Solutions Of Differential Equations Research Articles

Novel Multivariate Integral Operators Incorporating Trigonometric Transformations

The creation of new nonlinear multivariate integral operators is motivated by the need for mathematical tools that can handle the complex interdependencies that naturally arise in contemporary applications. From an abstract scientific point of view, it is also necessary to develop new operator theories beyond existing ones to offer original research perspectives. This article contributes to these complementary aspects. We present two nonlinear multivariate integral operators that have the particularity of incorporating trigonometric transformations of the main function. Thanks to their trigonometric nature, they completely stand out from existing operators, offering a new and complete framework. Therefore, we take advantage of advanced mathematical techniques for trigonometric functions to address the challenges they pose. In particular, we show that they have manageable integrals and series expansions, that they are solutions of specific differential and functional equations, and that they are involved in general inequalities of various types (Hölder-type, convex-type, etc.). In the application part, we use some of these properties to propose a wide collection of trigonometric inequalities that are both original and precise. Figures are produced to illustrate them for a direct visual check.

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Fixed Point Theorems: Exploring Applications in Fractional Differential Equations for Economic Growth

The aim of this research is to introduce two new notions, Θ-(Ξ,h)-contraction and rational (α,η)-ψ-interpolative contraction, in the setting of F-metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model.

Functional Solutions of Stochastic Differential Equations

We present an integration condition ensuring that a stochastic differential equation dXt=μ(t,Xt)dt+σ(t,Xt)dBt, where μ and σ are sufficiently regular, has a solution of the form Xt=Z(t,Bt). By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form Xt=Z(t,Yt), with Yt an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green’s Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point (t,x), we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process.

The Solution of Caputo Fractional Partial Differential Equations By Sumudu Transform

In this paper, we obtained an explicit solutions of the fractional diffusion-wave equations involving partial Caputo fractional derivative by using Sumudu transform. The solutions of the fractional diffusion-wave equations obtained in terms of Mittag-Leffler function and generalized Wright function.Some illustrative examples are also given.

Fractional calculus of interval-valued functions on time scales

This paper aims to develop fractional interval-valued calculus on time scales that unify the continuous and discrete cases. The definitions of fractional interval-valued calculus, which encompass the integral, Riemann–Liouville (R-L) derivative, and Caputo derivative of nabla type, are established within the framework of time scales. Moreover, some fundamental properties and nabla Laplace transform of these new operators are discussed. Based on these findings, explicit solutions of several nabla fractional interval-valued differential equations on time scales are presented.

SOME NEW TYPES OF GRONWALL-BELLMAN INEQUALITY ON FRACTAL SET

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature.

Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Fractional order iterative boundary value problem

In the present paper, we establish the existence and uniqueness solution for an iterative differential equation involving caputo fractional derivative of order $1\alpha2$. The existence results is proved by using Schauder fixed point. To prove the extremal solution, we demonstrate some fractional inequalities. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained result.

Fareeha Transform Performance In Solving Fractional Differential Telegraph Equations Combining Adomian Decomposition Method

Transformations have successfully outperformed a significant role in solving differential equations and have been applied in large-scale aspects of science. Fareeha transform has been illustrated effectively in data compression based on containing more information of the transform. In this paper, we expand the fractional Fareeha transform in the Caputo derivative sense combining the Adomian Decomposition Method to seek the solutions of fractional differential telegraph equations. The results of practical utilization have also been significantly shown successful in solving fractional telegraph differential equations.

Novel results for separate families of fuzzy-dominated mappings satisfying advanced locally contractions in b-multiplicative metric spaces with applications

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work.

Solutions for the unsteady motion of porous elastic solids within the context of an implicit constitutive theory

In this work we investigate well-posedness for the partial differential equation stemming from the balance of linear momentum for an implicit constitutive relation that can describe the response of porous elastic solids whose material moduli depend on the density. We study heteroclinic travelling waves and obtain closed form analytic solutions for the resulting ordinary differential equation. Finally, we consider some special solutions for the partial differential equation and solve the resulting equations numerically.

Numerical analysis of inclined magnetic force impact on cross liquid flowing with widening shallow and heat generating

ABSTRACT The creation, utilisation, conversion, and exchange of thermal energy are all covered under the thermal engineering subject of heat transfer. Heat transfer plays a major role in engineering and industrial sectors like food packaging, food additive manufacturing, electronic cooling, microturbine, etc. Scientists across the world have tried to explore the impact of heat transfer on fluid past an expandable surface due to its exciting applications in industries like polymer production, paper production, crystal glass manufacturing, etc. This article has unique exploration, and it emphasises interpreting the Cross fluid flow along stretching surface with convective heat transport analysis by using the Lie group methodology. We investigated the implication of inclined upper-convected Maxwell fluid magnetohydrodynamic (MHD) movement through a penetrable strained plate. To solve nonlinear differential equations and clearly determine their absolute invariants, the Lie group approach is introduced. For the numerical solution of non-linear ordinary differential equations with boundary conditions, the fourth-order Runge–Kutta procedure with shelling performance is utilised. Plots showing the impacts of motivated and non-motivated MHD are researched and examined for the alterations of many parameters, and statistical graphs are given to explain the physical values. It is evident that opposing behaviour occurs in connection to temperature captivation and generation parameters together with motivated and non-motivated MHD. Under an amplified, heat-generating parameter and Deborah number, the rate of heat transmission increases. The Lie group investigation of Cross liquid in the presence/absence of inclined and non-inclined has not been investigated yet in the available literature.

New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations

This work proposes a novel modified (G’/G2)-expansion method to construct the exact travelling wave solutions of three famous nonlinear partial differential equations namely, nanoionic currents along microtubules equation, Benney-Luke equation and the generalized Hirota-Satsuma coupled KdV system. In the result, these solutions can be divided into different types i.e. trigonometric, rational and exponential solutions. The computational tool Maple is used to arrive at these solutions. Applying the novel modified (G’/G2)-expansion approach yields results that are trustworthy, easy to understand, and useful in a variety of disciplines including physics, biology, finance, engineering, statistics, and more. Software is used to create the two- and three-dimensional graphs. The comparison of the obtained solutions is also given in the form of Tables 1, 2 and 3, to show the novelty of our obtained solutions.

Characteristic Mode Solution of Complex-Coefficient Complex-Solution Differential Equations

Computation of complex-coefficient complex-solution differential equations is a problem that arises in various domains of science and engineering. This paper aims at applying the Theory of Characteristic Modes (TCM) approach along with the Method of Moments (MoM) in solving these problems with emphasis on procedures for higher differential equations. Several available methods, known in literatures, are available for solving the problem. The complexity of the available methods differs based on the accuracy of the solution. In this paper, the general method is first presented and then a simplified version of it is proposed to solve high order differential equations. Two examples are illustrated, a third and a fourth order complex-coefficients complex-solution differential equations, to show the simplicity of the proposed method. The proposed approach can be also introduced along with other methods to solve these special occurrences differential equations and other boundary value problems.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L1 space is presented for the studied models using the proposed scheme.

On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions

Abstract This study investigates the existence of solutions for implicit fractional differential equations with fractional-order integral boundary conditions. We create the required conditions to ensure unique solution and Ulam-Hyers-Rassias stability. We also give examples to highlight the major findings.

Solving the fractional Fornberg-Whitham equation within Caputo framework using the optimal auxiliary function method

In this work, we solve the fractional-order Fornberg-Whitham (FW) problem in the context of the Caputo operator by using the Optimal Auxiliary Function Method. Tables and figures showing full numerical findings indicate the correctness and efficacy of this strategy. The results provide insights into the solution behavior of the FW equation and demonstrate the applicability of the Optimal Auxiliary Function Method. By giving insight on the behavior of the FW equation in a fractional context, this research advances the use of fractional calculus techniques in the solution of complicated differential equations.

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

Sensitivity and wave propagation analysis of the time-fractional (3+1)-dimensional shallow water waves model

AbstractThis study investigates the exact solutions of the time-fractional (3+1)-dimensional combined Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM) equation. The considered model describes the long surface gravity waves of small amplitude, which portrays the two-way propagation of waves. The modified generalized Kudryashov method and the exp(-$$\varphi (\xi $$ φ ( ξ ))-expansion methods are employed to resolve the aforementioned issue because of their effectiveness and simplicity. For generality, the time fractional version is studied; more advanced solutions that do not exist in the literature were obtained. As a result, a variety of the exact wave solutions of the conformable (3+1)-dimensional KdV-BBM equation are obtained. The dynamical behaviors of some obtained solutions are represented with the proper parameter values. The used methods yield noteworthy results in obtaining the analytical solutions of fractional differential equations under various conditions. Besides, the sensitivity of regarding dynamical system is assessed to show the numerical stability effects.