional derivative of order 𝛼 is defined as: 𝐷𝑓(𝑡)= ()∫()() ()𝑑𝜏   ,𝑛 −1<𝛼<𝑛 (7) The Caputo derivative can be considered a counterpart to the Riemann–Liouville (R-L) fractional derivative. The main difference between the R-L derivative and the Caputo derivative is the order in which the derivatives are taken: in the R-L derivative, the fractional derivative is applied first, followed by the integer-order derivative; in the Caputo derivative, the integer-order derivative is taken first, followed by the fractional derivative. The Riemann–Liouville fractional derivative is not always suitable for applied problems because the initial conditions are tied to fractional integral limits, which cannot be directly interpreted in a classical sense. The derivative of the constant in the R-L derivative is 𝐷(𝑐)=. (). The derivative of the constant in the Caputo derivative is zero [28]. M32-5 2nd Kocaeli Science Congress, November 19-21, 2025 2.7 Fourier Series 𝑓(𝑥)= +∑ (𝑎𝑐𝑜𝑠𝑛𝑥+𝑏𝑠𝑖𝑛𝑛𝑥)   (8) The function series f(x) with a period of 2𝜋 is called a Fourier Series. The coefficients 𝑎,𝑎 and 𝑏are called the Fourier coefficients. There are certain properties that a function must possess to be expanded into a Fourier series.The Fourier series must be uniformly convergent [30]. Definition 2.3 (Piecewise Continuity). A function is called a piecewise continuous function if it is continuous at all points except a finite number of points in the interval [a,b] where it is defined [30]. Theorem 2.1 (Dirichlet’s Theorem). If 𝑓(𝑥) and its derivatives are piecewise continuous on the interval [−𝜋,𝜋], then the Fourier series of 𝑓(𝑥) satisfies the following properties [30]: 1. At continuous points, the Fourier series converges to f(x). 2. At discontinuity points, it converges to the average of the right and left limits: ()() . 3. At the extreme points, it is equal to the average value of the right and left limits: ()() . To find the Fourier coefficient 𝑎 over the interval [−𝜋,𝜋], integrate both sides of Eq. (8) with respect to 𝑥 from −𝜋 to 𝜋: ∫𝑓(𝑥)𝑑𝑥   =∫ 𝑑𝑥   +∑𝑎∫𝑐𝑜𝑠𝑛𝑥𝑑𝑥   +𝑏∫𝑠𝑖𝑛𝑛𝑥𝑑𝑥      . Since for all 𝑛≥1: ∫𝑐𝑜𝑠𝑛𝑥𝑑𝑥   =0, ∫𝑠𝑖𝑛𝑛𝑥𝑑𝑥   =0. all terms in the summation vanish. This leaves: ∫𝑓(𝑥)𝑑𝑥   =  𝑑𝑥   =𝑎𝜋. Hence, the coefficient 𝑎is: 𝑎= ∫𝑓(𝑥)   𝑑𝑥. (9) To find the Fourier coefficient 𝑎, multiply both sides of Eq. (8) by cos (𝑘𝑥) and integrate over the interval [−𝜋,𝜋]: ∫𝑓(𝑥)𝑐𝑜𝑠𝑘𝑥𝑑𝑥   =∫ 𝑐𝑜𝑠𝑘𝑥𝑑𝑥   +∑𝑎∫𝑐𝑜𝑠𝑛𝑥𝑐𝑜𝑠𝑘𝑥𝑑𝑥   +𝑏∫𝑠𝑖𝑛𝑛𝑥𝑐𝑜𝑠𝑘𝑥𝑑𝑥      , 𝑠𝑖𝑛𝑛𝑥𝑐𝑜𝑠𝑘𝑥𝑑𝑥   =0,  𝑐𝑜𝑠𝑘𝑥𝑑𝑥=0,   𝑐𝑜𝑠𝑛𝑥𝑐𝑜𝑠𝑘𝑥𝑑𝑥   ={0,𝑘≠𝑛 𝜋,𝑘=𝑛. Finally, the Fourier coefficient 𝑎 is given by: M32-6 2nd Kocaeli Science Congress, November 19-21, 2025 𝑎= ∫𝑓(𝑥)𝑐𝑜𝑠𝑛𝑥𝑑𝑥   . (10) To find the Fourier coefficient 𝑏, multiply both sides of Eq. (8) by sin (𝑘𝑥) and integrate over the interval [−𝜋,𝜋]: ∫𝑓(𝑥)𝑠𝑖𝑛𝑘𝑥𝑑𝑥   =∫ 𝑠𝑖𝑛𝑘𝑥𝑑𝑥   +∑𝑎∫𝑐𝑜𝑠𝑛𝑥𝑠𝑖𝑛𝑘𝑥𝑑𝑥   +𝑏∫𝑠𝑖𝑛𝑛𝑥𝑠𝑖𝑛𝑘𝑥𝑑𝑥      , 𝑐𝑜𝑠𝑛𝑥𝑠𝑖𝑛𝑘𝑥𝑑𝑥   =0 , 𝑠𝑖𝑛𝑘𝑥𝑑𝑥   =0, 𝑠𝑖𝑛𝑛𝑥𝑠𝑖𝑛𝑘𝑥𝑑𝑥   ={0,𝑘≠𝑛 𝜋,𝑘=𝑛. Finally, the Fourier coefficient 𝑏 is given by: 𝑏= ∫𝑓(𝑥)𝑠𝑖𝑛𝑛𝑥𝑑𝑥   . (11) The coefficients given in Eqs. (9), (10), and (11) are called the Fourier series coefficients of Eq. (8) [30]. 2.8 Sturm-Liouville Problem The Sturm-Liouville equation is a second-order real linear differential equation. All its coefficients are continuous on the closed and finite interval [𝑎,𝑏], and the derivative of 𝑝(𝑥) is also continuous. If a function 𝑦is continuously differentiable on the interval (𝑎,𝑏) and satisfies Eq. (12) at every point, then 𝑦 is a solution of this equation.  󰇡𝑝(𝑥) 󰇢+𝑞(𝑥)𝑦=𝜆𝑟(𝑥)𝑦,(𝑥)>0,𝑟(𝑥)≥0. (12) The functions 𝑝(𝑥),𝑞(𝑥),𝑟(𝑥)are continuous in the interval[𝑎,𝑏]. Furthermore, the following boundary conditions are given. 𝑎𝑦(𝑎)+𝑎𝑝(𝑎)𝑦󰆒(𝑎)=0, 𝑏𝑦(𝑏)+𝑏𝑝(𝑏)𝑦󰆒(𝑏)=0. (13) The conditions 𝑎 +𝑎 ≠0 and 𝑏 +𝑏 ≠0 are met. The problem defined by Eqs. (12) and (13) is called a Sturm-Liouville problem. The nonzero solutions of this problem are called eigenvalues (𝜆), and the functions corresponding to these eigenvalues are called eigenfunctions [31]. 2.9 Boundary Conditions Consider the boundary conditions: 𝑦(𝑎)=𝑦(𝑏),𝑦󰆒(𝑎)=𝑦󰆒(𝑏) If the Sturm-Liouville problem is defined with these conditions, it is called a periodic boundary condition Sturm-Liouville problem [31]. Let us solve the Sturm-Liouville problem under periodic boundary conditions and determine its eigenvalues and eigenfunctions. Consider the differential equation: M32-7 2nd Kocaeli Science Congress, November 19-21, 2025 𝑥󰆒󰆒+𝜆𝑥=0,𝑥(0)=𝑥(𝑙),𝑥󰆒(0)=𝑥󰆒(𝑙). (14) Problem (14) is a second-order differential equation with constant coefficients. To solve it, we first find the eigenvalues 𝜆 from the characteristic equation. We will examine the three possible cases for the eigenvalue 𝜆: when 𝜆<0, 𝜆=0, and 𝜆>0. i) 𝜆<0, 𝜆=−𝑚. 𝑘+𝜆=0,𝑘=𝑚, 𝑘=∓𝑚, 𝑋(𝑥)=𝐴𝑒+𝐵𝑒 (15) Eq. (15) is the eigenfunction. When boundary conditions are applied to find the coefficients in the eigenfunction, 𝐴=0,𝐵=0 is found. 𝑋=0 is the obvious solution. ii) 𝜆=0, 𝑋󰆒󰆒 =0, 𝑋󰆒=𝑎 𝑋=𝑎𝑥+𝑏. When the boundary conditions are applied to the eigenfunction, we find 𝑎=0, 𝑏=0. 𝑋=0 is the obvious solution. iii) 𝜆˃0,𝜆=𝑚, 𝑘+𝜆=0, 𝑘=−𝑚, 𝑘=∓𝑚𝑖 𝑋(𝑥)=𝐴cos𝑚𝑥+𝐵𝑠𝑖𝑛𝑚𝑥. The derivative of the eigenfunction is given by: X󰆒(𝑥)=−𝑚𝐴sin𝑚𝑥+𝑚𝐵𝑐𝑜𝑠𝑚𝑥. Periodic boundary conditions, as given in Eq. (14), are applied, we obtain 𝐴=𝐴𝑐𝑜𝑠𝑚𝜋+𝐵𝑠𝑖𝑛𝑚𝜋, 𝐵=−𝐴𝑠𝑖𝑛𝑚𝜋+𝐵𝑐𝑜𝑠𝑚𝜋, 𝐴(1−𝑐𝑜𝑠𝑚𝜋)−𝐵𝑠𝑖𝑛𝑚𝜋=0, 𝐴𝑠𝑖𝑛𝑚𝜋+𝐵(1−𝑐𝑜𝑠𝑚𝜋)=0. For the solution of the system to be linearly independent, the coefficient determinant 𝐴 ,𝐵must be zero. 󰇻1−𝑐𝑜𝑠𝑚𝜋 −𝑠𝑖𝑛𝑚𝜋 𝑠𝑖𝑛𝑚𝜋 1−𝑐𝑜𝑠𝑚𝜋󰇻=0. (1−𝑐𝑜𝑠𝑚𝜋)+ (𝑠𝑖𝑛𝑚𝜋)=0, 1−2𝑐𝑜𝑠𝑚𝜋+𝑐𝑜𝑠𝑚𝜋+𝑠𝑖𝑛𝑚𝜋=0, M32-8 2nd Kocaeli Science Congress, November 19-21, 2025 𝑐𝑜𝑠𝑚𝜋=1, 𝑚𝜋=2𝑘𝜋, 𝑚=2𝑘. The eigenvalues are: 𝜆=(2𝑘). The corresponding eigenfunctions are: 𝑋=𝐴𝑐𝑜𝑠2𝑘𝑥+𝐵𝑠𝑖𝑛2𝑘𝑥. That is, the two linearly independent functions corresponding to this eigenvalue are: 𝜆=𝑐𝑜𝑠2𝑘𝑥 , 𝜆= 𝑠𝑖𝑛2𝑘𝑥. 3. Fourier Method for Fractional Differential Parabolic Equation 3.1 Solution of Fractional Differential Homogeneous Parabolic Equation by Fourier Method 𝐷𝑈(𝑥,𝑡)=𝑈, 𝑥˃0, 0˂𝛼˂1 (16) 𝑈(0,𝑡)=𝑈(𝜋,𝑡),𝑈(0,𝑡)=𝑈(𝜋,𝑡), 0≤t≤T (17) 𝑈(𝑥,0)=𝜑(𝑥),0≤𝑥≤𝜋. (18) The solution to the problem in the equation can be written as follows using Strum-Liouville. The solution depends on the variables x and t. Therefore; 𝑈(𝑥,𝑡;𝛼)=𝑋(𝑥)𝑇(𝑡;𝛼), (19) U =X󰆒󰆒T(t,α), D U(x,t;α)=XD T(t;α), XD T(t;α)=X󰆒󰆒T(t;α), ((;)) (;) =" =−𝜆. When the boundary conditions given in (17) are applied to the last equation, the following SturmLiouville problem is obtained. X󰆒󰆒(x)+λX(x)=0, (20) X󰆒(0)=X󰆒(π), (21) ((;)) (;) =−𝜆. Let’s solve the Sturm-Liouville problem (20) that we obtained using the boundary conditions (21). Since problem (20) is a second-order differential equation with constant coefficients, we examine the different cases of λ from the characteristic equation to find its eigenvalues. M32-9 2nd Kocaeli Science Congress, November 19-21, 2025 i) 𝜆˂0, 𝑘+𝜆=0,𝑘=𝑚, 𝑘=∓𝑚 𝑋(𝑥)=𝐴𝑒+𝐵𝑒 . (22) Eq. (22) is the eigenfunction. When boundary conditions are applied to find the coefficients in the eigenfunction 𝐴=0,𝐵=0, 𝑋=0 is the obvious solution. ii) 𝜆=0, 𝑋󰆒󰆒 =0, 𝑋󰆒=𝑎, 𝑋=𝑎𝑥+𝑏. (23) When the boundary conditions are applied to the eigenfunction, we find 𝑎=0, 𝑏=0 is the obvious solution. iii) 𝜆˃0, 𝜆=𝑚 .𝑘+𝜆=0, 𝑘=−𝑚, 𝑘=∓𝑚𝑖. 𝑋(𝑥)=𝐴cos𝑚𝑥+𝐵𝑠𝑖𝑛𝑚𝑥. The periodic boundary conditions given in (17) are applied. The eigenvalue is 𝜆=(2𝑘). There are two linearly independent functions corresponding to this eigenvalue: 𝑋=𝐴𝑐𝑜𝑠2𝑘𝑥+𝐵𝑠𝑖𝑛2𝑘𝑥 which are the eigenfunctions. Let’s express the eigenvalue λ in the function 𝑇: (;) (;)=−𝜆,0<𝛼<1, 𝜆>0 [29]. 𝑇(𝑡;𝛼)=𝐶𝐸,(−(2𝑘).𝑡). Now, combining 𝑋 and 𝑇, we can write the eigenfunctions in (19). 𝑈(𝑥,𝑡;𝛼)=(𝐴𝑐𝑜𝑠2𝑘𝑥+𝐵𝑠𝑖𝑛2𝑘𝑥).𝐶.𝐸,(−(2𝑘).𝑡) By superposition the sum of the solutions is also the solution. 𝑈(𝑥,𝑡)=() +∑[𝑈𝑐𝑜𝑠2𝑘𝑥+𝑈𝑠𝑖𝑛2𝑘𝑥]   .𝐸,(−(2𝑘).𝑡) (24) By substituting the initial condition given in (18) into expression (24), we obtain 𝑈(𝑥,0)=𝜑(𝑥)= +∑𝑈𝑐𝑜𝑠2𝑘𝑥+𝑈𝑠𝑖𝑛2𝑘𝑥   , 𝑈= ∫𝜑(𝑥)𝑑𝑥  , M32-16 2nd Kocaeli Science Congress, November 19-21, 2025 [2] Abbas, S., Benchohra, M., Lazreg, J. E., & N’Guérékata, G. (2017). "Hilfer and Hadamard functional random fractional differential inclusions." Cubo, 19, 17–38. [3] Abbas, S., Benchohra, M., Lazreg, J. E., & Zhou, Y. (2017). 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