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The mathematical efficiency of fractional-order differential equations in modeling real systems has been established. The first-order and second-order time derivatives are substituted in integer-order problems by a fractional derivative of order 0 < ω ≤ 1 , resulting in time-fractional heat and wave problems with variable coefficients. In this research, we analyze fractional-order wave and heat problems with variable coefficients within the framework of a Caputo derivative (CD) using the Elzaki residual power series method (ERPSM), which is a coupling of the residual power series method (RPSM) and the Elzaki transform (E-T). It relies on a novel form of fractional power series (FPS), which provides a convergent series as a solution. The accuracy and convergence rates have been proven by the relative, absolute, and recurrence error analyses, demonstrating the validity of the recommended approach. By employing the simple limit principle at zero, the ERPSM excels at calculating the coefficients of terms in a FPS, but other well-known approaches such as Adomian decomposition, variational iteration, and homotopy perturbation need integration, while the RPSM needs the derivative, both of which are challenging in fractional contexts. ERPSM is also more effective than various series solution methods due to the avoidance of Adomian's and He's polynomials to solve nonlinear problems. The results obtained using the ERPSM show excellent agreement with the natural transform decomposition method and homotopy analysis transform method, demonstrating that the ERPSM is an effective approach for obtaining the approximate and closed-form solutions of fractional models. We established that our approach for fractional models is accurate and straightforward and researcher can use this approach to solve various problems. [ABSTRACT FROM AUTHOR]

This paper focuses on a reduction technique to discover exact solutions for the generalized Camassa-Choi equation with temporal local M-derivative. The paper presents various types of exact solutions along with their corresponding first integrals. Furthermore, the interactions between the orders of α and β in the M-derivative are taken into account and depicted graphically for the derived solutions. Remarkably, the paper demonstrates that in certain situations, exact solutions can be obtained for any value of n, which holds significant mathematical intrigue. The authors note that Nucci's reduction technique has not previously been employed for differential equations with M-derivative, to the best of their knowledge. [ABSTRACT FROM AUTHOR]

Purpose: The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived. Design/methodology/approach: The main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal. Findings: Thus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained. Originality/value: Illustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems. [ABSTRACT FROM AUTHOR]

The aim of this study is to peruse the dispersive Schrödinger–Hirota equation (SH equation) with parabolic law linearity to get optical soliton solutions by utilizing the generalized Kudryashov method (GKM). Firstly, we introduced the solution algorithm of GKM. Then, we substituted the traveling wave transform to dispersive SH equation with parabolic law and gained nonlinear ordinary differential equation (NODE) form of real and imaginary parts. We successfully applied GKM to the NODE form of dispersive SH equation and obtained the appropriate solution sets. Using these sets, the solution of differential equation and necessary transformations, we created the analytical solution functions of dispersive SH equation and demonstrated the graphical simulations. Optical soliton solutions were obtained, graphical representations were made and interpreted, at the same time, it was observed that some parameters had an effect on the soliton behavior. The parabolic law form of the examined model has not been studied, and the obtained results have not been reported. [ABSTRACT FROM AUTHOR]

Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we discuss parameter estimation in black scholes model. A non-parametric estimation method and well known maximum likelihood estimator are considered. Our aim is to estimate the unknown parameters for stochastic differential equation with discrete time observation data. In simulation study we compare the non-parametric method with maximum likelihood method using stochastic numerical scheme named with Euler Maruyama. [ABSTRACT FROM AUTHOR]

In the article, we are concerned with the oscillatory solutions of a class of fractional differential equations. By using generalized Riccati function and Hardy inequalities, we present some oscillation criterias. As a result we give some examples that validity of the established results. [ABSTRACT FROM AUTHOR]

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation [ABSTRACT FROM AUTHOR]

In this paper, we implement fractional differential transform method (FDTM), which is a semi analytical numerical technique, to fractional differential-algebraic equations (FDAEs). The fractional derivatives are described in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. The method is illustrated by four examples of FDAEs and solutions are obtained. Comparisons are made between fractional differential transform method (FDTM), Homotopy Analysis Method (HAM) and the exact solutions. The results reveal that the proposed method is very effective and simple. [ABSTRACT FROM AUTHOR]

Abstract: In this study, numerical solution of differential-algebraic equations (DAEs) with index-3 has been presented by Padé Approximation. An index reduction for differential-algebraic equation with index-3 is suggested. First index-3 system is reduced to index-1 system. Then we have applied the method for solving differential-algebraic equations with index-3 numerically. [Copyright &y& Elsevier]

Abstract: In this paper, numerical solution of differential–algebraic equations (DAEs) with index-2 has been presented using Padé approximation method. An index reduction for differential–algebraic equation with index-2 is suggested. First index-2 system is reduced to index-1 system. Then we have applied the method for solving differential–algebraic equations with index-2, numerically. [Copyright &y& Elsevier]

In this paper, the method is developed to differential–algebraic equations systems. More effective method is presented and illustrated by numerical example. The solution for a differential–algebraic equation can be expanded up to arbitrary order using MAPLE computer algebra systems. First we calculate power series of the given equations system then transform it into Padé series form, which give an arbitrary order for solving differential–algebraic equation numerically. [Copyright &y& Elsevier]

In this paper, we have given numerical solution of the elasticity problem of settled on the wronkler ground with variable coefficient. First we calculate the basic successive substitute approximation of the given boundary value problem then transform it into Padé series form, which give an arbitrary order for solving differential equation numerically. We applied these methods to an example which is the elasticity problem of unit length homogeny beam, which is a special form of boundary value problem. [Copyright &y& Elsevier]

In this paper, we propose a new approach for solving boundary value problems a differential equation with retarded argument:x″(t)+a(t)x(t−τ(t))=f(t),x(t)=ϕ(t)(λ0⩽t⩽0), x(T)=xT ,where 0⩽t⩽T and a(t), f(t), τ(t)⩾0(0⩽t⩽T) and ϕ(t)(λ0⩽t⩽0) are known continuous functions. A differential equation with retarded argument is computed by converting the obtained series solution into padé series. First we calculate power series of the given equation system then transform it into padé (approximants) series form, which give an arbitrary order for solving differential equation numerically. [Copyright &y& Elsevier]

The aim of this paper is to present an efficient numerical procedure for solving boundary value problems for a differential equation with retarded argument:x′′(t)+a(t)x(t−τ(t))=f(t),x(t)=ϕ(t) (λ0&les;t&les;0), x(T)=xT,where 0&les;t&les;T and a(t),f(t),τ(t)&ges;0 (0&les;t&les;T) and ϕ(t) (λ0&les;t&les;0) are known continuous functions. A differential equation with retarded argument is computed by converting the obtained series solution into Pade´ (approximants) series. First we calculate power series of the given equation system then transform it into Pade´ (approximants) series form, which give an arbitrary order for solving differential equation numerically. [Copyright &y& Elsevier]

In this paper, the solution for a chemical differential algebraic equation (DAE) can be expanded up to arbitrary order using MAPLE computer algebra systems. First we calculate power series of the given equations system then transform it into Pade´ series form, which give an arbitrary order for solving chemical DAE numerically. [Copyright &y& Elsevier]

The principal purpose of this paper is to examine the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity which governs the propagation of dispersive pulses in optical fibers by proposing and using a direct algebraic form of the enhanced modified extended tanh expansion method for the first time. Our aim is not only restricted to obtaining different and more soliton solutions by proposed method for the first time in this study, but also includes examining the effect of the coefficients of self-steepening and nonlinear dispersion terms to the soliton propagation in the investigated problem. Utilizing a traveling wave transformation, the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity can be transformed into an nonlinear ordinary differential equation (NODE). Then, the NODE is convert into a set of algebraic equations by taking account into the Riccati differential equation. Solving the set of algebraic equations, we acquire the analytical soliton solutions of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. In the proposed method, the modified extended tanh function method is enhanced by presenting more solutions of Riccati differential equations with the direct algebraic form, is utilized. The more solutions have been established to the literature with new significant physical properties of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. We indicated that the presented method are effective, easily computable, and reliable in solving such nonlinear problems. Moreover, we demonstrate the dynamical behaviors and physical significance of some soliton solutions at appropriate values of parameters. A variety of soliton solutions to the perturbed Schrödinger–Hirota equation with Kerr law non-linearity by the direct algebraic form of enhanced modified extended tanh expansion method have been acquired. These solutions are dark–bright, trigonometric, hyperbolic, periodic, and singular soliton solutions. 3D, contour and 2D plots of some obtained solutions have been demonstrated to interpret the physical meaning of the equation. For some parameter values in the equation, the behavior of soliton solutions has been examined. The constraint conditions are established to confirm the existence of valid solutions. The obtained results can be effective in interpreting the physical meaning of this nonlinear system. We have seen that the proposed direct algebraic form of the enhanced modified extended tanh expansion method is a powerful mathematical technique which can be utilized to acquire the analytical solutions to different complex nonlinear mathematical models. [ABSTRACT FROM AUTHOR]