Forward Difference Formula Research Articles

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.

Impacts of Exponentially Growing/Decaying Pressure Gradient on Mixed Convection Flow of Viscous Reactive Fluid in a Vertical Tube: A Numerical Approach

This study examined the effects of a pressure gradient that was exponentially increasing or decreasing on the mixed convection flow of viscous reactive fluids in a vertical tube between two concentric tubes with r = 0 and r = b. The nonlinear partial differential equation governing the flow formation are solved using the implicit finite difference form where all time differentials were calculated using the forward difference formula, and second order central differences were utilized to approximate the first and second derivatives. The impacts of several physical parameters, including shear stress, the rate of heat transfer, mixed convection, viscous reactive fluid parameter, activation energy, Prandtl number, and exponential decaying/growing pressure gradient on skin friction and Nusselt number were investigated. It's intriguing to see that raising the values of Frank–Kamenetskii (λ), mixed convection (Gre) as well as exponential growing pressure term increases the velocity fluid, However, in the case of exponential decay, the parameter drop also causes the fluid's velocity to diminish. The results also showed that a small bump in the viscous reactive fluid parameter considerably improves the energy profile.

Resolving the spurious-state problem in the Dirac equation with the finite-difference method

To solve the Dirac equation with the finite difference method, one has to face up to the spurious-state problem due to the fermion doubling problem when using the conventional central difference formula to calculate the first-order derivative on the equal interval lattices. This problem is resolved by replacing the central difference formula with the asymmetric difference formula, i.e., the backward or forward difference formula. To guarantee the hermitian of the Hamiltonian matrix, the backward and forward difference formula should be used alternatively according to the parity of the wavefunction. This provides a simple and efficient numerical prescription to solve various relativistic problems in the microscopic world.

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

When analysing the thermal conductivity of magnetic fluids, the traditional Sharma-Tasso-Olver (STO) equation is crucial. The Sharma-Tasso-Olive equation’s approximate solution is the primary goal of this work. The quintic B-spline collocation method is used for solving such nonlinear partial differential equations. The developed plan uses the collocation approach and finite difference method to solve the problem under consideration. The given problem is discretized in both time and space directions. Forward difference formula is used for temporal discretization. Collocation method is used for spatial discretization. Additionally, by using Von Neumann stability analysis, it is demonstrated that the devised scheme is stable and convergent with regard to time. Examining two analytical approaches to show the effectiveness and performance of our approximate solution.

A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto\u2013Sivashinsky equation

A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In this work, an application of fifth degree basis spline functions is presented for a numerical investigation of the Kuramoto–Sivashinsky equation. The finite forward difference formula is used for temporal integration, whereas the basis splines, together with a new approximation for fourth order spatial derivative, are brought into play for discretization in space direction. In order to corroborate the presented numerical algorithm, some test problems are considered and the computational results are compared with existing methods.

A numerical investigation of Caputo time fractional Allen\u2013Cahn equation using redefined cubic B-spline functions

We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen–Cahn equation (ACE). We discretize the time fractional derivative of order alphain(0,1] by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O(h^{2}+Delta t^{2-alpha}) and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature.

Application of meshfree spectral method for the solution of multi-dimensional time-fractional Sobolev equations

The present study aims to formulate a meshfree spectral method to numerically solve multi-dimensional time-fractional Sobolev equations. In this method radial basis functions and point interpolation approach is utilized to build meshfree shape functions. These shape functions, having Kronecker delta function property, are then used for spatial discretization. Forward difference formula, in alignment to a quadrature rule, is used for temporal disctrezation. Validation of the proposed method is made by considering various test examples from literature. Simulated results reveal very good agreement with available exact solutions which are reported in graphical and tabulated forms. Approximation quality and efficiency of the current method is achieved in terms of e2, e∞ and erms error norms, number of collocation points as well as time-step size. Stability of the proposed method is thoroughly analyzed and discussed.

New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

In the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. For this purpose, first of all, the Schrodinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, $L_{2}$ and $L_{\infty}$ , as well as the two lowest invariants, $I_{1}$ and $I_{2}$ , have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.

An alternative approach to solve complex nonlinear least-squares problems

Abstract In this work, a hybrid electrochemical impedance spectroscopy (EIS) strategy was developed with the aim to solve complex nonlinear least-square (CNLS) problems. For the first time the CLNS problems were solved by an alternative approach, i.e. by using Python to merge the following tactics: circuit description code (CDC), central-difference formula, Nielsen's modification of the Levenberg–Marquardt algorithm (LMA) and visual inspection of the χ2-function minima. It was presented that the Nielsen's damping (λ) factor updating strategy can be used to design a topical fitting engine in which the change in the λ factor is continuous. The commonly used forward-difference formula was replaced with a more accurate central-difference formula to approximate the first derivates. It was elucidated that by combining the Nielsen's λ updating tactics and central-difference formula in the hybrid EIS strategy resulted in the design of a more robust fitting engine. The hybrid EIS strategy was enhanced by the CDC routine which allows researchers to effortlessly generate a great variety of electrical equivalent circuits (EECs). The credibility of the EEC parameters was increased by visual inspection of the χ2-function minima. Additionally, a novel convergence in the symmetry criterion was proposed to avoid errors during the visual inspection. The applicability of the hybrid EIS strategy was evaluated after it was integrated into a new software solution and compared to the more widely used MEISP and EQUIVCRT software packages. It was presented that the three fundamentally different software solutions yielded approximately equal EEC data. However, only EIS strategies with the similar LMA tactics produced the identical χ2-values in all case studies.

3-D gravity modeling of basement reliefs an offshore case study in the Persian gulf, Iran

In this paper we have proposed a three-dimensional approach to determine depth of basement in which the density contrast varies parabolically with depth. This program based on Newtons forward difference formula that with optimization of gravity anomalies calculate depths of basement reliefs indeed are anticlinal and synclinal structures has been buried under sediments. This structures are the causes of the positive and negative gravity anomalies. We assume the measured gravity fields have been distributed on a horizontal plane and also sedimentary basin is combined of juxtaposition 3-D cubic prisms. The measurement stations of the gravity field (grid nodes) coincide with center blocks. The initial depth is computed using gravity data and the estimated depths are adjusted with iteration. The advantage of the method is utilization of positive and negative gravity anomalies together as two inputs for written algorithm as well as application of a coded non-linear filter. The efficiency of the code is illustrated with a set of synthetic gravity anomalies. Further, the code is exemplified with the gravity anomalies of an offshore case study in the Persian Gulf, Iran. The purpose of exploratory project in this area will include the development of Hydrocarbon Fields. Keywords: Basement, Gravity Anomalies, Newtons Forward Difference Formula, Non-Linear Filter, Persian Gulf.

Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations

This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by first order forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one- and two-dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one- and two-dimensional test problems. It has been observed that the proposed method is more efficient in terms of less memory requirement and less computational efforts due to one time inversion of 5×5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.

An Explicit Numerical Method for the Fractional Cable Equation

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann‐Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first‐order derivative by a forward difference formula, the Riemann‐Liouville derivatives by the Grünwald‐Letnikov formula, and the spatial derivative by a three‐point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von‐Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.

3D gravity modeling of Büyük Menderes basin in Western Anatolia using parabolic density function

A method to model 3D sedimentary basins with parabolic density contrast is applied to Büyük Menderes basin in Western Anatolia. The measured gravity fields, reduced to a horizontal plane, are assumed to be available at grid nodes of a rectangular/square mesh. Juxtaposed 3D vertical prisms with their geometrical epicenters on top coincide with grid nodes of a mesh to approximate a sedimentary basin. The algorithm based on Newton’s forward difference formula automatically calculates the initial depth estimates of a sedimentary basin assuming that 2D infinite horizontal slabs can generate the measured gravity fields and among these slabs the density contrast varies with depth. The lower boundary of a sedimentary basin is formulated by estimating the depth values of the 3D prisms with in predetermined limits. Measured gravity fields pertaining to the Büyük Menderes basin, Turkey, where the density contrast varies with depth, are interpreted to show the applicability of the method.

Automatic 3-D gravity modeling of sedimentary basins with density contrast varying parabolically with depth

A method to model 3-D sedimentary basins with density contrast varying with depth is presented along with a code GRAV3DMOD. The measured gravity fields, reduced to a horizontal plane, are assumed to be available at grid nodes of a rectangular/square mesh. Juxtaposed 3-D rectangular/square blocks with their geometrical epicenters on top coincide with grid nodes of a mesh to approximate a sedimentary basin. The algorithm based on Newton's forward difference formula automatically calculates the initial depth estimates of a sedimentary basin assuming that 2-D infinite horizontal slabs among which, the density contrast varies with depth could generate the measured gravity fields. Forward modeling is realized through an available code GR3DPRM, which computes the theoretical gravity field of a 3-D block. The lower boundary of a sedimentary basin is formulated by estimating the depth values of the 3-D blocks within predetermined limits. The algorithm is efficient in the sense that it automatically generates the grid files of the interpreted results that can be viewed in the form of respective contour maps. Measured gravity fields pertaining to the Chintalpudi sub-basin, India and the Los Angeles basin, California, USA in which the density contrast varies with depth are interpreted to show the applicability of the method.

Local thermodynamic models used in sensitivity estimation of dynamic systems

Local thermodynamic models are used to accelerate dynamic optimization calculations. The optimization problem is solved with the sequential approach. The main calculation effort in this optimization is the evaluation of the sensitivities of the objective function and constraints w.r.t. the parameters. A new approach is tried out for using local thermodynamic models in the sensitivity estimation. The local model must be updated to ensure sufficient accuracy. The local model updates generate discontinuities which give trouble for the integrator. To avoid these problems the special purpose integrator DASSLM was developed. DASSLM simultaneously integrates one rigorous trajectory and the N parameter+1 local model trajectories needed for calculating the sensitivities by a forward difference formula. The simultaneous calculation of the local model and the rigorous model trajectories give at all times the error in the local model. This error can be used to update the local model according to a predefined tolerance. The steplength in DASSLM is determined only with information from the rigorous trajectory, the updates does thus not affect the steplength. The results show that the DASSLM local model approach is computationally efficient and stable, but that the accuracy of the gradients are reduced. In addition to the DASSLM integrator, the DASAC integrator is tried out with local thermodynamic models.

The Use of Nuclear Emulsions to Estimate Stray Neutron Spectra Near Particle Accelerators

The stray neutron field near a particle accelerator generates proton recoil tracks, of various lengths and orientations in photographic emulsions. The lengths are directly convertible to proton kinetic energies by use of five formulae. Three of these are least-square, straight-line fits of the experimental data of LATTES, FOWLER and CUER(1) while the other two are the empirical equations of BRADNER, SMITH, BARKAS and BISHOP(2) and W. H. BARKAS.(3) If the proton recoils are counted over all directions, then an analysis will show that the probability that a neutron of energy, En, will cause a proton to recoil with an energy, E < En, is σ/En. Therefore, the relative neutron flux density per unit energy interval per hydrogen atom, N(E), is related to the number of recoil protons per gram of absorbing material per unit energy interval, P(E), by: N(E) = −E/σ × dP(E)/dE where E is the incident neutron energy in MeV and σ is the n-p cross section for hydrogen at energy E. Because the proton track can only be counted if it both begins and ends in the emulsion, it is necessary to correct for the probability, increasing with energy, that a proton track will escape the emulsion. A correction based on the geometry of the emulsion has been derived.(4) Having established the proton distribution, as shown in Fig. 1, by measurement and correction for protons escaping the emulsion, the Gregory-Newton forward difference formula was used to determine dP/dE.(5) The initial use of this technique was to measure the stray neutron spectrum with Ilford G-5 emulsions, near the Brookhaven National Laboratory Cosmotron. The resulting neutron spectrum is exhibited as Fig. 2.