High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

• Published in: Journal of scientific computing Vol. 76; no. 2; pp. 867 - 887
• Main Authors: Chen, Minghua, Deng, Weihua
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2018; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Fourier transforms; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Partial differential equations; Power law; Queuing theory; Real numbers; Theoretical; Truncation errors; First passage time; 35R11; 65L20; Time-tempered fractional Feynman–Kac equation; Stability and convergence; Tempered fractional substantial derivative
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-018-0640-y

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in Wu et al. (Phys Rev E 93:032151, 2016 ), being called the time-tempered fractional Feynman–Kac equation named after Richard Feynman and Mark Kac who first considered the model describing the functional distribution of normal motion. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as S D t γ , λ ~ G ( x , p , t ) = D t γ , λ ~ G ( x , p , t ) - λ γ G ( x , p , t ) with λ ~ = λ + p U ( x ) , p = ρ + J η , J = - 1 , where D t γ , λ ~ G ( x , p , t ) = 1 Γ ( 1 - γ ) ∂ ∂ t + λ ~ ∫ 0 t t - z - γ e - λ ~ · ( t - z ) G ( x , p , z ) d z , and λ ≥ 0 , 0 < γ < 1 , ρ > 0 , and η is a real number. The designed schemes are unconditionally stable and have the global truncation error O ( τ 2 + h 2 ) , being theoretically proved and numerically verified in complex space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the ‘physical’ equation (without artificial source term).