Nonlinear Riccati Differential Equations Research Articles

Advancing differential equation solutions: A novel Laplace-reproducing kernel Hilbert space method for nonlinear fractional Riccati equations

Fractional ordinary differential equations are crucial for modeling various phenomena in engineering and physics. This study aims to enhance solution methodologies by combining the Laplace transform with the reproducing kernel Hilbert space method (RKHSM). This integration leads to a more effective approach compared to the classical RKHSM. We apply the Laplace-reproducing kernel Hilbert space method (L-RKHSM) to develop novel numerical solutions for nonlinear fractional Riccati differential equations. The L-RKHSM systematically produces both approximate and analytic solutions in series form. We present detailed results for four illustrative examples, showcasing the superior performance of the L-RKHSM over traditional methods. This innovative approach not only advances our understanding of nonlinear fractional ordinary differential equations but also demonstrates its effectiveness through significantly improved outcomes in various applications.

Computational Analysis of the Fractional Riccati Differential Equation with Prabhakar-type Memory

The key objective of the current work is to examine the behavior of the nonlinear fractional Riccati differential equation associated with the Caputo–Prabhakar derivative. An efficient computational scheme, that is, a mixture of homotopy analysis technique and sumudu transform, is used to solve the nonlinear fractional Riccati differential equation. The convergence and uniqueness analysis for the solution of the implemented technique is shown. In addition, the numerical consequences are demonstrated in the form of graphical representations to verify the reliability of the applied method in obtaining the solution to the mathematical model with Prabhakar-type memory.

Constructing Analytical Solutions of the Fractional Riccati Differential Equations Using Laplace Residual Power Series Method

In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to construct analytical series solutions of the target equation. The method is tested for eminent examples and the obtained results demonstrate the accuracy and efficiency of this technique by comparing it with other numerical methods.

A generalized analytical approach for highly accurate solutions of fractional differential equations

A generalized homotopy-based approach is developed to give highly accurate solutions of fractional differential equations. By introducing a scaling transformation, the computational domain of the nonlinear Riccati differential equations with fractional order changes from [0,+∞) to [0,1]. Analytical approximation of arbitrary accuracy is achieved, whose convergence is proved theoretically. The effectiveness and accuracy of our solution is strictly checked via error analysis. The proposed method is expected to be as a new and reliable analytical approach to give highly accurate solutions of strongly nonlinear problems in fractional calculus.

Numerical solution of nonlinear fractional Riccati differential equations using compact finite difference method

Numerical solution of nonlinear fractional Riccati differential equations using compact finite difference method

Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

APPROXIMATE SOLUTION OF FRACTIONAL RICCATI DIFFERENTIAL EQUATION USING SUMUDU DECOMPOSITION METHOD

In this paper, the Sumudu decomposition method is used to solve nonlinear Fractional Riccati differential equations. This method is a combination of the Sumudu transform and Adomian decomposition method. The analysis applied in this work forms a crucial step in the process of development of fractional calculus. The Caputo fractional derivative and Modified Riemann-Liouville derivative is applied for description of article. Numerical examples are explained to support our method. Outcomes are demonstrated graphically with tabulated forms to see the power of the proposed method.

A NEW PERSPECTIVE TO STUDY THE THIRD-ORDER MODIFIED KDV EQUATION ON FRACTAL SET

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform. 151(1–4) (2017) 409–417], we can get a class of exact solutions to the third-order modified KdV equation on the fractal set. These new type solutions can assist us to review different nonlinear phenomena better, which had been modeled via local fractional derivative.

Solution of Nonlinear Fractional Quadratic Riccati Differential Equations Using Perturbation Method

Homotopy perturbation transform method (HPSTM) in conjunction of Adomian and He’s polynomials has been devoted to obtain approximate solution of fractional order quadratic Riccati equations for arbitrary order $$ \alpha \in (0,1] $$. The suggested method finds the solution without any discontinuation and averts the round-off errors. We have taken three examples to justify the accuracy of the method with the help of Adomian polynomials based on Newton–Raphson’s scheme. The approximate solutions are compared with exact solutions.

Solving Parametric Fractional Differential Equations Arising from the Rough Heston Model Using Quasi-Linearization and Spectral Collocation

The rough Heston model has recently attracted the attention of many finance practitioners and researchers as it maintains the basic structure of the classical Heston model while having descriptive capabilities in terms of microstructural foundations of the market. Using the fact that the characteristic function of log-price in this model could be expressed in terms of the solution of a nonlinear parametric fractional Riccati differential equation not admitting a closed-form solution, devising efficient numerical schemes for pricing and calibration under this model has become a crucial need in the computational finance community. Although the fractional Adams method has been used in most of the recent studies on the rough Heston model, this method suffers from some stability and convergence issues in treating the problem. In this paper, we present a numerical method based on Newton--Kantorovich quasi-linearization to solve the nonlinearity issue followed by spectral collocation based on “polyfractonomials" to approximate the fractional derivative in an accurate and efficient manner. We provide sufficient conditions under which our method is convergent and the order of convergence is also obtained. In order to guarantee the specified convergence rate, we first prove some regularity results on the linearized problem and then employ the proposed scheme to solve a practical calibration problem from the SPX options market. The efficiency of the proposed method is illustrated by comparing the obtained results with those of the fractional Adams method as well as a fast hybrid scheme based on fractional power series expansion.

Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients

This paper presents a computational method for the approximate solution of arbitrary-order non-linear fractional Riccati differential equations with variable coefficients. Proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the Jacobi polynomials. Convergence analysis of the proposed method is provided. Numerical results for different fractional orders of the Riccati differential equations are discussed. Figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of Jacobi polynomials such as the Legendre polynomials, the Chebyshev polynomials of the second kind, the Chebyshev polynomials of the third kind, the Chebyshev polynomial of the fourth kind, and the Gegenbauer (or ultraspherical) polynomials. Numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods. It is observed that the results from the proposed method are more accurate. Maximum absolute error and the root-mean square error tables are given for all five kinds of polynomials for comparison purposes.

Synchronization of the vector state estimation methods with unmeasurable coordinates for intelligent water quality monitoring systems in the river

One of the main problems of signal processing is the issue of estimation, which consists in reproducing a useful signal on the basis of another one, e.g. one burdened with unwanted interference. The use of classical filtering to eliminate disturbances carries the risk of removing an important part of the information about the useful signal. In order to avoid this phenomenon, optimal filters are used, utilizing the statistical properties of signals. Such a case of consideration concerns the issue of estimation for stochastic systems. An example is the Kalman filter in which the main aspect of research is the differential equation of the state estimate in which there is an amplification factor dependent on the covariance matrix of the estimation error. The value of this factor is determined from the non-linear Riccati differential equation, depending on the characteristics of signals affecting the object under investigation and the measurement. The paper proposes an adaptive algorithm to determine the value of the filter amplification factor which allows to avoid the inconveniences resulting from the need to estimate the characteristics of signals affecting the object under investigation. The presented approach to estimation issues does not require the need to characterize random signals due to the application of the adaptive method of state estimation. The analyzed process refers to an object described by ordinary differential equations with continuous measurements representing the state of pollution of the river using the so-called interpretation “along the characteristics”. The method used in the module of the intelligent analytical system allows for quick determination of hard-to-measure parameters of the quality of water in the river and to react to upcoming ecological threats without any delays dangerous for the system.

Homotopy Perturbation Method for Solving Generalized Riccati Differential Equation

In this work, we have studied a class of nonlinear Riccati differential equation which is used as mathematical models in many physically significant fields and applied science. The homotopy perturbation method, has been modified for solving generalized linear and nonlinear Riccati differential equation, which is a second-order ordinary differential equation. Also, we have tested the modified homotopy perturbation method on the solving of different implementations which are show the efficiency and accuracy of the proposed method. The approximated solutions are agreed well with analytical solutions for the tested problems.

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

Operational matrices to solve nonlinear Riccati differential equations of arbitrary order

In this paper, an effective numerical method to achieve the numerical solution of nonlinear Riccati differential equations of arbitrary (integer and fractional) order has been developed. For this purpose, the fractional order of the Chebyshev functions (FCFs) based on the classical Chebyshev polynomials of the first kind have been introduced, that can be used to obtain the solution of these equations. Also, the operational matrices of fractional derivative and product for the FCFs have been constructed. The obtained results illustrated demonstrate that the suggested approaches are applicable and valid.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type $$_4F_{3}(1)$$ . With the aid of some standard reduction formulae such as Pfaff-Saalschutz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

Non-differentiable Solutions for Local Fractional Nonlinear Riccati Differential Equations

We investigate local fractional nonlinear Riccati differential equations (LFNRDE) by transforming them into local fractional linear ordinary differential equations. The case of LFNRDE with constant coefficients is considered and non-differentiable solutions for special cases obtained.

Variational Homotopy Perturbation Method for Solving Riccati Type Differential Problems

In this paper, a Variational homotopy perturbation method is proposed to solve nonlinear Riccati differential equation. By combining the Variational Iteration Method and the Homotopy Perturbation Method, this technique possesses a fast convergence rate with high accuracy. The results reveal that the proposed method is very effective and simple.

Effects of MHD on the Unsteady Thin Flow of Non-Newtonian Oldroyd –B Fluid over an Oscillating Inclined Belt Through a Porous Medium

The unsteady (MHD) thin film flow of an incompressible Oldroyd -B fluid over an oscillating inclined belt making a certain angle with the horizontal through a porous mediumis analyzed .The analytical solution of velocity field is obtained by a semi-numerical technique optimal homotopy asymptotic method (OHAM) .Finally the influence of various dimensionless parameters emerging in the model on velocity field is analyzed by graphical illustrations. I. Introduction In everyday life and in engineering flow of non-Newtonian fluid are frequently occurred .It is ubiquitous in nature and technologies .Therefore ,to understand there mechanics is essential in most application .Most of it problems appeared in several examples are polymer solution ,paints ,certain oils ,exocitic lubricants ,colloidal and suspension solutions , clay coatings and cosmetic products .In non-Newtonian fluids Thin film flows have a large varying of practical applications in nonlinear science and engineering industries . For modeling of non-Newtonian Fluid flow problem we used several model like (second grade, third grade , Maxwell fluid ,Oldroyd-B ) ..ect fluid model by which we got linear or nonlinear ordinary differential equation or partial differential equations which may be solved exactly, analytically or numerically. For solution of these problems different varieties of methods are used, in which (ADM,VAM,HAM,HPM,OHAM) are frequently used. In this work we have modeled a partial differential equations by using oldroyd-B fluid model. Solution of the problem is obtained by using OHAM. Fetecau et al. (3) studied the exact solutions of an Oldroyd-B fluid over a flat plate on constantly accelerating flow. In the following year, Fetecau et al. (4) obtained the exact solutions of the transient oscillating motion of an Oldroyd-B fluids in cylindrical domains. Haitao and Mingyu (5) investigated the series solution of an Oldroyd-B fluid by using the sine and Laplace transformations for the plane Poiseuille flowand plane couette flow. Lie L. (7) studied the effect of MHD flow for an Oldroyd-B fluid between two oscillating cylinders .Khan et al. Burdujan (2) discussed the solution for the flow of an incompressible Oldroyd-B fluid between two cylinders . Shahid et al. (12) studied the solution of the steady and unsteady flow of Oldroyd-B fluid by using Laplace and Fourier series. Shah et al. (13) studied the solution by using OHAM of thin film flow of third grade fluid on moving inclined plane. Marinca et al. (11) studied the approximate solution of non-linear steady flow of fourth grade fluid by using OHAM. Marinca et al (10) discussed the approximate solution of the unsteady viscous flow over a shrinking cylinder by using OHAM method . Kashkari (6) studied the OHAM solution of nonlinear Kawahara equation. For comparison HPM, VHPM and VIM method is used but OHAM ismore successful method. . Anakira et al. (1) the analytical solution of delay differential equation by using OHAM. Mabood et al. (8,9) investigated the approximate solution of non-linear Riccati differential equation by using OHAM . Taza Gul (14) studied the unsteady magnetohydrodynamics (MHD) thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt making a certain angle with the horizontal. In this paper , we study the unsteady (MHD) thin film flow of an incompressible Oldroyd -B fluid over an oscillating inclined belt making a certain angle with the horizontal in porous medium . the solution of velocity field is obtained by using a semi-numerical technique optimal homotopy asymptotic method (OHAM). Finally, we analyzed the influence of various dimensionless parameters emerging in the model on velocity field is analyzed by graphical illustrations.