Computational Methods for Differential Equations
Volume:11 Issue: 3, Summer 2023

Some problems in science and technology are modeled using ambiguous, imprecise, or lacking contextual data. In the modeling of some real-world problems, differential equations often involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty, rather than single bits. These types of differentials are not well represented by fuzzy differential equations or bipolar fuzzy differential equations. Therefore, m-pole fuzzy set theory can be applied to differential equations to deal with problems with multi-polar information. In this paper, we study differential equations in m extremely fuzzy environments. We introduce the concept gH-derivative of m-polar fuzzy valued function. By considering different types of differentiability, we propose some properties of the gH-differentiability of m-polar fuzzy-valued functions. We consider the m-polar fuzzy Taylor expansion. Using Taylor expansion, Euler's method and modified Euler's method for solving the m-polar fuzzy initial value problem are proposed. We discuss the convergence analysis of these methods. Some numerical examples are described to see the convergence and stability of the proposed method. We compare these methods by computing the global truncation error. From the numerical results, it can be seen that the modified Euler method converges to the exact solution faster than the Euler method.

We aim at presenting results including analytical solutions to linear fully fuzzy Caputo-Fabrizio fractional differential equations. In such linear equations, the coefficients are fuzzy numbers and, as a useful approach, the cross product has been considered as a multiplication between the fuzzy data. This approach plays an essential role in simplifying of computation of analytical solutions of linear fully fuzzy problems. The obtained results have been applied for deriving explicit solutions of linear Caputo-Fabrizio differential equations with fuzzy coefficients and of the corresponding initial value problems. Some of the topics which are needed for the results of this study from the point of view of the cross product of fuzzy numbers have been explained in detail. We illustrate our technique and compare the effect of uncertainty of the coefficients and initial value on the related solutions.

Nonlinear oscillations are an essential fact in physical science, mechanical structures, and other engineering problems. Some of the popular analytical solutions to analyze nonlinear differential equations governing the behavior of strongly nonlinear oscillators are the Energy Balance Method (EBM), and He's Amplitude Frequency Formulation (HAFF). The lack of precision and accuracy despite needing several computational steps to resolve the system frequency is the main demerit of these methods. This research creates a novel analytical approximation approach with a very efficient algorithm that can perform the calculation procedure much easier and with much higher accuracy than classic EBM and HAFF. The presented method's steps rely on Hamiltonian relations described in EBM and the de ned relationship between frequency and amplitude in HAFF. This paper demonstrates the substantial precision of the presented method compared to common EBM and HAFF applied in different and well-known engineering phenomena. For instance, the approximate solutions of the equations govern some strongly nonlinear oscillators, including the two-massspring systems, buckling of a column, and duffing relativistic oscillators are presented here. Subsequently, their results are compared with the Runge-Kutta method and exact solutions obtained from the previous research. The proposed novel approach resultant error percentages show an excellent agreement with the numerical solutions and illustrate a very quickly convergent and more precise than mentioned typical methods.

This article presents a numerical treatment of the singularly perturbed delay reaction diffusion problem with an integral boundary condition. In the considered problem, a small parameter ", is multiplied on the higher order derivative term. The presence of this parameter causes the existence of boundary layers in the solution. The solution also exhibits an interior layer because of the large spatial delay. Simpson's 1/3 rule is applied to approximate the integral boundary condition given on the right end plane. A standard finite difference scheme on piecewise uniform Shishkin mesh is proposed to discretize the problem in the spatial direction, and the Crank-Nicolson method is used in the temporal direction. The developed numerical scheme is parameter uniformly convergent, with nearly two orders of convergence in space and two orders of convergence in time. Two numerical examples are considered to validate the theoretical results.

The present study focuses on the two new hybrid methods variational iteration J-transform technique (J-VIT) and J-transform method with optimal homotopy analysis (OHAJTM) for analytical assessment of space-time fractional Fokker-Planck equations (STF-FPE), appearing in many realistic physical situations, e.g., in ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of Ribonucleic acid, deoxyribonucleic acid, continuous random movement, and formation of wave patterns. OHAJTM is developed via optimal homotopy analysis after implementing the properties of J-transform while (J-VIT) is produced by implementing properties of the J-transform and the theory of variational iteration. Banach approach is utilized to analyze the convergence of these methods. In addition, it is demonstrated that J-VIT is T-stable. Computed new approximations are reported as a closed form expression of the Mittag-Leffler function, and in addition, the effectiveness/validity of the proposed new approximations is demonstrated via three test problems of STF-FPE by computing the error norms: L2 and absolute errors. The numerical assessment demonstrates that the developed techniques perform better for STF-FPE and for a given iteration, and OHAJTM produces new approximations with better accuracy as compared to J-VIT as well as the techniques developed recently.

In this paper, we propose an explicit diffuse the split-step truncated Euler-Maruyama (DSSTEM) method for stochastic differential equations with non-global Lipschitz coefficients. We investigate the strong convergence of the new method under local Lipschitz and Khasiminskii-type conditions. We show that the newly proposed method achieves a strong convergence rate arbitrarily close to half under some additional conditions. Finally, we illustrate the efficiency and performance of the proposed method with numerical results.

In this study,  firstly, the residual method, which was developed for initial value problems, is improved to find unknown coefficients without requiring for any system solution. Later, the adaptation of improved residual method is given to find approximate solutions of boundary value problems. Finally, the method improved and adapted for boundary value problems is used to find both critical eigenvalue and eigenfunctions of the one-dimensional Bratu problem. The most significant advantage of the method is finding approximate solutions of nonlinear problems without any linearization or solving any system of equations. Error analysis of the adapted method is given and an upper bound on the approximation error is derived for the eigenfunctions. The numerical results obtained are compared with the theoretical findings. Comparisons and theoretical observations show that the improved and adapted method is very convenient and successful in solving boundary value problems and eigenvalue problems approximately with high accuracy.

This manuscript investigates a computational method based on fractional-order Fibonacci functions (FFFs) for solving distributed-order (DO) fractional differential equations and DO time-fractional diffusion equations. Extra DO fractional derivative operator and pseudo-operational matrix of fractional integration for FFFs are proposed. To evaluate the unknown coefficients in the FFF expansion, utilizing the matrices, an optimization problem relating to considered equations is formulated. This approach converts the original problems into a system of algebraic equations. The approximation error is proposed. Several problems are proposed to investigate the applicability and computational efficiency of the scheme. The approximations achieved by some existing schemes are also tested conforming to the efficiency of the present method. Also, the model of the motion of the DO fractional oscillator is solved, numerically.

This paper addresses the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with fractional derivative definition. Initially, conformable derivative definitions and their features are presented. Then, by submitting exp({φ())-expansion, generalized (G′=G)-expansion, and Modified Kudryashov methods, exact solutions of this equation are generated. The 3D, contour, and 2D surfaces, as well as the related contour plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The physical meaning of the geometrical structures for some of these solutions is discussed. For the observation of the physical activities of the problem, achieved exact solutions are vital. The acquired results can help to demonstrate the physical application of the investigated models and other nonlinear physical models found in mathematical physics. Therefore, it would appear that these approaches might yield noteworthy results in producing the exact solutions to fractional differential equations in a wide range.

In this paper, we deal with second-order fuzzy linear two-point boundary value problems (BVP) under Hukuhara derivatives. Considering the first-order and second-order Hukuhara derivatives, four types of fuzzy linear two-point BVPs can be obtained where each may or may not have a solution. Therefore a fuzzy two-point (BVP) may have one, two, three, or four different kinds of solutions concerning this kind of derivative. To solve this fuzzy linear two-point (BVP), we convert each to two cases of crisp boundary value problems. We apply a standard method(numerical or analytical) to solve crisp two-point BVPs in their domain. Subsequently, the crisp solutions are combined to obtain a fuzzy solution to the fuzzy problems, and the solutions are checked to see if they satisfy the fuzzy issues. Conditions are presented under which fuzzy problems have the fuzzy solution and illustrated with some examples.

In this paper, a spectral collocation method for solving nonlinear pantograph type delay differential equations is presented. The basis functions used for the spectral analysis are based on Chebyshev, Legendre, and Jacobi polynomials. By using the collocation points and operations matrices of required functions such as derivative functions and delays of unknown functions, the method transforms the problem into a system of nonlinear algebraic equations. The solutions of this nonlinear system determine the coefficients of the assumed solution. The method is explained by numerical examples and the results are compared with the available methods in the literature. It is seen from the applications that our method gives more efficient results than that of the reported methods.

A mathematical modeling of contaminate transportation has been presented in the current paper. The time-dependent dispersion has been considered in the transportation of contaminant in a  nite homogeneous porous medium. The study of contaminants concentration has been presented for the uniform unsteady flow of ground-water. Instead of a constant dispersion, in order to consider the effect of groundwater velocity on contaminant transportation, dispersion has been considered as a groundwater velocity-dependent quantity. As found in the many practical aspects, a linear increase in concentration at a source point with time has been assumed for the present modeling. The spread of the initial contaminant concentration has been considered linearly decreasing along the direction of one-dimensional  flow. The contaminant transport equation for the above-mentioned conditions and environment has been solved. The Laplace transform variation iteration method (LVIM) has been adopted to obtain a solution. Spatial and temporal variations of concentration for a developed model have been presented graphically by varying dispersion. The LVIM has been found suitable for the present study of contaminant transport modeling. The MATHEMATICA package has been used for the present study.

This study deals with a numerical solution of a nonlinear Volterra integral equation of the first kind. The method of this research is based on a new kind of orthogonal wavelets, called the Chebyshev cardinal wavelets. These wavelets known as new basis functions contain numerous beneficial features like orthogonality, spectral accuracy, and cardinality. In addition, we assume an expansion of the terms of Chebyshev cardinal wavelets within unknown coefficients as a substitute for an unknown solution. Relatively, considering the mentioned expansion and the cardinality feature within the generated operational matrix of the introduced wavelets, a system of nonlinear algebraic equations is extracted for the stated problem. Finally, by solving the yielded system, the estimated solution results.

In this article, the spectral collocation method based on radial basis functions is used to solve the mentioned models. The advantage of this method is that it converts the equations into a system of algebraic equations. Therefore, we can solve this problem with Newton's method. The purpose of this article is to numerically solve stochastic models such as the Heston model, Vasicek model, Cox-Ingersoll and Ross model, and a model of the Black-Scholes called the Genral Stock model. The method is computationally attractive, and numerical examples confirm the validity and efficiency of the proposed method.

In this work, a non-classical sinc-collocation method is used to  find numerical solution of third-order boundary value problems. The novelty of this approach is based on using the weight functions in the traditional sinc- expansion. The properties of sinc-collocation are used to reduce the boundary value problems to a nonlinear system of algebraic equations which can be solved numerically. In addition, the convergence of the proposed method is discussed by preparing the theorems which show exponential convergence and guarantee its applicability. Several examples are solved and the numerical results show the efficiency and applicability of the method.