Research

12 pages, 292 KiB

Open AccessArticle

On the Positive Decreasing Solutions of Half-Linear Delay Differential Equations of Even Order

by Ghada AlNemer, Waed Muhsin, Osama Moaaz and Elmetwally M. Elabbasy

Mathematics 2023, 11(6), 1282; https://doi.org/10.3390/math11061282 - 07 Mar 2023

Viewed by 700

In this paper, we derive new properties for the decreasing positive solutions of half-linear delay differential equations of even order. The positive-decreasing solutions have a great influence on the study of qualitative properties, which include oscillation, convergence, etc.; therefore, we take care of [...] Read more.

In this paper, we derive new properties for the decreasing positive solutions of half-linear delay differential equations of even order. The positive-decreasing solutions have a great influence on the study of qualitative properties, which include oscillation, convergence, etc.; therefore, we take care of finding sufficient conditions to exclude these solutions. In addition, we present new criteria for testing the oscillation of the studied equation. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

13 pages, 300 KiB

Open AccessArticle

Mean-Field and Anticipated BSDEs with Time-Delayed Generator

by Pei Zhang, Nur Anisah Mohamed and Adriana Irawati Nur Ibrahim

Mathematics 2023, 11(4), 888; https://doi.org/10.3390/math11040888 - 09 Feb 2023

Cited by 3 | Viewed by 1138

Abstract

In this paper, we discuss a new type of mean-field anticipated backward stochastic differential equation with a time-delayed generator (MF-DABSDEs) which extends the results of the anticipated backward stochastic differential equation to the case of mean-field limits, and in which the generator considers [...] Read more.

In this paper, we discuss a new type of mean-field anticipated backward stochastic differential equation with a time-delayed generator (MF-DABSDEs) which extends the results of the anticipated backward stochastic differential equation to the case of mean-field limits, and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we shall demonstrate the existence and uniqueness of the solutions to these equations. Finally, we shall establish a comparison theorem for the solutions. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

11 pages, 841 KiB

Open AccessArticle

Two Analytical Techniques for Fractional Differential Equations with Harmonic Terms via the Riemann–Liouville Definition

by Ragwa S. E. Alatwi, Abdulrahman F. Aljohani, Abdelhalim Ebaid and Hind K. Al-Jeaid

Mathematics 2022, 10(23), 4564; https://doi.org/10.3390/math10234564 - 02 Dec 2022

Cited by 1 | Viewed by 1327

Abstract

This paper considers a class of non-homogeneous fractional systems with harmonic terms by means of the Riemann–Liouville definition. Two different approaches are applied to obtain the dual solution of the studied class. The first approach uses the Laplace transform (LT) and the solution [...] Read more.

This paper considers a class of non-homogeneous fractional systems with harmonic terms by means of the Riemann–Liouville definition. Two different approaches are applied to obtain the dual solution of the studied class. The first approach uses the Laplace transform (LT) and the solution is given in terms of the Mittag-Leffler functions. The second approach avoids the LT and expresses the solution in terms of exponential and periodic functions which is analytic in the whole domain. The current methods determine the solution directly and efficiently. The results are applicable for other problems of higher order. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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9 pages, 261 KiB

Open AccessEditor’s ChoiceArticle

Blow-Up Time of Solutions for a Parabolic Equation with Exponential Nonlinearity

by Yanjin Wang and Jianzhen Qian

Mathematics 2022, 10(16), 2887; https://doi.org/10.3390/math10162887 - 12 Aug 2022

Viewed by 1275

Abstract

This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed [...] Read more.

This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed the solution’s lower and upper bound of the blow-up time for the equation. Our results complement the existing results. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

11 pages, 259 KiB

Open AccessArticle

A Novel Projection Method for Cauchy-Type Systems of Singular Integro-Differential Equations

by Saeed Althubiti and Abdelaziz Mennouni

Mathematics 2022, 10(15), 2694; https://doi.org/10.3390/math10152694 - 29 Jul 2022

Cited by 4 | Viewed by 1351

Abstract

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution [...] Read more.

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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11 pages, 279 KiB

Open AccessArticle

Semi-Hyers–Ulam–Rassias Stability via Laplace Transform, for an Integro-Differential Equation of the Second Order

by Daniela Inoan and Daniela Marian

Mathematics 2022, 10(11), 1893; https://doi.org/10.3390/math10111893 - 01 Jun 2022

Cited by 4 | Viewed by 1301

Abstract

The Laplace transform method is applied to study the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of the second order. A general equation is formulated first; then, some particular cases for the function from the kernel are considered. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

12 pages, 285 KiB

Open AccessArticle

New Monotonic Properties of Positive Solutions of Higher-Order Delay Differential Equations and Their Applications

by Ali Muhib, Osama Moaaz, Clemente Cesarano, Shami A. M. Alsallami, Sayed Abdel-Khalek and Abd Elmotaleb A. M. A. Elamin

Mathematics 2022, 10(10), 1786; https://doi.org/10.3390/math10101786 - 23 May 2022

Cited by 6 | Viewed by 1603

Abstract

In this work, new criteria were established for testing the oscillatory behavior of solutions of a class of even-order delay differential equations. We follow an approach that depends on obtaining new monotonic properties for the decreasing positive solutions of the studied equation. Moreover, [...] Read more.

In this work, new criteria were established for testing the oscillatory behavior of solutions of a class of even-order delay differential equations. We follow an approach that depends on obtaining new monotonic properties for the decreasing positive solutions of the studied equation. Moreover, we use these properties to provide new oscillation criteria of an iterative nature. We provide an example to support the significance of the results and compare them with the related previous work. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

7 pages, 251 KiB

Open AccessArticle

Hyers–Ulam–Rassias Stability of Hermite’s Differential Equation

by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu

Mathematics 2022, 10(6), 964; https://doi.org/10.3390/math10060964 - 17 Mar 2022

Cited by 2 | Viewed by 1245

Abstract

In this paper, we studied the Hyers–Ulam–Rassias stability of Hermite’s differential equation, using Pachpatte’s inequality. We compared our results with those obtained by Blaga et al. Our estimation for

|z (x) - y (x)|

, where z is an approximate solution and y [...] Read more.

In this paper, we studied the Hyers–Ulam–Rassias stability of Hermite’s differential equation, using Pachpatte’s inequality. We compared our results with those obtained by Blaga et al. Our estimation for

|z (x) - y (x)|

, where z is an approximate solution and y is an exact solution of Hermite’s equation, was better than that obtained by the authors previously mentioned, in some parts of the domain, especially in a neighborhood of the origin. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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9 pages, 1719 KiB

Open AccessEditor’s ChoiceArticle

The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method

by Farah M. Al-Askar, Wael W. Mohammed, Abeer M. Albalahi and Mahmoud El-Morshedy

Mathematics 2022, 10(5), 817; https://doi.org/10.3390/math10050817 - 04 Mar 2022

Cited by 30 | Viewed by 2080

Abstract

The stochastic (2+1)-dimensional breaking soliton equation (SBSE) is considered in this article, which is forced by the Wiener process. To attain the analytical stochastic solutions such as the polynomials, hyperbolic and trigonometric functions of the SBSE, we use the tanh–coth method. The results [...] Read more.

The stochastic (2+1)-dimensional breaking soliton equation (SBSE) is considered in this article, which is forced by the Wiener process. To attain the analytical stochastic solutions such as the polynomials, hyperbolic and trigonometric functions of the SBSE, we use the tanh–coth method. The results provided here extended earlier results. In addition, we utilize Matlab tools to plot 2D and 3D graphs of analytical stochastic solutions derived here to show the effect of the Wiener process on the solutions of the breaking soliton equation. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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9 pages, 255 KiB

Open AccessEditor’s ChoiceArticle

Semi-Hyers–Ulam–Rassias Stability of the Convection Partial Differential Equation via Laplace Transform

by Daniela Marian

Mathematics 2021, 9(22), 2980; https://doi.org/10.3390/math9222980 - 22 Nov 2021

Cited by 12 | Viewed by 1841

Abstract

In this paper, we study the semi-Hyers–Ulam–Rassias stability and the generalized semi-Hyers–Ulam–Rassias stability of some partial differential equations using Laplace transform. One of them is the convection partial differential equation. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

25 pages, 596 KiB

Open AccessArticle

Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

by Natalia P. Bondarenko and Andrey V. Gaidel

Mathematics 2021, 9(20), 2617; https://doi.org/10.3390/math9202617 - 17 Oct 2021

Cited by 5 | Viewed by 1897

Abstract

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general [...] Read more.

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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17 pages, 460 KiB

Open AccessArticle

The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations

by Bogdan Căruntu and Mădălina Sofia Paşca

Mathematics 2021, 9(18), 2324; https://doi.org/10.3390/math9182324 - 19 Sep 2021

Viewed by 1909

Abstract

We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the [...] Read more.

We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the simplicity and accuracy of the method. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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16 pages, 1778 KiB

Open AccessArticle

Arbitrary Coefficient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays

by Vasilii Zaitsev and Inna Kim

Mathematics 2021, 9(17), 2158; https://doi.org/10.3390/math9172158 - 04 Sep 2021

Cited by 3 | Viewed by 1620

Abstract

We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and [...] Read more.

We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function by linear static output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for the solvability of the arbitrary coefficient assignment problem by the static output feedback controller. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We provide an example illustrating our results. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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10 pages, 323 KiB

Open AccessArticle

Application of Said Ball Curve for Solving Fractional Differential-Algebraic Equations

by Fateme Ghomanjani and Samad Noeiaghdam

Mathematics 2021, 9(16), 1926; https://doi.org/10.3390/math9161926 - 12 Aug 2021

Cited by 5 | Viewed by 1987

Abstract

The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is [...] Read more.

The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is proved. Some examples are presented to show the efficiency and accuracy of the method. Based on the obtained results, the SBC is more accurate than the Bezier curve method. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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24 pages, 835 KiB

Open AccessArticle

Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws

by Shijian Lin, Qi Luo, Hongze Leng and Junqiang Song

Mathematics 2021, 9(16), 1885; https://doi.org/10.3390/math9161885 - 08 Aug 2021

Cited by 4 | Viewed by 2181

Abstract

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by [...] Read more.

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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18 pages, 343 KiB

Open AccessFeature PaperArticle

Local Dynamics of Logistic Equation with Delay and Diffusion

by Sergey Kashchenko

Mathematics 2021, 9(13), 1566; https://doi.org/10.3390/math9131566 - 03 Jul 2021

Cited by 3 | Viewed by 2172

Abstract

The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small [...] Read more.

The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of the boundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

22 pages, 11348 KiB

Open AccessArticle

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

by Chih-Yu Liu, Cheng-Yu Ku, Li-Dan Hong and Shih-Meng Hsu

Mathematics 2021, 9(13), 1535; https://doi.org/10.3390/math9131535 - 30 Jun 2021

Cited by 4 | Viewed by 1976

Abstract

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving [...] Read more.

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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9 pages, 269 KiB

Open AccessArticle

Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

by Shyam Sundar Santra, Rami Ahmad El-Nabulsi and Khaled Mohamed Khedher

Mathematics 2021, 9(12), 1323; https://doi.org/10.3390/math9121323 - 08 Jun 2021

Cited by 6 | Viewed by 2010

Abstract

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. [...] Read more.

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

8 pages, 227 KiB

Open AccessArticle

Generalization of Quantum Ostrowski-Type Integral Inequalities

by Muhammad Aamir Ali, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2021, 9(10), 1155; https://doi.org/10.3390/math9101155 - 20 May 2021

Cited by 6 | Viewed by 1695

Abstract

In this paper, we prove some new Ostrowski-type integral inequalities for q-differentiable bounded functions. It is also shown that the results presented in this paper are a generalization of know results in the literarure. Applications to special means are also discussed. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

13 pages, 933 KiB

Open AccessArticle

On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

by Haifa Bin Jebreen and Fairouz Tchier

Mathematics 2021, 9(1), 78; https://doi.org/10.3390/math9010078 - 31 Dec 2020

Viewed by 1951

Abstract

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the

θ

-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a [...] Read more.

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the

θ

-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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27 pages, 493 KiB

Open AccessArticle

Nonlinear Observability Algorithms with Known and Unknown Inputs: Analysis and Implementation

by Nerea Martínez and Alejandro F. Villaverde

Mathematics 2020, 8(11), 1876; https://doi.org/10.3390/math8111876 - 29 Oct 2020

Cited by 6 | Viewed by 2765

Abstract

The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence, the availability of computational [...] Read more.

The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence, the availability of computational tools capable of analysing the observability of nonlinear systems with unknown inputs has been limited until lately. Two symbolic algorithms based on differential geometry, ORC-DF and FISPO, have been recently proposed for this task, but their critical analysis and comparison is still lacking. Here we perform an analytical comparison of both algorithms and evaluate their performance on a set of problems, while discussing their strengths and limitations. Additionally, we use these analyses to provide insights about certain aspects of the relationship between inputs and observability. We found that, while ORC-DF and FISPO follow a similar approach, they differ in key aspects that can have a substantial influence on their applicability and computational cost. The FISPO algorithm is more generally applicable, since it can analyse any nonlinear ODE model. The ORC-DF algorithm analyses models that are affine in the inputs, and if those models have known inputs it is sometimes more efficient. Thus, the optimal choice of a method depends on the characteristics of the problem under consideration. To facilitate the use of both algorithms, we implemented the ORC-DF condition in a new version of STRIKE-GOLDD, a MATLAB toolbox for structural identifiability and observability analysis. Since this software tool already had an implementation of the FISPO algorithm, the new release allows modellers and model users the convenience of choosing between different algorithms in a single tool, without changing the coding of their model. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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20 pages, 599 KiB

Open AccessArticle

Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

by Innocent Simbanefayi and Chaudry Masood Khalique

Mathematics 2020, 8(6), 1012; https://doi.org/10.3390/math8061012 - 20 Jun 2020

Cited by 5 | Viewed by 1956

Abstract

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general [...] Read more.

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

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