Computational Methods for Differential Equations
Volume:10 Issue: 1, Winter 2022

In this paper, we investigate the solutions of a class of ψ-Hilfer fractional differential equations with the initial values in the sense of ψ-fractional integral by using the successive approximation techniques. Next, the continuous dependence of a solution for the given Cauchy-type problem is presented.

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order α, (0 < α < 1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−2,y(tj−2)),(tj−1,y(tj−1)), and (tj, y(tj)) on each interval [tj−1,tj] for (j ≥ 2), while the linear interpolation approximation are applied on the first interval [t0,t1]. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for y(t). Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.

In this paper, an approximate solution of a nonlinear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary conditions is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme are shown by numerical experiments. We have compared the approximate solutions with that in the literature.

The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of an interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.

This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative, and a suitable numerical integration rule. Error estimate of the method is investigated in detail. In the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.

In this paper, the numerical solution of an algebraic complex fuzzy equation of degree n, based on the parametric fuzzy numbers, is discussed. The unknown variable and right-hand side of the equation are considered as fuzzy complex numbers, whereas, the coefficients of the equation, are considered to be real crisp numbers. The given method is a numerical method and proposed based on the separation of the real and imaginary parts of the equation and using the parametric forms of the fuzzy numbers in the form of polynomials of degree at most m. In this case, a system of nonlinear equations is achieved. To get the solutions of the system, we used the Gauss-Newton iterative method. We also very briefly explain the conjugate of the solution of such equations. Finally, the efficiency and quality of the given method are tested by applying it to some numerical examples.

In this paper, we first establish two new identities for differentiable function involving generalized fractional integrals. Then, by utilizing these equalities, we obtain some midpoint type inequalities involving generalized fractional integrals for mappings whose derivatives in absolute values are convex. We also give several results as special cases of our main results.

In this paper, the synchronization between complex fractional-order chaotic systems and the integer-order hyperchaotic system has been investigated. Due to increased complexity and the presence of additional variables, it seems to be very interesting and can be associated with real-life problems. Based on the idea of tracking control and nonlinear control, we have designed the controllers to obtain the synchronization between the chaotic systems. To establish the efficacy of the methods computations have been carried out. Excellent agreement between the analytical and computational studies has been observed. The achieved synchronization is illustrated in the field of secure communication. The results have been compared with published literature.

In this paper, a reliable numerical scheme is developed and reviewed in order to obtain an approximate solution of time-fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time-fractional derivative is employed in the Caputo sense. The L2 convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of the spectral method and justification the theoretical analysis and to compare with other existing methods in the literatures.

In this article, we use the collocation method based on the radial basis functions with symmetric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional-differential equations with proportional delays. In this method, we control the absolute errors and the condition number of the system matrix through the program prepared with Maple 18.0 by increasing the number of collocation points that have a direct effect on the defined shape parameter. Also, we present the tables of the rate of the convergence (ROC) to investigate and show the convergence rate of this method compared to the RBF method with constant shape parameter. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs).

In this article, we will implement the(G0/G)-expansion method which is used for the first time to obtain new optical soliton solutions of the thin-film ferroelectric materials equation (TFFME). Also, the numerical solutions of the suggested equation according to the variational iteration method (VIM) are demonstrated effectively. A comparison between the achieved exact and numerical solutions has been established successfully.

This paper proposes an optimal control method for the chaotic attitude of the satellite when it is exposed to external disturbances. When there is no control over the satellite, its chaotic attitude is investigated using Lyapunov exponents (LEs), Poincare diagrams, and bifurcation diagrams. In order to overcome the problem of singularity in the great maneuvers of satellite, we consider the kinematic equations based on quaternion parameters instead of Euler angles, and obtain control functions by using the Pontryagin maximum principle (PMP). These functions are able to reach the satellite attitude to its equilibrium point. Also the asymptotic stability of these control functions is investigated by Lyapunov’s stability theorem. Some simulation results are given to visualize the effectiveness and feasibility of the proposed method.

The current study utilizes the extended sinh-Gordon equation expansion and ( G'/G2)-expansion function methods in constructing various optical soliton and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schrodinger’s equation which describes the elevation of water wave surface for slowly modulated wave trains in deep water in hydrodynamics. We secure different kinds of solutions like optical dark, bright, singular, combo solitons as well as hyperbolic and trigonometric functions solutions. Moreover, singular periodic wave solutions are recovered and the constraint conditions which provide the guarantee to the soliton solutions are also reported. In order to shed more light on these novel solutions, graphical features 3D, 2D and contour with some suitable choice of parameter values have been depicted. We also discuss the stability analysis of the studied nonlinear model with aid of modulation instability analysis.

In this article, we study the existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Banach fixed point theorem. In the later case we explore the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval.

Linear impulsive fractional differential-algebraic systems (LIFDAS) in a finite dimensional space are studied. We obtain the solution of LIFDAS. Using Gramian matrices, necessary and sufficient conditions for controllability and observability of time varying LIFDAS are established. We acquired the criterion for time-invariant LIFDAS in the form of rank conditions. The results are more generalized than the results that are obtained for various differential-algebraic systems without impulses.

In this paper, we prove the existence of solution of two nonlinear integral inclusions by using generalization of Krasnoselskii fixed point theorem for set-valued mappings. As an application, we prove the existence of solution of the boundary-valued problem of ordinary differential inclusion.

Medical ultrasound images are usually degraded by a specific type of noise, called ”speckle”. The presence of speckle noise in medical ultrasound images will reduce the image quality and affect the effective information, which can potentially cause a misdiagnosis. Therefore, medical image enhancement processing has been extensively studied and several denoising approaches have been introduced and developed. In the current work, a robust fractional partial differential equation (FPDE) model based on the anomalous diffusion theory is proposed and used for medical ultrasound image enhancement. An efficient computational approach based on a combination of a time integration scheme and localized meshless method in a domain decomposition framework is performed to deal with the model. In order to evaluate the performance of the proposed de-speckling approach, it is used for speckle noise reduction of a synthetic ultrasound image degraded by different levels of speckle noise. The results indicate the superiority of the proposed approach in comparison with classical anisotropic diffusion denoising model (Catte’s pde model).

In this paper, we study the monotonicity and convexity of the period function associated with centers of a specific class of symmetric Newtonian systems of degree 8. In this regard, we prove that if the period annulus surrounds only one elementary center, then the corresponding period function is monotone; but, for the other cases, the period function has exactly one critical point. We also prove that in all cases, the period function is convex.

Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.

In this paper, a completely new statistical-based approach is developed for solving the system of nonlinear equations. The developed approach utilizes the characteristics of the normal distribution to search the solution space. The normal distribution is generally introduced by two parameters, i.e., mean and standard deviation. In the developed algorithm, large values of standard deviation enable the algorithm to escape from a local optimum, and small values of standard deviation help the algorithm to find the global optimum. In the following, six benchmark tests and thirteen benchmark case problems are investigated to evaluate the performance of the Normal Distribution-based Algorithm (NDA). The obtained statistical results of NDA are compared with those of PSO, ICA, CS, and ACO. Based on the obtained results, NDA is the least time-consuming algorithm that gets high-quality solutions. Furthermore, few input parameters and simple structure introduce NDA as a user friendly and easy-to-understand algorithm.