Computational Methods for Differential Equations
Volume:9 Issue: 4, Autumn 2021

The cardiovascular system is an extremely intelligent and dynamic system which adjusts its performance depending on the individual's physical and environmental conditions. Some of these physical and environmental conditions may create slight disruptions in the cardiovascular system leading to a variety of diseases. Since prevention has always been preferable to treatment, this paper examined the Instantaneous Pressure-Volume Relation (IPVR) and also the pressure of the artery root. Fuzzy mathematics as a powerful tool is used to evaluate and predict the status of an individual's blood pressure. The arterial pressure is modeled as a first-order fuzzy differential equation and an analytical solution for this equation is obtained and an example shows the behavior of the solution. The risk factors using fuzzy rules are assessed, which help diagnose the status of an individual's blood pressure. Using the outcome by drawing the individual's attention to these risk factors, the individual's health is improved. Moreover, in this study, adaptive neuro-fuzzy inference system (ANFIS) models are evaluated to predict the status of an individual's blood pressure on the basis of the inputs.

In this paper, we consider the iterative system of singular Rimean-Liouville fractional-order boundary value problems with Riemann-Stieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of an infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of a unique nonnegative solution to the addressed problem by fixed point theorem in complete metric space. As an application, we present an example to illustrate the main results.

One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation ψ αf = g, on T n. We prove that if f is not analytic, then fractional integro-differential equation I 1−α t Dα x u(x, t) + iI1−α x Dα t u(x, t) = f(t) has no solution in C1 (B) with 0 < α ≤ 1. We also obtain solutions for the space-time fractional heat equations on S 1 and T n. At the end of this article, there are examples of fractional partial differential equations and a fractional integral equation together with their solutions.

Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in L2 , H1 , and W1,p norms for 2 ≤ p < ∞, are obtained. In this paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.

The aim of this work is to prove the existence and uniqueness of the positive solutions for a fractional boundary value problem by a parameterized integral boundary condition with p-Laplacian operator. By using iteration sequence, the existence of two solutions is proved. Also by applying a fixed point theorem on solid cone, the result for the uniqueness of a positive solution to the problem is obtained. Two examples are given to confirm our results.

‎This paper is devoted to study dynamical behaviors of the fractional-order BazykinBerezovskaya model and its discretization. The fractional derivative has been described in the Caputo sense. We show that the discretized system, exhibits more complicated dynamical behaviors than its corresponding fractional-order model. Specially, in the discretized model Neimark-Sacker and flip bifurcations and also chaos phenomena will happen. In the final part, some numerical simulations verify the analytical results.

Nowadays, the fixed interest rate financing method is commonly used in the capitalist financial system and in a wide range of financial liability instruments, the most important of which is bond. In the Islamic financial system, using these instruments is considered as usury and has been prohibited. In fact, Islamic law, Shariah, forbids Muslims from receiving or paying off the Riba. Therefore, using customary financial instruments such as bond is not acceptable or applicable in countries which have a majority of Muslim citizens. In this paper, we introduce one financial instrument, Sukuk, as a securities-based asset under stochastic income. These securities can be traded in secondary markets based on the Shariah law. To this end, this paper will focus on the most common structure of the Islamic bond, the Ijarah and its negotiation mechanism. Then, by presenting the short-term stochastic model, we solve fixed interest rate and model the securities-based asset by the stochastic model. Finally, we approximate the resulting model by radial basis function method, as well as utilizing the Matlab software.

In this literature, we study the existence and stability of the solution of the boundary value problem of fractional differential equations with Φp-Laplacian operator. Our problem is based on Caputo fractional derivative of orders σ, ϵ, where k − 1 < σ, ϵ ≤ k, and k ≥ 3. By using the Schauder fixed point theory and properties of the Green function, some conditions are established which show the criterion of the existence and non-existence solution for the proposed problem. We also investigate some adequate conditions for the Hyers-Ulam stability of the solution. Illustrated examples are given as an application of our result.

This paper introduces a novel method for estimation of the parameters of the constant elasticity of variance model. To do this, the likelihood function will be constructed based on the approximate density function. Then, to estimate the parameters, some optimization algorithms will be applied.

In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time fractional derivative in the Caputo sense with the order in (0, 1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.

‎‎‎‎‎In this paper, we present a numerical technique to deal with the one-dimensional forward-backward heat equations. First, the physical domain is divided into two non-overlapping subdomains resulting in two separate forward and backward subproblems, and then a meshless method based on multiquadric radial basis functions is employed to treat the spatial variables in each subproblem using the Kansa’s method. We use a time discretization scheme to approximate the time derivative by the forward and backward finite difference formulas. In order to have adequate boundary conditions for each subproblem, an initial approximate solution is assumed on the interface boundary, and the solution is improved by solving the subproblems in an iterative way. The numerical results show that the proposed method is very useful and computationally efficient in comparison with the previous works.

In the present study, we propose an effective nonlinear anisotropic diffusion-based hyperbolic parabolic model for image denoising and edge detection. The hyperbolicparabolic model employs a second-order PDEs and has a second-time derivative to time t. This approach is very effective to preserve sharper edges and better-denoised images of noisy images. Our model is well-posed and it has a unique weak solution under certain conditions, which is obtained by using an iterative finite difference explicit scheme. The results are obtained in terms of peak signal to noise ratio (PSNR) as a metric, using an explicit scheme with forward-backward diffusivities.

The collocation method based on the exponential cubic B-splines (ECB-splines) together with the Crank Nicolson formula is used to solve nonlinear coupled Burgers’ equation (CBE). This method is tested by studying three different problems. The proposed scheme is compared with some existing methods. It produced accurate results with the suitable selection of the free parameter of the ECB-spline function. It produces accurate results. Stability of the fully discretized CBE is investigated by the Von Neumann analysis.

In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the twodimensional heat equation.

In this paper, we consider a fractional Sturm-Liouville equation of the form, − cDα 0+ ◦ Dα 0+ y(t) + q(t)y(t) = λy(t), 0 < α < 1, t ∈ [0, 1], with Dirichlet boundary conditions I 1−α 0+ y(t)|t=0 = 0, and I 1−α 0+ y(t)|t=1 = 0, where, the sign ◦ is composition of two operators and q ∈ L2 (0, 1), is a real valued potential function. We use a recursive method based on Picard’s successive method to find the solution of this problem. We prove the method is convergent and show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.

In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.

In this work, the collocation method based on B-spline functions is used to obtain a numerical solution for a one-dimensional hyperbolic telegraph equation. The proposed method is consists of two main steps. As the first step, by using a finite difference scheme for the time variable, a partial differential equation is converted to an ordinary differential equation by the space variable. In the next step, for solving this equation collocation method is used. In the analysis section of the proposed method, the convergence of the method is studied. Also, some numerical results are given to demonstrate the validity and applicability of the presented technique. The L∞, L2, and Root-Mean Square(RMS) in the solutions show the efficiency of the method computationally.

In this study, the existence of positive solutions is considered for second-order boundary value problems on any time scales even in the case when y ≡ 0 may also be a solution.

The simplified phenomenological model of long-crested shallow-water wave propagations is considered without/with the Coriolis effect. Symmetry analysis is taken into consideration to obtain exact solutions. Both classical wave transformation and transformations are obtained with symmetries and solvable equations are kept thanks to these transformations. Additionally, the exact solutions are obtained via various methods which are ansatz-based methods. The obtained results have a major role in the literature so that the considered equation is seen in a large scale of applications in the area of geophysical.