Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

(SI10-123) Comparison Between the Homotopy Perturbation Method and Variational Iteration Method for Fuzzy Differential Equations

In this article, the authors discusses the numerical simulations of higher-order differential equations under a fuzzy environment by using Homotopy Perturbation Method and Variational Iteration Method. The fuzzy parameter and variables are represented by triangular fuzzy convex normalized sets. Comparison of the results are obtained by the homotopy perturbation method with those obtained by the variational iteration method. Examples are provided to demonstrate the theory

Solutions of fractional gas dynamics equation by a new technique

[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Akgül, A.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). Solutions of fractional gas dynamics equation by a new technique. Mathematical Methods in the Applied Sciences. 43(3):1349-1358. https://doi.org/10.1002/mma.5950S13491358433Singh, J., Kumar, D., & Kılıçman, A. (2013). Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform. Abstract and Applied Analysis, 2013, 1-8. doi:10.1155/2013/934060Momani, S. (2005). Analytic and approximate solutions of the space- and time-fractional telegraph equations. Applied Mathematics and Computation, 170(2), 1126-1134. doi:10.1016/j.amc.2005.01.009Hajipour, M., Jajarmi, A., Baleanu, D., & Sun, H. (2019). On an accurate discretization of a variable-order fractional reaction-diffusion equation. Communications in Nonlinear Science and Numerical Simulation, 69, 119-133. doi:10.1016/j.cnsns.2018.09.004Meng, R., Yin, D., & Drapaca, C. S. (2019). Variable-order fractional description of compression deformation of amorphous glassy polymers. Computational Mechanics, 64(1), 163-171. doi:10.1007/s00466-018-1663-9Baleanu, D., Jajarmi, A., & Hajipour, M. (2018). On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dynamics, 94(1), 397-414. doi:10.1007/s11071-018-4367-yJajarmi, A., & Baleanu, D. (2018). A new fractional analysis on the interaction of HIV withCD4+T-cells. Chaos, Solitons & Fractals, 113, 221-229. doi:10.1016/j.chaos.2018.06.009Baleanu, D., Jajarmi, A., Bonyah, E., & Hajipour, M. (2018). New aspects of poor nutrition in the life cycle within the fractional calculus. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1684-xJajarmi, A., & Baleanu, D. (2017). Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control, 24(12), 2430-2446. doi:10.1177/1077546316687936Singh, J., Kumar, D., & Baleanu, D. (2018). On the analysis of fractional diabetes model with exponential law. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1680-1Kumar, D., Singh, J., Tanwar, K., & Baleanu, D. (2019). A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. International Journal of Heat and Mass Transfer, 138, 1222-1227. doi:10.1016/j.ijheatmasstransfer.2019.04.094Kumar, D., Singh, J., Al Qurashi, M., & Baleanu, D. (2019). A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Advances in Difference Equations, 2019(1). doi:10.1186/s13662-019-2199-9Kumar, D., Singh, J., Purohit, S. D., & Swroop, R. (2019). A hybrid analytical algorithm for nonlinear fractional wave-like equations. Mathematical Modelling of Natural Phenomena, 14(3), 304. doi:10.1051/mmnp/2018063Kumar, D., Tchier, F., Singh, J., & Baleanu, D. (2018). An Efficient Computational Technique for Fractal Vehicular Traffic Flow. Entropy, 20(4), 259. doi:10.3390/e20040259Goswami, A., Singh, J., Kumar, D., & Sushila. (2019). An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A: Statistical Mechanics and its Applications, 524, 563-575. doi:10.1016/j.physa.2019.04.058Mohyud-Din, S. T., Bibi, S., Ahmed, N., & Khan, U. (2018). Some exact solutions of the nonlinear space–time fractional differential equations. Waves in Random and Complex Media, 29(4), 645-664. doi:10.1080/17455030.2018.1462541Momani, S., & Shawagfeh, N. (2006). Decomposition method for solving fractional Riccati differential equations. Applied Mathematics and Computation, 182(2), 1083-1092. doi:10.1016/j.amc.2006.05.008Hashim, I., Abdulaziz, O., & Momani, S. (2009). Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14(3), 674-684. doi:10.1016/j.cnsns.2007.09.014Yıldırım, A. (2010). He’s homotopy perturbation method for solving the space- and time-fractional telegraph equations. International Journal of Computer Mathematics, 87(13), 2998-3006. doi:10.1080/00207160902874653Momani, S., & Odibat, Z. (2007). Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons & Fractals, 31(5), 1248-1255. doi:10.1016/j.chaos.2005.10.068Rida, S. Z., El-Sayed, A. M. A., & Arafa, A. A. M. (2010). On the solutions of time-fractional reaction–diffusion equations. Communications in Nonlinear Science and Numerical Simulation, 15(12), 3847-3854. doi:10.1016/j.cnsns.2010.02.007Machado, J. A. T., & Mata, M. E. (2014). A fractional perspective to the bond graph modelling of world economies. Nonlinear Dynamics, 80(4), 1839-1852. doi:10.1007/s11071-014-1334-0Raja Balachandar, S., Krishnaveni, K., Kannan, K., & Venkatesh, S. G. (2018). Analytical Solution for Fractional Gas Dynamics Equation. National Academy Science Letters, 42(1), 51-57. doi:10.1007/s40009-018-0662-xWang, Y.-L., Liu, Y., Li, Z., & zhang, H. (2018). Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method. International Journal of Computer Mathematics, 96(3), 585-593. doi:10.1080/00207160.2018.1455091Gumah, G. N., Naser, M. F. M., Al-Smadi, M., & Al-Omari, S. K. (2018). Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1937-8Al-Smadi, M. (2018). Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation. Ain Shams Engineering Journal, 9(4), 2517-2525. doi:10.1016/j.asej.2017.04.006Kashkari, B. S. H., & Syam, M. I. (2018). Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation. Complexity, 2018, 1-7. doi:10.1155/2018/2304858Akgül, A., & Grow, D. (2019). Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces. Differential Equations and Dynamical Systems, 28(3), 715-744. doi:10.1007/s12591-019-00453-3Akgül, A., Khan, Y., Akgül, E. K., Baleanu, D., & Al Qurashi, M. M. (2017). Solutions of nonlinear systems by reproducing kernel method. The Journal of Nonlinear Sciences and Applications, 10(08), 4408-4417. doi:10.22436/jnsa.010.08.33Karatas Akgül, E. (2018). Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(2), 145-151. doi:10.11121/ijocta.01.2018.00568Akgül, A., Inc, M., & Karatas, E. (2015). Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 8(6), 1055-1064. doi:10.3934/dcdss.2015.8.1055Akgül, A., Inc, M., Karatas, E., & Baleanu, D. (2015). Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Advances in Difference Equations, 2015(1). doi:10.1186/s13662-015-0558-8Aronszajn, N. (1950). Theory of reproducing kernels. Transactions of the American Mathematical Society, 68(3), 337-337. doi:10.1090/s0002-9947-1950-0051437-7Bergman, S. (1950). The Kernel Function and Conformal Mapping. Mathematical Surveys and Monographs. doi:10.1090/surv/005Inc, M., & Akgül, A. (2014). Approximate solutions for MHD squeezing fluid flow by a novel method. Boundary Value Problems, 2014(1). doi:10.1186/1687-2770-2014-18Inc, M., Akgül, A., & Geng, F. (2014). Reproducing Kernel Hilbert Space Method for Solving Bratu’s Problem. Bulletin of the Malaysian Mathematical Sciences Society, 38(1), 271-287. doi:10.1007/s40840-014-0018-8Wang, Y., & Chao, L. (2008). Using reproducing kernel for solving a class of partial differential equation with variable-coefficients. Applied Mathematics and Mechanics, 29(1), 129-137. doi:10.1007/s10483-008-0115-yWu, B. Y., & Li, X. Y. (2011). A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Applied Mathematics Letters, 24(2), 156-159. doi:10.1016/j.aml.2010.08.036Yao, H., & Lin, Y. (2009). Solving singular boundary-value problems of higher even-order. Journal of Computational and Applied Mathematics, 223(2), 703-713. doi:10.1016/j.cam.2008.02.01

MHD free convective flow past a vertical plate

The free convective flow in incompressible viscous fluid past a vertical plate is studied under the presence of magnetic field. The flow is considered along the vertical plate at x-axis in upward direction and y-axis is taken normal to it. The governing equations are written in vector form. Afterwards, the equations are solved numerically using finite element method with automated solution techniques. Later, the effects of magnetic field strength to the velocity and temperature of the fluid are obtained. It is found that for heated plate, the velocity and the temperature of the fluid decreases when the magnetic field strength increases. Meanwhile for cooled plate, the velocity decreases but the temperature increases when the magnetic field strength increases

Numerical Solution of First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application

The numerical algorithm for solving “first-order linear differential equation in fuzzy environment” is discussed. A scheme, namely, “Runge-Kutta-Fehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i)-gH and (ii)-gH differential (exact solutions concepts) system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio