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In this manuscript, we study and examine the time‐fractional modified anomalous sub‐diffusion model of distributed‐order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second‐order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full‐discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the δ$$ \delta $$‐Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table. [ABSTRACT FROM AUTHOR]

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and $ \mathcal {Z} $ Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are $ \mathcal {O}(\tau ^{p}+h^2) $ O (τ p + h 2) for $ 1 \leq p \leq ~4 $ 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes (B-S) equation to price American options is studied which is governed by a partial differential problem. A nonlinear partial differential equation (PDE) is resulted by applying a penalty approach for this problem. To numerically solve this PDE, an equipped finite difference method with variable step size (VSS) in the time direction is designed and implemented, coupled with a boundary value solver based on a multiple shooting method (MSM) in the spatial direction. It is completely proven that the proposed approximate option prices satisfy the early exercise constraint. Moreover, the convergence of the proposed method is analysed. The extracted numerical results indicate the punctuality and effectiveness of the present scheme comparing the previous works. [ABSTRACT FROM AUTHOR]

In this paper, the explicit forms of the fractional second linear multistep methods (FSLMMs) are introduced for solving fractional differential equations (FDEs) of the fractional-order in (1 , 2). These explicit FSLMMs are constructed based on fractional backward difference formulas 1, 2, and 3 (FBDF1, FBDF2, and FBDF3) with the first, second, third, and fourth orders of convergence. Also, the monotonicity of these FBDFs is considered when the order of fractional derivatives lies into (1 , 2). The order of consistency, linear stability, and the order of convergence of these explicit methods are analysed. Moreover, the stability regions of the proposed methods are completely studied in the stability topic. Finally, four experimental examples are presented to confirm the proposed theories. [ABSTRACT FROM AUTHOR]

In this paper, a novel image enhancement model based on shock filter for image deblurring is proposed in three cases. For the weight of shock filter, the fractional‐order derivative of initial blurry image is used. This fractional‐order weight can be adjusted adaptively according to the gradient of blurred image. Compared with the traditional integer‐order shock filter, the proposed model creates less staircase effect, whereas the enhancement and deblurring of the resulted images are efficiently good. Stability properties of the proposed method are studied theoretically, and also to show the validity of stability studies results, some numerical experiments are presented. The three cases of proposed method are compared with together and with other methods. Different test images are used to show the validity of these cases. Numerical experiments confirm the efficiency of these cases. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and fractional derivatives of the dual flatlet multiwavelets by multiplying some matrices is suggested. The system is then used to solve a fractional convection–diffusion equation. The biorthogonality and high vanishing moments properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from the other bases. [ABSTRACT FROM PUBLISHER]