## Temporal Variable

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### Manju Sharma - One of the best experts on this subject based on the ideXlab platform.

• ##### a parameter uniform difference scheme for parabolic partial differential equation with a retarded argument
Applied Mathematical Modelling, 2010
Co-Authors: Aditya Kaushik, Kapil K Sharma, Manju Sharma
Abstract:

The paper deals with the analysis of a non-stationary parabolic partial differential equation with a time delay. The highest order derivative term is affected by the small parameter. This is precisely the case when the magnitude of the convective term becomes much larger compare to that of diffusion term. The solution of problem exhibits steep gradients in the narrow intervals of space and short interval of times. In these cases a dissipative loss turned out to be more complex. Even for the one spatial dimension and one Temporal Variable, not all difference scheme can capture these steep variation. Although the analysis is restricted to the model in one space dimension, the technique and comparison principles developed should prove useful in assessing the merits of numerical solution of other nonlinear model equations too.

### Mostafa Abbaszadeh - One of the best experts on this subject based on the ideXlab platform.

• ##### meshless upwind local radial basis function finite difference technique to simulate the time fractional distributed order advection diffusion equation
Engineering With Computers, 2021
Abstract:

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the Temporal Variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

• ##### crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation
Journal of Computational and Applied Mathematics, 2020
Co-Authors: Mostafa Abbaszadeh, Mehdi Dehghan, Yong Zhou
Abstract:

Abstract The fractional PDEs based upon the distributed-order fractional derivative have several applications in physics. The two-dimensional time-space distributed-order weakly singular integro-partial differential model is investigated by a combination of finite difference and Galerkin spectral methods. A second-order finite difference formula is employed to approximate the Temporal Variable. In this stage, the stability and convergence of the semi-discrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space Variable. Also, in this step, the error estimate of the full-discrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.

• ##### A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives
Engineering with Computers, 2020
Abstract:

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The Temporal Variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time Variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$ O ( τ 3 2 ) . Also, the space Variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.

• ##### error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives
Journal of Computational and Applied Mathematics, 2019
Abstract:

Abstract In the current investigation, an error estimate has been proposed to solve the two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives based on the finite element/finite difference scheme. The time and space derivatives are based on the Riemann–Liouville and Riesz fractional derivatives, respectively. At first, the Temporal Variable has been discretized by a second-order difference scheme and then the space Variable has been approximated by the finite element method (FEM). The analytical study shows that the presented scheme is unconditionally stable and convergent. Finally, some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique.

### Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.

• ##### meshless upwind local radial basis function finite difference technique to simulate the time fractional distributed order advection diffusion equation
Engineering With Computers, 2021
Abstract:

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the Temporal Variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

• ##### crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation
Journal of Computational and Applied Mathematics, 2020
Co-Authors: Mostafa Abbaszadeh, Mehdi Dehghan, Yong Zhou
Abstract:

Abstract The fractional PDEs based upon the distributed-order fractional derivative have several applications in physics. The two-dimensional time-space distributed-order weakly singular integro-partial differential model is investigated by a combination of finite difference and Galerkin spectral methods. A second-order finite difference formula is employed to approximate the Temporal Variable. In this stage, the stability and convergence of the semi-discrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space Variable. Also, in this step, the error estimate of the full-discrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.

• ##### A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives
Engineering with Computers, 2020
Abstract:

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The Temporal Variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time Variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$ O ( τ 3 2 ) . Also, the space Variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.

• ##### error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives
Journal of Computational and Applied Mathematics, 2019
Abstract:

Abstract In the current investigation, an error estimate has been proposed to solve the two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives based on the finite element/finite difference scheme. The time and space derivatives are based on the Riemann–Liouville and Riesz fractional derivatives, respectively. At first, the Temporal Variable has been discretized by a second-order difference scheme and then the space Variable has been approximated by the finite element method (FEM). The analytical study shows that the presented scheme is unconditionally stable and convergent. Finally, some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique.

### Mateusz Kwaśnicki - One of the best experts on this subject based on the ideXlab platform.

• ##### fluctuation theory for levy processes with completely monotone jumps
Electronic Journal of Probability, 2019
Co-Authors: Mateusz Kwaśnicki
Abstract:

We study the Wiener–Hopf factorization for Levy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the Temporal Variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.

• ##### fluctuation theory for l evy processes with completely monotone jumps
arXiv: Probability, 2018
Co-Authors: Mateusz Kwaśnicki
Abstract:

We study the Wiener-Hopf factorization for Levy processes $X_t$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the Temporal Variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_t$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_t$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi)$ of $X_t$, including a peculiar structure of the curve along which $f(\xi)$ takes real values.

### Aditya Kaushik - One of the best experts on this subject based on the ideXlab platform.

• ##### a parameter uniform difference scheme for parabolic partial differential equation with a retarded argument
Applied Mathematical Modelling, 2010
Co-Authors: Aditya Kaushik, Kapil K Sharma, Manju Sharma
Abstract:

The paper deals with the analysis of a non-stationary parabolic partial differential equation with a time delay. The highest order derivative term is affected by the small parameter. This is precisely the case when the magnitude of the convective term becomes much larger compare to that of diffusion term. The solution of problem exhibits steep gradients in the narrow intervals of space and short interval of times. In these cases a dissipative loss turned out to be more complex. Even for the one spatial dimension and one Temporal Variable, not all difference scheme can capture these steep variation. Although the analysis is restricted to the model in one space dimension, the technique and comparison principles developed should prove useful in assessing the merits of numerical solution of other nonlinear model equations too.