On the Approximate Solution of the Multi-Term Time-Fractional Diffusion Problem through Hermite Polynomials

On the Approximate Solution of the Multi-Term Time-Fractional Diffusion Problem through Hermite Polynomials Bekir Aktan1, Mine Aylin Bayrak1, Ali Demir1 1Department of Mathematics, Kocaeli University, Izmit/Kocaeli, 41001, Turkey Corresponding author: beki[email protected] ORCID IDs: First Author: 0009-0003-5134-6650 Second Author: 0000-0001-7716-3455 Third Author: 0000-0003-3425-1812 DOI : 10.5281/zenodo.18012104 Abstract TThis research presents a novel approach for constructing approximate solutions to multi-term time fractional diffusion problems through Hermite polynomials, the Residual Power Series Method (RPSM), and collocation techniques. Initially, the multi-term time fractional diffusion equation is transformed into a single-term equation using the Riemann integral, simplifying the original problem. Next, Hermite polynomials are employed to express the solution as a series, which, through the application of collocation points, reduces the problem to a system of fractional differential and algebraic equations. The RPSM is then applied to determine the unknown coefficient functions in the series representation. By substituting these coefficients back into the series, an approximate solution is constructed. To illustrate the applicability and accuracy of the proposed method, three numerical examples are provided, demonstrating its effectiveness in solving complex fractional diffusion problems. Keywords: Multi-term time-fractional diffusion equation, Hermite Collocation method, Residual power series method. 1 Introduction Fractional diffusion equations with various boundary conditions are widely used to analyze anomalous diffusion processes, such as super-diffusion and sub-diffusion. In particular, timefractional diffusion equations have gained significant attention across multiple scientific disciplines, including mathematics, physics, biochemistry, finance, and medicine [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. In the mathematical modeling of diffusion processes, fractional differential equations play a substantial role to analyze the processes in detail. These equations have a wide range of applications in various fields of science and engineering, as the order of the fractional derivative and Riemann integration could be any real number, such as m− 1 ≤α, β ≤m : m∈N . As a result, fractional-order derivatives have been extensively studied and developed to interpret real-life phenomena such as biological systems, thermal conductivity, and chemical diffusion more accurately [ 9 , 10 , 11 ]. The M9-1 KOSC-2025 Proceedings main concern of the current research is to establish approximate solutions to multi-term time fractional diffusion problem through a proposed method, a combination of Hermite polynomials, RPSM [12,13,14,15,16] and collocation method. In this study, the following the multi-term time-fractional diffusion problem is considered: Dα tu(x, t)+Dβ tu(x, t) = uxx(x, t)+F(x, t, u), x ∈(0, ℓ), t ∈(0, T)(1) subject to the initial and boundary conditions u(x, 0) = ϕ(x), x ∈[0, ℓ](2) u(0, t) = µ1(t), t ∈[0, T](3) u(ℓ, t) = µ2(t), t ∈[0, T](4) The organization of the rest of the paper is given as follows. In Section 2, related preliminaries of the problem are presented. Hermite polynomials and their properties are given in Section 3. In Section 4, the algorithms of the developed methods are presented. Some examples are demonstrated to verify the advantages of the proposed methods in Section 5. Finally, a brief conclusion is presented to summarize the outcomes in Section 6. 2 Preliminaries In this section, fundamental definitions and notions are presented. Definition 2.1. The Riemann-Liouville integral for αis [17,18,19,20]: Jαf(x) =    1 Γ(α)Rx 0(x−τ)α−1f(τ)dτ, α > 0 f(x), α = 0.(5) Definition 2.2. The αth order fractional derivative in Caputo sense is given by [17,18,19]: Dαf(x) = Jm−α(Dmf(x)) =    1 Γ(m−α)Rx 0(x−τ)m−α−1f(m)(τ)dτ, m −1< α < m, m ∈N, d(m) dx(m)f(x), α =m. (6) which has the following properties: •DαC= 0 , (Cis a constant) •DαJαf(x) = f(x), M9-2 2nd Kocaeli Science Congress, November 19-21, 2025 •JαDαf(x) = f(x)− m−1 P i=0 f(i)(0) xi Γ(i+1) , • Jβ(Dαf(x)) =    Dα−βf(x)−tβ−α Γ(β+1−α)f(0), α < β Jβ+1−αf′(x), otherwise. (7) Definition 2.3. A power series expansion of the form ∞ X n=0 cn(t−t0)nα =c0+c1(t−t0)α+c2(t−t0)2α+..., (8) 0≤m−1< α ≤m, t ≥t0 is called fractional power series about t=t0[19]. Theorem 2.1. Suppose that fhas a fractional power series representation at t0of the form f(t) = ∞ X n=0 cn(t−t0)nα,0⩽m−1< α ⩽1, t0⩽t < t0+R. (9) If f ( t ) ∈C [ t0, t0 + R )and Dnα t0f ( t ) ∈C ( t0, t0 + R )for n = 0 , 1 , 2 , ... , then the coefficients cn in Eq. (10) will take the form of cn=Dnα t0f(t0) Γ(1 + nα),(10) where Dnα t0=Dα t0Dα t0...Dα t0(ntimes) [19]. 3 Hermite polynomials (HP) Hermite polynomials Hn, n ∈N are the solutions of the following differential equations [ 21 , 22 , 23]: Hn+1(x)+H′ n(x)−2xHn(x)=0, H′ n(x)=2nHn−1(x)(11) which can be represented in the series form as follows: Hn(x) = Γ(n+ 1) [n/2] X k=0 (−1)k(2x)n−2k 2kΓ(k+ 1)Γ(n−2k+ 1), n ∈N0(12) Moreover, the recursion formula for Hermite polynomials are presented as: H0(x)≡1, H1(x)=2x, Hn+1(x)=2xHn(x)−2nHn−1(x), n ≥1,(13) 2nd Kocaeli Science Congress, November 19-21, 2025 M9-3 KOSC-2025 Proceedings A significant property of Hermite polynomials is that they form an orthogonal system in L2 ( R, w ): Z∞ −∞ Hn(x)Hm(x)ω(x)dx =√2π2nΓ(n+ 1)δnm (14) where ω ( x ) = exp ( −x2 )and δnm is the Kronecker delta. Hence one can obtain a system of orthonormal Hermite polynomials ψn(x), n = 1,2, ... Z∞ −∞ ψn(x)ψm(x)ω(x)dx =δnm.(15) Since Hermite polynomials are smooth functions, the solution can be interpolated and the problem transformed into an algebraic system of equations which is easily handled by computational software such as MATLAB and many others. 4 Algorithmic Framework and Implementation of the Proposed Method In order to reduce multi-term time fractional diffusion problem into a single term time fractional diffusion problem, Riemann integral of order β is applied to the fractional differential equation : Dα tu(x, t)+Dβ tu(x, t) = uxx(x, t)+F(x, t, u),(16) which yields the following Dα−β tu(x, t) = Jβ tuxx(x, t)+u(x, 0)1 + tβ−α Γ(β+1−α)−u(x, t)+Jβ tF(x, t, u).(17) At this stage, the steps of the algorithm for the method can be listed as: Step 1. We assume that u ( x, t )is sufficiently smooth functions to be approximated by truncating the series in terms of Hermite polynomials as follows: u(x, t)≈ M−1 X n=0 cn(t)Hn(x),(18) Step 2.Plugging the Mth degree approximation of Eq.(18) into the Eq.(17) yields the following: M−1 X n=0 Dα−β tcn(t)Hn(x) = M−1 X n=0 Jβ tcn(t)H′′ n(x) + M−1 X n=0 cn(0)Hn(x)1 + tβ−α Γ(β+1−α) − M−1 X n=0 cn(t)Hn(x)+Jβ tFx, t, M−1 X n=0 cn(t)Hn(x),(19) M9-4 2nd Kocaeli Science Congress, November 19-21, 2025 Step 3.Employing the collocation points at xr = 2j−1 M, j = 1 , 2 , ..., M − 1leads to the following system of fractional ordinary differential equations: M−1 X n=0 Dα−β tcn(t)Hn(xr) = M−1 X n=0 Jβ tcn(t)H′′ n(xr) + M−1 X n=0 cn(0)Hn(xr)1 + tβ−α Γ(β+1−α) − M−1 X n=0 cn(t)Hn(xr)+Jβ tFxr, t, M−1 X n=0 cn(t)Hn(xr),(20) Step 4. Initial and boundary conditions Eq.(2)-(4) in the reduced problem give rise to the following system of (M+ 1) algebraic equations : M−1 X n=0 cn(0)Hn(xr)=ϕ(xr)(21) M−1 X n=0 cn(t)Hn(0) = µ1(t)(22) M−1 X n=0 cn(t)Hn(1) = µ2(t)(23) which allows us to determine the initial values of coefficients for cn ( t ) , n = 0 , 1 , ..., M − 1in the series representation of the solution . Step 5. The utilization of RPSM leads to establish the unknown functions cn ( t ) , n = 0 , 1 , ..., M − 1 . Moreover, placing the obtained coefficient functions cn ( t ) , n = 0 , 1 , ..., M − 1back into the series leads to approximate solution uM(x, t). 5 Illustrative Examples This section provides three examples to confirm the accuracy and effectiveness of the proposed method as well as its implementation. Example 1. Consider the multi-term time-fractional diffusion equation Dα tu(x, t)+Dβ tu(x, t) = uxx(x, t)+F(x, t),0< β < α < 1, x ∈(0,1), t ∈(0,1) (24) subject to the initial and boundary conditions u(x, 0) = 0, x ∈[0,1] (25) u(0, t) = u(1, t) = 0, t ∈[0,1] (26) where F ( x, t ) = Γ(1 + 2 α− 2 β ) tα−2β Γ(1+α−2β) + t2α−3β Γ(1+2α−3β)x (1 −x ) + 2 t2α−2β . The analytical solution of the considered problem is obtained as u(x, t) = x(1 −x)t2α−2β. In Fig. 1, the graphs of exact and approximate solutions of u ( x, t )at t = 0 . 8for various values of α, β and m = 3 are presented. In Fig 2., the 3D graphs of absolute errors for α = 0 . 9 , β = 0 . 4 and m = 3 are given. In Table 1., the exact solution and absolute errors for various values of 2nd Kocaeli Science Congress, November 19-21, 2025 M9-5 KOSC-2025 Proceedings 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 Figure 1: The graphs of exact and approximate solutions of u ( x, t )for various values of α, β of Ex.1 . 0 1 0.2 0.9 0.4 0.8 0.6 1 0.7 0.9 10-16 0.8 0.6 0.8 1 0.7 0.5 0.6 1.2 0.4 0.5 1.4 0.3 0.4 0.3 0.2 0.2 0.1 0.1 00 Figure 2: The 3D graphs of absolute errors for α= 0.9,β= 0.4of Ex.1. x, t with m = 3 at α = 0 . 9, β = 0 . 4are presented. Example 2. Consider the multi-term time-fractional diffusion equation Dα tu(x, t)+Dβ tu(x, t) = uxx(x, t)+u(x, t)+F(x, t),0< β < α < 1, x ∈(0,1), t ∈(0,1) (27) subject to the initial and boundary conditions u(x, 0) = 0, x ∈[0,1] (28) u(0, t) = t2α−2β−tα−β, t ∈[0,1] (29) u(1, t) = 4(t2α−2β−tα−β), t ∈[0,1] (30) Table 1: The values of absolute errors for m= 3 and α= 0.9,β= 0.4of Ex. 1. x/t 0.1 0.3 0.5 0.7 0.9 0.1 0 6.9389e-18 2.0817e-17 0 0 0.3 0 0 2.7756e-17 2.7756e-17 2.7756e-17 0.5 6.9389e-18 1.3878e-17 0 5.5511e-17 2.7756e-17 0.7 1.3878e-17 2.7756e-17 2.7756e-17 2.7756e-17 2.7756e-17 0.9 1.7347e-18 3.4694e-18 6.9389e-18 1.3878e-17 8.3267e-17 M9-6 2nd Kocaeli Science Congress, November 19-21, 2025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 Figure 3: The graphs of exact and approximate solutions of u(x, t)of Ex.1 . 0 1 0.5 0.9 1 0.8 1.5 1 0.7 0.9 10-16 2 0.6 0.8 2.5 0.7 0.5 0.6 3 0.4 0.5 3.5 0.3 0.4 0.3 0.2 0.2 0.1 0.1 00 Figure 4: The 3D graphs of absolute errors for α= 0.9,β= 0.4of Ex.1. where F ( x, t ) = (3 x3 +1)  Γ(1+2 α− 2 β ) tα−2β Γ(1+α−2β)− Γ(1+ α−β ) t−β Γ(1−β) +Γ(1+2 α− 2 β ) t2α−3β Γ(1+2α−3β)− Γ(1+ α−β ) tα−2β Γ(1+α−2β)− (3 x3 +18 x +1)( t2α−2β−tα−β ). The analytical solution of the considered problem is obtained u(x, t) = (3x3+ 1)(t2α−2β−tα−β). In Fig. 3, the graphs of exact and approximate solutions of u ( x, t )at t = 0 . 8for various values of α, β and m = 4 are presented. In Fig 4., the 3D graphs of absolute errors for α = 0 . 9 , β = 0 . 4 and m = 4 are given. In Table 2., the exact solution and absolute errors for various values of x, t with m= 4 at α= 0.9,β= 0.4are presented. Example 3. Consider the multi-term time-fractional diffusion equation Dα tu(x, t)+Dβ tu(x, t) = uxx(x, t)+xux(x, t)+F(x, t),0< β < α < 1, x ∈(0,1), t ∈(0,1) (31) subject to the initial and boundary conditions u(x, 0) = cos(x), x ∈[0,1] (32) Table 2: The values of absolute errors for m= 4 at t= 0.8and α= 0.9,β= 0.4of Ex. 1. x/t 0.1 0.3 0.5 0.7 0.1 5.5511e-17 0 2.7756e-17 2.7756e-17 0.3 2.7756e-17 0 8.3267e-17 8.3267e-17 0.5 5.5511e-17 5.5511e-17 0 1.9429e-16 0.7 1.6653e-16 1.1102e-16 5.5511e-17 2.2204e-16 0.9 1.1102e-16 1.1102e-16 0 2.2204e-16 2nd Kocaeli Science Congress, November 19-21, 2025 M9-7 KOSC-2025 Proceedings 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Figure 5: The graphs of exact and approximate solutions of u(x, t)of Ex.1 . 0 1 1 0.9 2 0.8 1 3 0.7 0.9 10-9 0.6 0.8 4 0.7 0.5 5 0.6 0.4 0.5 6 0.3 0.4 0.3 0.2 0.2 0.1 0.1 00 Figure 6: The 3D graphs of absolute errors for α= 0.9,β= 0.4of Ex.1. u(0, t) = tα−β+ 1, t ∈[0,1] (33) u(1, t) = (tα−β+ 1)cos(1), t ∈[0,1] (34) where F ( x, t ) = cos ( x )  Γ(1 + α−β )( t−β Γ(1−β) + tα−2β Γ(1+α−2β) ) + ( t−α Γ(1−α) + t−β Γ(1−β) )  + ( cos ( x ) + xsin ( x ))( tα−β + 1). The analytical solution of the considered problem is obtained u ( x, t ) = cos(x)(tα−β+ 1) . In Fig. 5, the graphs of exact and approximate solutions of u ( x, t )at t = 0 . 8for various values of α, β and m = 8 are presented. In Fig 6., the 3D graphs of absolute errors for α = 0 . 9 , β = 0 . 4 and m = 8 are given. In Table 3., the exact solution and absolute errors for various values of x, t with m= 8 at α= 0.9,β= 0.4are presented. Table 3: The values of absolute errors for m= 7 at t= 0.8and α= 0.9,β= 0.4of Ex. 1. x/t 0 0.2 0.4 0.6 0.8 1 0 1.8974e-19 4.9031e-17 1.5244e-16 5.8382e-17 9.8251e-17 4.0146e-17 0.2 2.4769e-10 3.5425e-10 3.9839e-10 4.3226e-10 4.6081e-10 4.8596e-10 0.4 5.3825e-10 7.8255e-10 8.8374e-10 9.6138e-10 1.0268e-09 1.0845e-09 0.6 5.6719e-10 8.2117e-10 9.2637e-10 1.0071e-09 1.0752e-09 1.1351e-09 0.8 1.7391e-10 2.5385e-10 2.8697e-10 3.1237e-10 3.3379e-10 3.5267e-10 1 1.0391e-16 1.4621e-16 8.3338e-18 1.9409e-17 1.8851e-16 1.4425e-16 M9-8 2nd Kocaeli Science Congress, November 19-21, 2025 6 Conclusion The main contribution of this work is the development of a novel approach for solving multi-term time fractional diffusion problems using the Riemann integral, Hermite polynomials, the Residual Power Series Method (RPSM), and collocation techniques. This method enables the construction of approximate solutions for a wide range of multi-term fractional problems. The core idea is to reduce the original multi-term fractional problem to a single-term fractional problem via the Riemann integral. Subsequently, Hermite polynomials, in conjunction with collocation points and RPSM, are employed to derive an approximate solution. Furthermore, Hermite polynomials can be replaced with other special polynomial families to potentially improve the accuracy of the solution, making the proposed method highly adaptable and versatile for tackling various multi-term fractional problems. 7 Acknowledgments The authors thank for valuable discussions and suggestions. 2nd Kocaeli Science Congress, November 19-21, 2025 M9-9