Determination of Initial Value Function in Time-Fractional Diffusion Equation with Robin Boundary Condition through Hermite Polynomials

Determination of Initial Value Function in Time-Fractional Diffusion Equation with Robin Boundary Condition through Hermite Polynomials Bekir Aktan1,*, Mine Aylin Bayrak1, Ali Demir1 1Department of Mathematics, Kocaeli University, Izmit/Kocaeli, 41001, Turkey Corresponding author: b[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.18012047 Abstract This study contributes to establishment of initial value function in a time fractional diffusion equation with robin boundary condition by means of a new approach involving Hermite polynomials, Residual power series method (RPSM) and collocation methods together. First of all, robin boundary condition is made homogeneous by a suitable transformation which reduce the problem to a simpler one. The solution of reduced problem is written in series form in terms of Hermite polynomials which yields a system of fractional differential and algebraic equations. Secondly, RPSM is employed to accomplish approximate solution to the reduced problem which includes some unknowns. At this stage, collocation points and additional data are used together to determine the values of unknown initial value function at collocation points which are the unknowns in the approximate solution. Finally, the initial value of approximate solution yields the unknown initial value function. Two examples are presented to confirm the accuracy and effectiveness of the proposed method. Keywords: Inverse problem, Time Fractional diffusion equation, The robin boundary condition, Hermite Collocation method, Residual power series method. 1 Introduction Fractional diffusion equations with various boundary conditions have been used to analyze the diffusion processes such as super diffusion and sub diffusion processes. In this respect time fractional diffusion equations have gained popularity in various areas of science such as mathematics [ 1 , 2 ], physics [ 3 ], biochemistry [ 4 ], finance [ 5 ] and medicine [ 6 ]. In the mathematical modelling of diffusion processes the inverse problems play a substantial role in the determination of unknown functions such as initial value function or source function. Therefore, the inverse problems have a broad range of applications in biological processes, thermal conductivity, chemical M8-1 KOSC-2025 Proceedings diffusion [ 7 , 8 , 9 ], and etc. As a result, inverse problems become an indispensable part for mathematical modelling of a number of processes in various disciplines. The additional condition is provided to deal with inverse problems. The main concern of the current study is to determine the initial value function in a time fractional diffusion equation with robin boundary condition by developing a new approach which exploits Hermite polynomials together with RPSM [ 10 , 11 , 12 , 13 , 14 ] and collocation method. In this study, the following inverse initial value problem for the time-fractional diffusion equation with robin boundary condition is considered: Dα tu(x, t) = uxx(x, t) + 1 xux+F(x, t), x ∈(0, ℓ), t ∈(0, T)(1) subject to the initial condition u(x, 0) = ϕ(x), x ∈[0, ℓ](2) where Dα turepresents the Caputo fractional derivative of order α. The Robin boundary condition at x=ℓ u(ℓ, t)+ηux(ℓ, t) = µ(t), t ∈[0, T ](3) where λ > 0denotes the heat transfer coefficient, models surface convection, since the convective heat conduction between the body and its surroundings [ 15 ] is modelled by robin boundary condition. Generally, the right hand side function µ ( t )equals to λu∞ where u∞ represents the ambient temperature. Problem (1)-(3) is the classic forward problem when the source term F ( x, t ), initial value ϕ ( x ) and function µ ( t )are given. The inverse initial value problem we study is to recover the initial value ϕ(x)from the additional data u(x, T)=E(x), x ∈[0, ℓ](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 [16,17,18,19]: Jαf(x) =    1 Γ(α)Rx 0(x−τ)α−1f(τ)dτ, α > 0 f(x), α = 0.(5) M8-2 2nd Kocaeli Science Congress, November 19-21, 2025 Definition 2.2. The αth order fractional derivative in Caputo sense is given by [ 16 , 17 , 18 , 19 ]: Dαf(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) 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+..., (7) 0≤m−1< α ≤m, t ≥t0 is called fractional power series about t=t0[18]. 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. (8) 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),(9) where Dnα t0=Dα t0Dα t0...Dα t0(ntimes) [18]. 3 Hermite polynomials (HP) Hermite polynomials Hn, n ∈N are the solutions of the following differential equations [ 20 , 21 , 22,23]: Hn+1(x)+H′ n(x)−2xHn(x)=0, H′ n(x)=2nHn−1(x)(10) 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(11) 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,(12) 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 (13) 2nd Kocaeli Science Congress, November 19-21, 2025 M8-3 KOSC-2025 Proceedings 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.(14) 4 Implementation and algorithm of the proposed method In order to make inhomogeneous robin boundary condition into homogeneous robin boundary condition, we set: u(x, t) = u1(x, t)+u2(x, t)(15) where u1(x, t)is the solution of the following problem: Dα tu1(x, t) = ∂2u1 ∂x2(x, t) + 1 x ∂u1 ∂x (x, t) + ˆ F(x, t)(16) u1(x, 0) = ˆ ϕ(x), x ∈[0, ℓ](17) u1,x(ℓ, t)+ηu1(ℓ, t) = 0, t ∈[0, T ](18) u1(x, T) = E(x), t ∈[0, T].(19) Moreover, u2(x, t)is given in the form u2(x, t) = µ(t) ηℓ + 1x(20) satisfying the following conditions: u2(x, 0) = ϕ(x)−ˆ ϕ(x), x ∈[0, ℓ](21) u2,x(ℓ, t)+ηu2(ℓ, t) = µ(t), t ∈[0, T ](22) where ˆ F(x, t)=F(x, t)−Dα tu2(x, t) + ∂2u2 ∂x2(x, t) + 1 x ∂u2 ∂x (x, t).(23) At this stage, the algorithm of the method takes the following steps: Step 1. We assume that u1 ( x, t )is sufficiently smooth functions to be approximated by truncating the series as follows: u1(x, t)≈ M−1 X n=0 cn(t)Hn(x),(24) M8-4 2nd Kocaeli Science Congress, November 19-21, 2025 Step 2.Plugging the Mth degree approximation of Eq.(16) into the Eq.(24) in the first problem yields the following: M−1 X n=0 Dα tcn(t)Hn(x) = M−1 X n=0 cn(t)H′′ n(x) + 1 x M−1 X n=0 cn(t)H′ n(x) + ˆ F(x, t),(25) 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 tcn(t)Hn(xr) = M−1 X n=0 cn(t)H′′ n(xr) + 1 xr M−1 X n=0 cn(t)H′ n(xr) + ˆ F(xr, t),(26) Step 4. Initial and robin boundary conditions Eq.(17)-(18) 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)(27) M−1 X n=0 cn(t)H′ n(ℓ)+η M−1 X n=0 cn(t)Hn(ℓ) = 0,(28) 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, ϕ ( xr )is obtained by using truncating the series of the solution uM ( x, t )in additional data as follows: M−1 X n=0 cn(T)Hn(x)dx =E(x)(29) Step 6. The initial value of approximate solution provides initial value function. 5 Illustrative Examples In this section, some examples are presented to verify the accuracy and effectiveness of the proposed approach. Example 1. Assuming that the functions η = 1, ℓ = 2, µ ( t ) = t2α + 1, F ( x, t ) = ( x2−x− 4)Γ(1 + 2α)tα Γ(1+α)−2(t2α+ 1),E(x) = 2(x2−x−4) are smooth functions. The analytical solution and initial value function are obtained as u ( x, t ) = ( x2−x− 4)( t2α + 1) and ϕ(x)=(x2−x−4), respectively . In Fig. 1, the graphs of exact and approximate solutions of ϕ ( x )are presented. In Fig 2-3., the 3D graphs of approximate solutions for u1 ( x, t )and u2 ( x, t )for α = 0 . 8, respectively. In Table 1-2., the exact solution and absolute errors for M= 3 at t= 0.8for various values of α. Example 2. Assuming that the functions η= 1,F(x, t)=xΓ(1 + 2α)tα Γ(1+α)), 2nd Kocaeli Science Congress, November 19-21, 2025 M8-5 KOSC-2025 Proceedings 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -4.5 -4 -3.5 -3 -2.5 -2 Exact Approximate Figure 1: The graphs of exact and approximate solutions of ϕ(x)of Ex.1 . -9 1 -8 0.9 -7 0.8 -6 2 0.7 1.8 -5 0.6 1.6 -4 1.4 0.5 1.2 -3 0.4 1 -2 0.3 0.8 0.6 0.2 0.4 0.1 0.2 00 0 1 0.2 0.9 0.4 0.8 0.6 2 0.7 1.8 0.8 0.6 1.6 1 1.4 0.5 1.2 1.2 0.4 1 1.4 0.3 0.8 0.6 0.2 0.4 0.1 0.2 00 Figure 2: The 3D graphs of u1(x, t)and u2(x, t)for α= 0.8of Ex.1, respectively . -9 1 -8 0.9 -7 0.8 -6 2 0.7 1.8 -5 0.6 1.6 -4 1.4 0.5 1.2 -3 0.4 1 -2 0.3 0.8 0.6 0.2 0.4 0.1 0.2 00 -9 1 -8 0.9 -7 0.8 -6 2 0.7 1.8 -5 0.6 1.6 -4 1.4 0.5 1.2 -3 0.4 1 -2 0.3 0.8 0.6 0.2 0.4 0.1 0.2 00 Figure 3: The 3D graphs of exact and approximate solutions of u(x, t)for α= 0.8of Ex.1. Table 1: The values of exact solution and absolute errors for M = 3 at t = 0 . 8for various values of αof Ex. 1. x Exact α = 1 α= 0.9α= 0.8α= 0.6α= 0.4α= 0.2 0 -6.5600 4.4409e-15 6.3061e-14 1.2967e-13 1.3323e-14 1.2257e-13 2.6645e-14 0.2 -6.8224 1.5099e-14 7.8160e-14 1.8208e-13 6.2172e-15 1.6964e-13 1.9540e-14 0.4 -6.9536 3.1086e-14 9.0594e-14 2.2382e-13 0 2.0961e-13 1.4211e-14 0.6 -6.9536 4.3521e-14 9.8588e-14 2.5668e-13 5.3291e-15 2.3981e-13 8.8818e-15 0.8 -6.8224 5.4179e-14 1.0481e-13 2.7889e-13 9.7700e-15 2.6024e-13 3.5527e-15 1 -6.5600 6.1284e-14 1.0747e-13 2.9310e-13 1.3323e-14 2.7356e-13 0 1.2 -6.1664 6.5725e-14 1.0569e-13 2.9576e-13 1.5987e-14 2.7711e-13 1.7764e-15 1.4 -5.6416 6.6613e-14 1.0303e-13 2.8866e-13 1.6875e-14 2.7089e-13 3.5527e-15 1.6 -4.9856 6.4837e-14 9.4147e-14 2.7356e-13 1.7764e-14 2.5580e-13 6.2172e-15 1.8 -4.1984 6.0396e-14 8.3489e-14 2.4780e-13 1.7764e-14 2.3181e-13 7.1054e-15 2 -3.2800 5.1514e-14 7.0166e-14 2.1227e-13 1.6875e-14 1.9940e-13 6.2172e-15 M8-6 2nd Kocaeli Science Congress, November 19-21, 2025 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Figure 4: The graphs of exact and approximate solutions of ϕ(x)of Ex.2 . 0.5 1 0.9 0.8 0.7 2 0.6 1 1.8 0.5 1.6 1.4 0.4 1.2 0.3 1 0.8 0.2 0.6 0.4 0.1 1.5 0.2 00 2 0 1 0.5 0.9 0.8 1 2 0.7 1.8 1.5 0.6 1.6 1.4 0.5 2 1.2 0.4 1 2.5 0.3 0.8 0.6 0.2 0.4 0.1 0.2 00 Figure 5: The 3D graphs of u1(x, t)and u2(x, t)for α= 0.8of Ex.2, respectively . µ(t) = (3+2t2α,0≤x≤1 3t2α,1≤x≤2,E(x) = (1+2x, 0≤x≤1 3,1≤x≤2are non-smooth functions. The analytical solution and initial value function are obtained as u(x, t) = (1+x(t2α+ 1),0≤x≤1 3+x(t2α−1),1≤x≤2and ϕ(x) = (1+x, 0≤x≤1 3−x, 1≤x≤2, respectively. In Fig. 4., the graphs of exact and approximate solutions of ϕ ( x )are presented. In Fig 5-6., the 3D graphs of approximate solutions for u1 ( x, t )and u2 ( x, t )for α = 0 . 8, respectively. In Table 3-4., the exact solution and absolute errors for M= 2 at t= 0.8for various values of α. 1 1 1.5 0.8 2 0.6 2.5 0.4 3 2 1.8 0.2 1.6 1.4 1.2 1 0.8 0.6 00.4 0.2 0 1 1.2 1 1.4 1.6 1.8 2 0.8 2.2 2.4 2.6 2.8 3 0.6 0.4 2 1.8 1.6 0.2 1.4 1.2 1 0.8 0.6 00.4 0.2 0 Figure 6: The 3D graphs of exact and approximate solutions of u(x, t)for α= 0.8of Ex.2. 2nd Kocaeli Science Congress, November 19-21, 2025 M8-7 KOSC-2025 Proceedings Table 2: The values of exact solution and absolute errors values for M = 2 at t = 0 . 8for various values of αof Ex. 2. x Exact α = 1 α= 0.9α= 0.8α= 0.6α= 0.4α= 0.2 0 1.000 0 0 0 0 0 0 0.2 1.328 0 0 0 0 0 0 0.4 1.656 0 0 0 2.2204e-16 0 0 0.6 1.984 0 0 0 2.2204e-16 0 0 0.8 2.312 0 0 0 4.4409e-16 0 0 1 2.640 0 0 0 0 0 0 1.2 2.568 0 0 0 0 0 0 1.4 2.496 0 0 0 4.4409e-16 0 0 1.6 2.424 0 0 4.4409e-16 0 0 0 1.8 2.352 0 0 0 0 0 0 2 2.280 0 0 0 0 0 0 6 Conclusion The contribution of this study is to provide a new approach to recover initial value function in a time fractional diffusion equation with robin boundary condition. Hermite polynomials, RPSM and collocation methods are main parts of this approach. By some transformation, we make robin boundary conditions homogeneous and obtain a simpler problem. The solution of the simpler problem is formed in series form in terms of Hermite polynomials which leads to a system of fractional differential and algebraic equations. RPSM is utilized to obtain approximate solution to the simpler problem including some unknowns because of unknown initial function. At this stage, employment of collocation method and additional condition yield the value of initial value function at collocation points which are the unknown in the approximate solution. Finally, the initial value of approximate solution reveal the unknown the initial value function. The outcomes are also verified by provided examples. 7 Acknowledgments The authors thank for valuable discussions and suggestions. 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