Computational Methods for Differential Equations
Volume:10 Issue: 2, Spring 2022

In this manuscript, a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.

The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It’s of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−1, 1] × [−1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.

In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions,$max{S^klnbig(frac{S}{K}big),0}$ and $max{S^klnbig(frac{K}{S}big),0}‎, ‎(kin mathbb{R^{+}}cup {0})$‎. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM are quite accurate to the closed form solutions.

In the current paper, for the economic growth model, an efficient numerical approach on arbitrary collocation points is described according to Radial Basis Functions (RBFs) interpolation to approximate the solutions of optimal control problems. The proposed method is based on parametrizing the solutions with any arbitrary global RBF and transforming the optimal control problem into a constrained optimization problem using arbitrary collocation points. The superiority of the method is its flexibility to select between different RBF functions for the interpolation and also parametrization an extensive range of arbitrary nodes. The Lagrange multipliers method is employed to convert the constrained optimization problem into a system of algebraic equations. Numerical results approve the accuracy and performance of the presented method for solving optimal control problems in the economic growth model.

In this work, a direct computational method has been developed for solving the thermal analysis of porous fins with a rectangular cross-section with the aid of Chebyshev polynomials. The method transforms the nonlinear differential equation into a system of nonlinear algebraic equations and then solved using a novel technique. The solution of the system gives the unknown Chebyshev coefficients. An algorithm for solving this nonlinear system is presented. The results are obtained for different values of the variables and a comparison with other methods is made to demonstrate the effectiveness of the method.

This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson’s differential equation. The considered problem may or may not be accompanied by a control box constraint. The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve them. A fuzzy system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.

‎This study explores the time-dependent flow of MHD Oldroyd-B fluid under the effect of ramped wall velocity and temperature‎. ‎The flow is confined to an infinite vertical plate embedded in a permeable surface with the impact of heat generation and thermal radiation‎. ‎Solutions of velocity‎, ‎temperature‎, ‎and concentration are derived symmetrically by applying non-dimensional parameters along with Laplace transformation $(LT)$ and numerical inversion algorithm‎. Graphical results for different physical constraints are produced for the velocity‎, ‎temperature‎, ‎and concentration profiles‎. ‎Velocity and temperature profile decrease by increasing the effective Prandtl number‎. ‎The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity‎. ‎Velocity is decreasing for $kappa$‎, ‎$M$‎, ‎$Pr_{reff,}$ and $S_{c}$ while increasing for $G_{r}$ and $G_{c}$‎. ‎Temperature is an increasing function of the fractional parameter‎. ‎Additionally‎, ‎Atangana-Baleanu $(ABC)$ model is good to explain the dynamics of fluid with better memory effect as compared to other fractional operators‎.

In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDEs) and some more results about Laplace transform introduced by Abdeljawad are investigated. The Laplace transform method is developed to get the exact solution of CFDEs. The aim of this paper is to convert the CFDEs into ordinary differential equations (ODEs), this is done by using the fractional Laplace transform of (α + β) order.

In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods.

In this paper, option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.

This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package.

‎In this work‎, ‎we established some exact solutions for the‎ $(1+1)$-dimensional and $(2+1)$-dimensional fifth-order integrable‎ equations ($(1+1)$D and $(2+1)$D FOIEs) which is considered based on‎ the improved $tanh(phi(xi)/2)$ expansion method (IThEM)‎, ‎by‎ utilizing Maple software‎. ‎We obtained new periodic solitary wave‎ ‎solutions‎. ‎The obtained solutions include soliton‎, ‎periodic‎, ‎kink‎, kink-singular wave solutions‎. ‎Comparing our new results with Wazwaz‎ results‎, ‎namely‎, ‎the Hereman-Nuseri method shows that our results give‎ further solutions‎. ‎Many other such types of nonlinear equations‎ arise in fluid dynamics‎, ‎plasma ‎physics,‎ and nonlinear physics‎.

In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact, stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally, European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.

‎In this study‎, ‎a robust computational method involving exponential cubic spline for solving singularly perturbed parabolic convection-diffusion equations arising in the modeling of neuronal variability has been presented‎. ‎Some numerical examples are considered to validate the theoretical findings‎. ‎The proposed scheme is shown to be an $varepsilon-$uniformly convergent accuracy of order $ Oleft( left( Delta tright)‎ +‎h^2 right) $‎.

In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different sizes and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm.

This paper deals with the numerical treatment of singularly perturbed delay differential equations having a delay on the first derivative term. The solution of the considered problem exhibits boundary layer behavior on the left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting in an asymptotically equivalent singularly perturbed boundary value problem. The uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.

In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, π). By taking the Mochizuki-Trooshin’s method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.

The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two-dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the timedependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.

‎In this paper‎, ‎we propose an exponential Euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion $ B^{ a‎, ‎b}$ under some assumptions on $a$ and $b$‎. ‎We obtain also the convergence rate of the method to the true solution after proving an $L^{ 2}$-maximal bound for the stochastic integrals in this case‎.

In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractionalorder Schrödinger equation in fractional time derivative form of order α. Some exact solutions of the fractionalorder fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that unstable nonlinear Schrödinger equation exist different dynamical behaviors. In addition, the physical behaviors of these new exact solutions are given with two and three dimensional graphs.