Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
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Oke Davies Adeyemo
Abstract
This research article analytically investigates a soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences and engineering. The soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences along with engineering, is examined with a view to securing various pertinent results of interest. For the first time, the conserved currents of an integrodifferential equation (especially those of higher dimensions) are calculated using a detailed optimal system of one-dimensional subalgebras. Infinitesimal generators of diverse structures ascribed to Lie point symmetries of the understudy model are first calculated via Lie group analysis technique. Additionally, we construct various commutations along Lie-adjoint representation tables connected to the nine-dimensional Lie algebra achieved. Further to that, detailed and comprehensive computation of the optimal system of one-dimensional subalgebras linked to the algebra is also unveiled for the under-investigated model. This, in consequence, engenders the calculation of abundant conserved currents for the soliton equation through Ibragimov’s conserved vector theorem by utilizing its formal Lagrangian. Later, the applications of our results are highlighted.
1 Introduction
Nonlinear equations with dispersive property picture a class of mathematical equations refereed usually to as partial differential equations (PDEs) [1–22]. These PDEs are key in delineating a number of physical models inclusive of waves residing in a shallow water channel, confinement of Bose–Einstein condensate, light propagation in an optical waveguide, and so forth.
Furthermore, it has been observed that these equations simply put are non-solvable explicitly, yet mathematical techniques with analysis, together with computational capabilities, are divulging the means of engendering these models as prognostic tools with a high level of efficiency. This provides a rich phenomenon, for instance, balancing nonlinearity alongside dispersion produces coherent structures like vortices, and solitons – metastable states that are long-lived, found out to be waves localized and propagates with little or no disfigurement. The applicability of such structures can be seen in optical communication, within which solitons are engaged in conveying information. Additionally, these structures also have physical interesting features as a result of their particle-like actions.
In the recent times, investigations have largely been turned to nonlinear partial differential equations (NLNPDEs) as well as exact travelling wave results associated with these NLNPDEs. As a result, diverse complex physical happenstances are depicted via these NLNPDEs. A few of these NLNPDEs including the Boussinesq–Burgers-type system recounting shallow water waves and also emerging near ocean beaches and lakes were given attention in this article [1]. Moreover, Adeyemo et al. [2] examined another generalized NLNPDEs called advection–diffusion equation with power law nonlinearity in fluid mechanics. This generalized equation characterized buoyancy-propelled plume movement embedded in a medium that is bent on nature. Additionally, the vector bright solitons, alongside their various interaction attributes related to the coupled Fokas–Lenells system [3], was studied in the given reference. The femtosecond optical pulses embedded in a double-refractive optical fiber, modeled into an NLNPDEs, were further investigated. Recently, Adeyemo et al. [4] examined a (3+1)-D nonlinear generalized type of potential Yu–Toda–Sasa–Fukuyama model existent in Physics alongside Engineering. Besides, Du et al. [5] investigated the modified as well as generalized Zakharov–Kuznetsov model, delineating the ion-acoustic meandering solitary waves resident in a magneto-plasma and possessive of electron–positron–ion observable in the autochthonous universe. This model was utilized in representing dust-magneto-acoustic, and ion-acoustic, together with dust-ion-acoustic waves in the laboratory dusty plasmas. Further to that, a generalized structure of the Korteweg–de Vries–Zakharov–Kuznetsov model was investigated by Khalique and Adeyemo [6]. The dilution of warm isentropic fluid alongside cold static framework species together with hot isothermal, applicable in fluid dynamics was recounted via the use of the model. The list continues unending, see more in previous studies [4,7–15].
Sophus Lie (1842–1899) with his quintessential work on Lie Algebras [17–20] which is essentially a unified approach for the treatment of a wide class of differential equations (DEs). With the inspiration of Galois theory, Sophus Lie, a Norwegian mathematician, established symmetry methods and demonstrated that many of the known ad hoc methods of integration of DEs could be obtained in a systematic manner. The approach has evolved into a helpful tool for solving DEs, classifying them, and preserving the solution set.
Furthermore, it has been observed that conservation laws are established and entrenched natural laws that have been studied by many researchers in various scientific fields. Conservation laws that are commonly used in this context include conservation of linear momentum in an isolated system, conservation of electric charge, conservation of energy, conservation of mechanical energy in the absence of dissipative forces, and many others. Conservation laws are deliberated to be basic laws of nature, with extensive application in physics and numerous other fields. Some of the important criteria of conservation laws are as follows [21]:
the stability analysis and the global behavior of solutions.
the development of numerical methods and provide an essential starting point for finding potential variables and nonlocally related systems.
the investigation of integrability and linearization mappings.
Securing soliton solutions to NLNPDEs, as a result of its pertinence, is thus becoming a crucial point of interest and active space of investigation to scientists. Consequently, in a bid to gain the soliton solutions, travelling wave solutions, and other interesting exact solutions to NLNPDEs, sturdy approaches have been developed in the literature by scientists. We have some of them as power series solution method [22], simplest equation method [23], Darboux transformation [24], multiple exponential function method [25], just to mention a few. Others include bifurcation technique [26], Painlevé expansion [27], homotopy perturbation technique [28], tanh–coth approach [29], extended homoclinic test approach [30], Cole–Hopf transformation technique [31], Adomian decomposition approach [32], Bäcklund transformation [33], F-expansion technique [34], rational expansion technique [35], extended simplest equation approach [36], Kudryashov’s technique [37], Hirota technique [38], Darboux transformation [39], tanh-function technique [40],
There have emerged NLNPDEs that have been solved by using already arisen mathematical techniques. Nevertheless, not all the emergent modeled equations are solvable. Examination of analytic explicit outcomes to soliton equations is already in the limelight and as such highly significant with influence in physics of mathematics among others [9–11]. Such soliton model includes the 3D soliton-modeled Jimbo–Miwa-type [52] given as
investigated under the variable-dependent-Cole–Hopf transformation that reads
Additionally, mapping (1.1) and Hirota bilinear relation to each other results in
Thus, bilinear differential operators unveiled in (1.2), that is,
where
Furthermore, Asaad [52] engaged the Pfaffian technique to achieve closed-form results to the soliton equation (1.4).
Next, the high dimensions soliton model divulged as
where
Hence, in our study, we seek to explicitly examine the optimal solutions assented to the three-dimensional soliton equation (1.5) via robust Lie group theoretic technique. Consequently, the integral function emergent in (1.5) is first eliminated via the representation
denoted as HD-SOLeqn for short. We observed that recently Khalique and Adeyemo [61,62] bought into play Lie-symmetric approach to gain various abundant invariant solutions of system (1.6). In addition, copious solutions in terms of closed-formed travelling waves of the under-study model via the systematic polynomial complete discriminant alongside elementary integral approaches. Meanwhile, homotopy formula was employed in computing some conserved quantities of (1.6).
Nonetheless, this study invokes the optimal Lie algebraic systematic approach to compute various vectors via a nine-dimensional Lie subalgebras ascribed to (1.6) to gain more extensive conserved quantities of the system with various applications in sciences and engineering. We state for the purpose of emphasis that this research engage a detailed computation of one-dimensional optimal system of Lie subalgebras, obtained from a nine-dimensional Lie algebra, to generate abundant conservation laws to (1.6). Moreover, for the first time, the significance of the associated conserved quantities are highlighted within the fields of physical sciences. All these attest to the fact that the work is novel and original.
Now, we catalog the rest of the research article as follows. Section 2 supplies well-thought-out steps adhered-to in calculating the Lie symmetries ascribed to the soliton equation. Moreover, one parametric transformation groups alongside a Lie-sub-algebraic optimal system is computed for the gained Lie algebra. In addition, Section 3 achieves conserved current calculations for optimal system of Lie subalgebras of solitonic system (1.6) in conjunction with the formal Lagrangian using Ibragimov’s conserved quantities theorem after which conclusions are furnished.
2 Lie algebra and optimal system of the HD-SOLeqn (1.6)
Computations of Lie-point symmetries of HD-SOLeqn (1.6) are first done from where optimal systems of sub-algebras are contrived. In consequence, engagement of the gained symmetries is taken into consideration to attain diverse possible conserved quantities ascribed to system (1.6).
2.1 Conspectus of infinitesimal generators of (1.6)
We suppose first that Lie transformation group of infinitesimal generators be explicated in the following format:
We now contemplate an infinitesimal dimensional Lie algebra covered by vector fields
where coefficient functions
Theorem 2.1
Suppose vector w is assumed to be the infinitesimal generators ascribed to classical point symmetric group of HD-SOLeqn (1.6), where
where constants
See the comprehensive proof of Theorem 2.1 in the study of Khalique and Adeyemo [61].
Suppose one assumes in gained solution (2.3) that arbitrary
Corollary 2.1
Lie algebra represented by
Thus, HD-SOLeqn system (1.6) admits Lie algebra of nine dimensions whose basis are formatted as
One observes quickly here that the instituted infinitesimal generators can be explicated as a linearly combined vectors formatted as
The next phase of the research engenders the Lie transformation-groups connected to Lie-algebra
2.2 One-parameter Lie group transformation of (2.4)
On involving the Lie equations in previous studies [17,63] alongside the related initial conditions in the calculation of one parameteric transformation group ascribed to the gained generators (2.4). Thus, we institute a theorem given shortly:
Theorem 2.2
Suppose that transformation group
with
We direct the reader to the designated references with a view to gaining a much better under-standing of the ascribed proof of Theorem 2.2. Now, by extension, the subsequent theorem suffices, that is:
Theorem 2.3
If
with
2.3 Optimal system of Lie subalgebras of (1.6)
This section tends to utilize the accessible chance of a symmetry group in the computation of one-dimensional Lie optimal system ascribed to Lie subalgebra [17,64] concerning (1.6). Therefore, deciding on subgroups of a symmetry group allows one to gain various kinds of solutions with a well-standardized approach which is stressed. The piece of work involved in sub-algebraic classification that is single-dimensional is a connective factor compared to the signalized one with orbit codification of the Lie adjoint-representation [17]. Now, an subalgebraic optimal set is achieved through the choice of single representative of any given equivalent sub-algebraic class. Additionally, it is possible to decipher the involved issue in orbit classification via the engagement of a Lie algebra general member. Thereafter, simplification is done via diverse transformations of adjoint. So, grounded on a well-known algorithm outlined by Hu et al. [64], we institute an optimal system of one dimension for HD-SOLeqn (1.6). Now, we seek first the principal invariant function associated with (1.6), next, we calculate the transformation matrix and on the final analysis, classification of the Lie algebraic finite dimension [17,64] is computed for (1.6) under consideration.
2.3.1 Principal invariants of System (1.6)
In achieving the one dimensional sub-algebra optimal system of Lie algebra
Lie algebra commutator table for HD-SOLeqn (1.6)
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We observe that Table 1 is skew symmetric having got zero diagonal elements. The real function
with function
Now, for any
Thus, by equating the coefficients of all same powers of
On solving the system of equations displayed in (2.9), one secures the value of invariant
2.3.2 Adjoint transformation matrix of (1.6)
Let
whereby we have
Adjoint representation table of Lie algebra for HD-SOLeqn (1.6)
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Therefore, by adopting the same process, one secures the other eight adjoint transformation matrices as
In consequence, we secure the general adjoint transformation matrix as
where
2.3.3 Adjoint transformation equation and Lie subalgebras of (1.6)
Here, the adjoint transformation equation associated with parameters
where
Remark 2.1
It is noteworthy to state clearly here that the existence of solution of system (2.10) with regards to real constants
Next, we begin the optimal system computations proper in the subsequent part of the study by first contemplating the invariants earlier secured on the basis of its sign as presented in the algorithm adopted [64] in two stages as
Case 1.
We choose in this situation, the representative element
Case 2.
This case occasions three scenarios
Case 2.1.
We select in this subcase, the representative element
Case 2.2.
We choose the representative element
Case 2.3.
Here, we insert
In consequence, we treat this invariants accordingly in the succeeding part of this work by contemplating three basic cases where the invariants are
Case 2.3.1.
In this situation, we have
Case 2.3.2.
Now, just as earlier demonstrated, here, we also have
Case 2.3.3.
In third subcase of Case 2.3., we note that
Case 2.3.4.
This scenario presents
Case 2.3.5.
Here,
Furthermore, for
Remark 2.2
We note that other possible representatives from Case 2.3.4. and Case 2.3.5. have been obtained earlier thereby contributing no additional subalgebra to the optimal system.
Case 2.3.6.
In this situation, we take
Case 2.3.6.1.
Now, we contemplate
Conversely, we consider
Moreover, contemplating
Next, we contemplate the case
Case 2.3.6.2.
Here, we examine the case
Moreover, we study the case when
Case 2.3.6.3.
Reckoning
Besides, for
Considering
In the same vein, for
Case 2.3.6.4.
Regarding the case of
On the contrary, if we reckon
Remark 2.3
It is noteworthy to state that some of the representative elements obtained with both positive and negative signs are equivalent. For instance,
Finally, in view of the detailed calculations and analysis presented alongside remarks (2.2) and (2.3), we reduce the list of the representatives slightly by admitting that the discrete symmetry
Theorem 2.4
The optimal system of one-dimensional Lie subalgebra of HD-SOLeqn (1.6) comprises the list:
We note that various invariant solutions associated with the subalgebras presented in Theorem 2.4 have been copiously explored [61,62]. Thus, we compute the conserved vectors related to the vectors in the subsequent part of the research paper.
3 Conserved currents of system (1.6) with applications
This part of the article reveals the calculation of conserved vectors related to (1.6) by invoking the Ibragimov’s theorem [65,66] for determining conserved quantities. Hence, some salient information are divulged in order to understand the technique.
3.1 Preliminaries
A new theorem in [65] was introduced by Ibragimov for the computations of conserved vectors associated with a given DE. In addition, availability of classical Lagrangian is not demanded for the theorem to function. Ibragimov’s technique suggests fundamentally that infinitesimal generators are uniquely associated with their conserved current. Furthermore, the concept stands on theavailability of adjoint equation related to nonlinear DEs. Thus, a detailed outline of the theorem is furnished shortly.
Formal Lagrangian and adjoint equation
Theorem 3.1
[65] The system of adjoint relations given as
which exists for a known system of
where
owns the symmetries inherited by set of relations (3.2). The complete derivative
Noteworthy, it is to declare that suppose set of relations (3.2) admit a point transformation group, having a generator convey as
then the system of adjoint relations (3.1) admit the operator (3.5) whose extension to a variable
with coefficient
which is selected approximately and the included variable
Hence, the formal Lagrangian is explicated as
with the adjoint relation to (3.2) stated as
together with criteria (3.1) holding.
Theorem 3.2
Every nonlocal symmetry (3.5), Lie–Bäcklund, as well as Lie point, admitted by the system of (3.2) produce a conserved current for relations (3.2) alongside the adjoint (3.10), with the conserved current
where
Remark 3.1
We assert here that a given system of (3.2) is said to be self-adjoint if after replacing
3.2 Derivation of conservation laws via Ibragimov’s theorem
We give the Ibragimov’s conservation theorem [65] in this part of the article to find the conserved vectors of the HD-SOLeqn (1.6). Utilizing the highlighted information earlier presented, one achieves the theorem:
Theorem 3.3
The HD-SOLeqn (1.6) as well as its adjoint equation is accordingly explicated as
with a second-order formal Lagrangian given as
Obviously, one can frankly state that from the adjoint relation (3.13) and remark (3.1), HD-SOLeqn (1.6) is not self-adjoint. By applying of the earlier highlighted facts in Theorem 3.2, we calculate the conserved currents associated with the 12 elements of the achieved optimal system of one-dimensional subalgebras as well as other Lie point symmetries attained for (1.6). These conserved currents are
3.2.1 Physical meaning of some of the obtained conservation laws in physical sciences and engineering
Local conservation laws for the higher dimensional integrodifferential soliton Eq. (1.6) observe a divergence criterion on the whole space
where
as can be obtained in this case, where
Theorem 3.4
The set of 19 non-trivial local conservation laws generated by HD-SOLeqn (1.6) via Ibragimov’s theorem comprise components
Consequently, in the case of HD-SOLeqn (1.6), for example, conserved vectors
for solution
which relates to the corresponding space translation symmetry. In addition, one obtains conserved quantity of dilation energy as
which correspond to
corresponding to
3.3 Application of conserved quantities in physical sciences and engineering
Frankly speaking, conservation law, also referred to as the law of conservation, in physics, depicts a principle that states that a certain physical property (i.e., a measurable quantity) remains unchanged with time within an isolated physical system. Thus, in physical sciences and engineering mathematics, conservation laws state that a particular measurable property associated with an isolated physical system remains constant (that is does not change) as the system evolves over time [70].
Conservation laws [71–73] alternatively, are an area of applicable engrossment in engineering and physics, with the inclusion of theoretical vis-à-vis quantum-mechanics. This part of our research scrutinizes the conserved quantities of model HD-SOLeqn (1.6) with an observable trait that showcases results unveiling the availability of conservation of momentum alongside that of energy. Physical quantities resident in isolated systems, namely, mass, charge, angular momentum, energy, along with linear momentum are conserved. Meanwhile, it is unveiled that conserved quantities invoke an advantageous feature of DE ntegrability check. Further to that, conserved quantities significantly engender the establishment of existence-uniqueness characteristic for linearization mappings and solutions. Besides, they bring to play stability analysis and to ascertain global behavior of solutions.
In addition, conserved quantities play a leading role in the evolution of numerical techniques. They also furnish a crucial starting point in securing non-locally related systems and potential variables. In particular, a conserved quantity is fundamental in the investigation of a given DE, which implies that it holds for any posed data whether initial conditions and/or boundary conditions. Furthermore, the conservation law structure is such that it is not depending on co-ordinates since it involves a contact transformation mapping one to the other. Momentum is simply referred to as the resistance of a given object with regards to an alteration in the object’s velocity. Engineers employ this concept to make lives safer for people through the design of products in lengthening the time over which a deceleration happens.
Moreover, momentum is crucial in Physics due to the fact that it recounts the connection that is existing between mass, speed, as well as direction. Besides, it also delineates the force required to put an object to a halt and to further keep them in motion. It is to be noted that an apparently small object can deploy a large amount of force, provided it possesses sufficient momentum. Figures 1 and 2 present some practical applications of momentum.
![Figure 1
In an experiment recently conducted at Glasgow University, Heisenberg’s uncertainty principle is manifested in which a light beam that initially possesses no orbital angular momentum was observed. We define
Δ
ϕ
\Delta \phi
as the range of azimuthal angular positions appearing for a photon in a cross-sectional part of the beam through a sector aperture. Moreover, at the upstream of the said aperture, the beam is observed to be in an
ℓ
=
0
\ell =0
eigen state of orbital angular momentum (i.e., upper panel). Nonetheless, the uncertainty principle states that the limitation in
ϕ
\phi
causes a spread in orbital angular momentum represented as
L
=
ℓ
ℏ
L=\ell \hslash
for every photon. Additionally, for strait apertures, the connection is portrayed as
Δ
ϕ
Δ
L
≥
ℏ
∕
2
\Delta \phi \Delta L\ge \hslash /2
[74].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_001.jpg)
In an experiment recently conducted at Glasgow University, Heisenberg’s uncertainty principle is manifested in which a light beam that initially possesses no orbital angular momentum was observed. We define
![Figure 2
In the models of atom introduced by Rutherford and Thomson. The later postulated a model that predicted that virtually at small angles, all of the incident alpha particles would be dispersed. Geiger and Rutherford discovered that almost none of the alpha particles were dispersed. However, via a very large angle, a few were found to deflect. Thus, the outcome of the Thomson model was in disagreement with Rutherford’s experiments. Rutherford engaged conservation of energy alongside momentum in the development of a new, and much better model of the atom, and that is the nuclear model [75].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_002.jpg)
In the models of atom introduced by Rutherford and Thomson. The later postulated a model that predicted that virtually at small angles, all of the incident alpha particles would be dispersed. Geiger and Rutherford discovered that almost none of the alpha particles were dispersed. However, via a very large angle, a few were found to deflect. Thus, the outcome of the Thomson model was in disagreement with Rutherford’s experiments. Rutherford engaged conservation of energy alongside momentum in the development of a new, and much better model of the atom, and that is the nuclear model [75].
Consequently, understanding momentum makes it possible for engineers to have an understanding of various kinds of collisions (Figure 3). As a result, having the knowledge can assist in ensuring that cars are much safer, predict the results observed when two objects bang into each other, or investigate the proof of a traffic accident. One typical example is the utilization of air bags in automobiles. Using air bags in automobiles makes it possible for the effect of the force exerted on an object that is involved in a collision to be minimized. Air bags attain this by lengthening the required time in stopping the momentum of the driver as well as passenger. It is to be acknowledged that whether it is achieved or otherwise, situations involving momentum and impulse can be experienced everywhere.
![Figure 3
We have a uni-dimensional inextensible collision in play between two objects. Momentum is conserved, whereas kinetic energy is found not to be conserved. In (a), two objects that are initially having equal mass, move directly toward each other at a uniform speed. In the case of (b), the objects cohere, making way for a perfectly inductile collision. Therefore, the consolidated objects stop and that is the instance revealed in this figure. However, this is untrue for all unyielding collisions [76].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_003.jpg)
We have a uni-dimensional inextensible collision in play between two objects. Momentum is conserved, whereas kinetic energy is found not to be conserved. In (a), two objects that are initially having equal mass, move directly toward each other at a uniform speed. In the case of (b), the objects cohere, making way for a perfectly inductile collision. Therefore, the consolidated objects stop and that is the instance revealed in this figure. However, this is untrue for all unyielding collisions [76].
Mass can be defined as a measure of the quantum of material present in an object, being directly associated with the type and number of atoms that is found in the object. Mass remains constant with regards to the position of the involved body, its movement or alteration of its shape, except material is removed or added.
In engineering science, mass is used to signal the size of something. Mass which is measured in grams or kilograms likewise referred to as drive is a measure of anxiety, which makes it important in human lives.
Moreover, weight and mass are essential in engineering due to the fact that the greater the mass of any given object is, the greater the force required in accomplishing the same change in motion needed. Thus, for a given object, a larger force initiates a larger change in motion. Figure 4 shows a set-up relaying the relationship between force and acceleration. Besides, Figure 5 portrays some connections among force, weight, and mass.
![Figure 4
A graphical depiction of experimental relationship between force and acceleration – Newton’s law [77].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_004.jpg)
A graphical depiction of experimental relationship between force and acceleration – Newton’s law [77].
![Figure 5
A diagrammatic representation of relationship between force, weight, and mass [78].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_005.jpg)
A diagrammatic representation of relationship between force, weight, and mass [78].
Besides, mass is crucial in science as a result of two major factors affecting the movement of things in space: gravity and inertia. The more mass, an object possesses, the more experience it has regarding both properties. That is the reason heavy things (things with large mass) are difficult to move. Next, energy quantity implies energy quantum found in a certain volume of natural gas expressed in kilowatt hour (kWh).
People walk, ride bicycles, move cars along roads as well as boats through water, cook food on stoves, make ice in freezers, light our homes together with offices, manufacture products, and also send astronauts into space using energy. There are various different forms of energy, inclusive of heat, mechanical, electrical, and so on.
Therefore, energy systems are utilized on daily basis by humans to make life easy. Some of these ways include washing clothes, watching television, taking a shower, heating and lighting the home, working from home on desktop computers or laptop, running appliances, cooking, and so on. Residential uses of energy on a global scale account for almost 40% of total energy utilized.
More applications of energy is found in the phosphagen system (Figure 6) that is active during all-out exercise lasting for about 5–10 s including a 100-m dash, diving, lifting a heavy weight, jumping, dashing up a flight of stairs, or any other scheme that engages a maximal, short burst of power [79].
![Figure 6
A diagrammatic representation revealing the Phosphagen system [80].](/document/doi/10.1515/phys-2023-0155/asset/graphic/j_phys-2023-0155_fig_006.jpg)
A diagrammatic representation revealing the Phosphagen system [80].
4 Conclusion
In this article, we clearly purveyed a conspectus investigation carried out on the HD-SOLeqn (1.6). We engage the universal technique, namely, Lie group analysis to which when engaged to solve a DE, it occasions the methodical procedures of generating Lie point symmetries of such equation. In the study, a demonstration of the robust usefulness of the aforementioned technique assisted us in performing a detailed and comprehensive construction of a one-dimensional optimal system of the Lie subalgebras for the nine-dimensional Lie algebra obtained for the equation, which affords us the chance to obtain various more general and robust combinations of the symmetries. Moreover, owing to the relevance of conserved quantities in the study of DEs, we attained diverse associated conservation laws to the HD-SOLeqn using the Lie subalgebras, where various quantities such as conservation of energy are derived. In consequence, this study clearly highlighted the importance of soliton solutions of higher-dimensional NLNPDEs in physics and engineering mathematics and the robust application of the Lie group theory of DEs in proffering solutions to them. Therefore, this research can be beneficial in various fields and in particular in the research area of physical sciences and engineering. Particularly, in an area where further analysis of the result could be of immense usefulness.
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Funding information: The authors state no funding involved.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
References
[1] Gao XY, Guo YJ, Shan WR. Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq–Burgers system, auto-and non-auto-Bäcklund transformations. Appl Math Lett. 2020;104:106170. 10.1016/j.aml.2019.106170Search in Google Scholar
[2] Adeyemo OD, Motsepa T, Khalique CM. A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences. Alex Eng J. 2022;61:185–94. 10.1016/j.aej.2021.04.066Search in Google Scholar
[3] Zhang CR, Tian B, Qu QX, Liu L, Tian HY. Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber. Z Angew Math Phys. 2020;71:1–19. 10.1007/s00033-019-1225-9Search in Google Scholar
[4] Adeyemo OD, Khalique CM, Gasimov YS, Villecco F. Variational and non-variational approaches with Lie algebra of a generalized (3+1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation in engineering and physics. Alex Eng J. 2023;63:17–43. 10.1016/j.aej.2022.07.024Search in Google Scholar
[5] Du XX, Tian B, Qu QX, Yuan YQ, Zhao XH. Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma. Chaos Solitons Fract. 2020;134:109709. 10.1016/j.chaos.2020.109709Search in Google Scholar
[6] Khalique CM, Adeyemo OD. A study of (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation via Lie symmetry approach. Results Phys. 2020;18:103197. 10.1016/j.rinp.2020.103197Search in Google Scholar
[7] Younis M. Optical solitons in (n+1) dimensions with Kerr and power law nonlinearities. Mod Phys Lett B. 2017;31:1750186. 10.1142/S021798491750186XSearch in Google Scholar
[8] Liu MM, Yu JP, Ma WX, Khalique CM, Sun YL. Dynamic analysis of lump solutions based on the dimensionally reduced generalized Hirota bilinear KP-Boussinesq equation. Mod Phys Lett B. 2023;37:2250203. 10.1142/S0217984922502037Search in Google Scholar
[9] Bilal M, Seadawy AR, Younis M. Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin-Gottwald-Holm system and modulation instability analysis. Math Methods Appl Sci. 2021;44:4094–104. 10.1002/mma.7013Search in Google Scholar
[10] Simbanefayi I, Gandarias ML, Khalique CM. Travelling wave solutions, symmetry reductions and conserved vectors of a generalized hyper-elastic rod wave equation. Partial Differ Equ Appl Math. 2023;7:10050110.1016/j.padiff.2023.100501Search in Google Scholar
[11] Younis M, Ali S, Rizvi STR, Tantawy M, Tariq KU. Investigation of solitons and mixed lump wave solutions with (3+1)-dimensional potential-YTSF equation. Commun Nonlinear Sci Numer Simul. 2021;94:105544. 10.1016/j.cnsns.2020.105544Search in Google Scholar
[12] Benoudina N, Zhang Y, Khalique CM. Lie symmetry analysis, optimal system, new solitary wave solutions and conservation laws of the Pavlov equation. Commun Nonlinear Sci Numer Simulat. 2021;94:105560. 10.1016/j.cnsns.2020.105560Search in Google Scholar
[13] Kudryashov NA. Analytical theory of nonlinear differential equations. Moskow - Igevsk: Institute of Computer Investigations; 2004. Search in Google Scholar
[14] Bluman GW, Cheviakov AF, Anco SC. Applications of symmetry methods to partial differential equations. New York: Springer; 2010. 10.1007/978-0-387-68028-6Search in Google Scholar
[15] Kumar S, Kumar D, Kumar A. Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation. Chaos Solit Fractals. 2021;142:110507. 10.1016/j.chaos.2020.110507Search in Google Scholar
[16] Kumar S, Kumar D, Wazwaz AM. Group invariant solutions of (3+1)-dimensional generalized B-type Kadomstsev Petviashvili equation using optimal system of Lie subalgebra. Phys Scr. 2019;94:065204. 10.1088/1402-4896/aafc13Search in Google Scholar
[17] Olver PJ. Applications of Lie groups to differential equations. In: Graduate Texts in Mathematics. Vol. 107. 2nd edition. Berlin: Springer-Verlag; 1993. 10.1007/978-1-4612-4350-2Search in Google Scholar
[18] Ovsiannikov LV. Group analysis of differential equations. New York, USA: Academic Press; 1982. 10.1016/B978-0-12-531680-4.50012-5Search in Google Scholar
[19] Zhang L, Kwizera S, Khalique CM. A study of a new generalized Burgers’ equation: symmetry solutions and conservation laws. Adv Math Models Appl. 2023;8:2. Search in Google Scholar
[20] Adeyemo OD, Zhang L, Khalique CM. Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+1)-dimensional generalized Zakharov-Kuznetsov equation type I. Eur Phys J Plus. 2022;137:95410.1140/epjp/s13360-022-03100-zSearch in Google Scholar
[21] Anco SC, Bluman GW. Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Eur J Appl Math. 2002;13:545–66. 10.1017/S095679250100465XSearch in Google Scholar
[22] Feng L, Tian S, Zhang T, Zhou J. Lie symmetries, conservation laws and analytical solutions for two-component integrable equations. Chinese J Phys. 2017;55:996–1010. 10.1016/j.cjph.2017.03.008Search in Google Scholar
[23] Yu J, Wang D, Sun Y, S, Wu. Modified method of simplest equation for obtaining exact solutions of the Zakharov-Kuznetsov equation, the modified Zakharov-Kuznetsov equation, and their generalized forms. Nonlinear Dyn. 2016;85:2449–65. 10.1007/s11071-016-2837-7Search in Google Scholar
[24] Zhang Y, Ye R, Ma WX. Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations. Math Meth Appl Sci. 2020;43:613–27. 10.1002/mma.5914Search in Google Scholar
[25] Adem AR, Yildrim Y, Sar EY. Complexiton solutions and soliton solutions: (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Pramana. 2019;92:1–12. 10.1007/s12043-018-1707-xSearch in Google Scholar
[26] Zhang L, Khalique CM. Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs. Discrete and Continuous dynamical systems Series S. 2018;11:777–90. 10.3934/dcdss.2018048Search in Google Scholar
[27] Weiss J, Tabor M, Carnevale G. The Painlevé property and a partial differential equations with an essential singularity. Phys Lett A. 1985;109:205–8. 10.1016/0375-9601(85)90303-2Search in Google Scholar
[28] Chun C, Sakthivel R. Homotopy perturbation technique for solving two point boundary value problems-comparison with other methods. Comput Phys Commun. 2010;181:1021–4. 10.1016/j.cpc.2010.02.007Search in Google Scholar
[29] Wazwaz AM. Traveling wave solution to (2+1)-dimensional nonlinear evolution equations. J Nat Sci Math. 2007;1:1–13. Search in Google Scholar
[30] Darvishi MT, Najafi M. A modification of extended homoclinic test approach to solve the (3+1)-dimensional potential-YTSF equation. Chin Phys Lett. 2011;28:040202. 10.1088/0256-307X/28/4/040202Search in Google Scholar
[31] Salas AH, Gomez CA. Application of the Cole–Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Math Probl Eng. 2010;2010. 10.1155/2010/194329Search in Google Scholar
[32] Wazwaz AM. Partial differential equations. Boca Raton, Florida, USA: CRC Press; 2002. Search in Google Scholar
[33] Gu CH. Soliton theory and its application. Zhejiang, China: Zhejiang Science and Technology Press; 1990. Search in Google Scholar
[34] Zhou Y, Wang M, Wang Y. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys Lett A. 2003;308:31–6. 10.1016/S0375-9601(02)01775-9Search in Google Scholar
[35] Zeng X, Wang DS. A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation. Appl Math Comput. 2009;212:296–304. 10.1016/j.amc.2009.02.020Search in Google Scholar
[36] Kudryashov NA, Loguinova NB, Extended simplest equation method for nonlinear differential equations. Appl Math Comput. 2008;205:396–402. 10.1016/j.amc.2008.08.019Search in Google Scholar
[37] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fract. 2005;24:1217–3110.1016/j.chaos.2004.09.109Search in Google Scholar
[38] Hirota R. The direct method in soliton theory. Cambridge, UK: Cambridge University Press; 2004. 10.1017/CBO9780511543043Search in Google Scholar
[39] Matveev VB, Salle MA. Darboux transformations and solitons. New York, USA: Springer; 1991. 10.1007/978-3-662-00922-2Search in Google Scholar
[40] Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Appl Math Comput. 2005;169:321–38. 10.1016/j.amc.2004.09.054Search in Google Scholar
[41] Wang M, Li X, Zhang J. The (G′∕G)-expansion method and travelling wave solutions for linear evolution equations in mathematical physics. Phys Lett A. 2005;24:1257–68. Search in Google Scholar
[42] Chen Y, Z Yan. New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method. Chaos Solitons Fract. 2005;26:399–406. 10.1016/j.chaos.2005.01.004Search in Google Scholar
[43] Osman MS. One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation. Nonlinear Dynam. 2019;96:1491–6. 10.1007/s11071-019-04866-1Search in Google Scholar
[44] He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos Solitons Fract. 2006;30:700–8. 10.1016/j.chaos.2006.03.020Search in Google Scholar
[45] Date Y, Jimbo M, Kashiwara M, Miwa T. Operator approach of the Kadomtsev-Petviashvili equation - Transformation groups for soliton equations III. JPSJ. 1981;50:3806–12. 10.1143/JPSJ.50.3806Search in Google Scholar
[46] Kuo CK, Ma WX. An effective approach to constructing novel KP-like equations. Waves random complex media. 2020;32:629–40. 10.1080/17455030.2020.1792580Search in Google Scholar
[47] Ma WX, Fan E. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl. 2011;61:950–9. 10.1016/j.camwa.2010.12.043Search in Google Scholar
[48] Wazwaz AM. Multiple-soliton solutions for a (3+1)-dimensional generalized KP equation. Commun Nonlinear Sci Numer Simul. 2012;17:491–5. 10.1016/j.cnsns.2011.05.025Search in Google Scholar
[49] Ma WX. Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A. 2015;379:1975–8. 10.1016/j.physleta.2015.06.061Search in Google Scholar
[50] Zhao Z, Han B. Lump solutions of a (3+1)-dimensional B-type KP equation and its dimensionally reduced equations. Anal Math Phys. 2017;9:119–30. 10.1007/s13324-017-0185-5Search in Google Scholar
[51] Simbanefayi I, Khalique CM. Group invariant solutions and conserved quantities of a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Mathematics. 2020;8:1012. 10.3390/math8061012Search in Google Scholar
[52] Asaad MG. Soliton solution in (3+1)-dimensions. The 2015 International Academic Research Conference. September University of Nevada, USA. 2015. Vol. 59. p. 15–18. Search in Google Scholar
[53] Geng X. Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J Phys Math Gen. 2003;36:2289. 10.1088/0305-4470/36/9/307Search in Google Scholar
[54] Liu J, Zhang Y. Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation. Results Phys. 2018;10:94–8. 10.1016/j.rinp.2018.05.022Search in Google Scholar
[55] Liu J, Wu P, Zhang Y. New periodic wave solutions of (3+1)-dimensional soliton equation. Therm Sci. 2017;21:169–76. 10.2298/TSCI17S1169LSearch in Google Scholar
[56] Geng X, Ma Y. N-soliton solution and its Wronskian form of a (3+1)-dimensional nonlinear evolution equation. Phys Lett A. 2007;369:285–9. 10.1016/j.physleta.2007.04.099Search in Google Scholar
[57] Jian-Ping W, Xian-Guo G. Grammian determinant solution and Pfaffianization for a (3+1)-dimensional soliton equation. Commun Theor Phys. 2009;52:791. 10.1088/0253-6102/52/5/05Search in Google Scholar
[58] Jian-Ping W. A bilinear Bäcklund transformation and explicit solutions for a (3+1)-dimensional soliton equation. Chinese Phys Lett. 2008;25:4192. 10.1088/0256-307X/25/12/002Search in Google Scholar
[59] Wang X, Wei J, Geng X. Rational solutions for a (3+1)-dimensional nonlinear evolution equation. Commun Nonlinear Sci Numer Simul. 2020;83:105116. 10.1016/j.cnsns.2019.105116Search in Google Scholar
[60] Wang X, Wei J. Antidark solitons and soliton molecules in a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 2020;102:363–77. 10.1007/s11071-020-05926-7Search in Google Scholar
[61] Khalique CM, Adeyemo OD. Soliton solutions, travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics. Commun Theor Phys. 2021;73:125003. 10.1088/1572-9494/ac27a1Search in Google Scholar
[62] Adeyemo OD, Khalique CM, Dynamical soliton wave structures of one-dimensional Lie subalgebras via group-invariant solutions of a higher-dimensional soliton equation in ocean physics and mechatronics engineering, Commun Appl Math Comput. 2022;4:1531–82. 10.1007/s42967-022-00195-0Search in Google Scholar
[63] Ibragimov NH, CRC handbook of lie group analysis of differential equations. Vol. 1–3. Boca Raton, Florida: CRC Press; 1994–1996. Search in Google Scholar
[64] Hu X, Li Y, Chen Y. A direct algorithm of one-dimensional optimal system for the group invariant solutions. J Math Phys. 2015;56:053504. 10.1063/1.4921229Search in Google Scholar
[65] Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007;333:311–28. 10.1016/j.jmaa.2006.10.078Search in Google Scholar
[66] Ibragimov NH. Integrating factors, adjoint equations and Lagrangians. J Math Anal Appl. 2006;318:742–57. 10.1016/j.jmaa.2005.11.012Search in Google Scholar
[67] Márquez AP, Bruzón MS. Lie point symmetries, traveling wave solutions and conservation laws of a non-linear viscoelastic wave equation. Mathematics. 2021;9:2131. 10.3390/math9172131Search in Google Scholar
[68] Adeyemo OD, Khalique CM. Lie group theory, stability analysis with dispersion property, new soliton solutions and conserved quantities of 3D generalized nonlinear wave equation in liquid containing gas bubbles with applications in mechanics of fluids, biomedical sciences and cell biology, Commun Nonlinear Sci Numer Simul. 2023;123:107261. 10.1016/j.cnsns.2023.107261Search in Google Scholar
[69] https://www.britannica.com/science/conservation-law. Search in Google Scholar
[70] https://en.wikipedia.org/wiki/Conservationlaw. Search in Google Scholar
[71] Khalique CM, Adeyemo OD. Closed-form solutions and conserved vectors of a generalized (3. 1)-dimensional breaking soliton equation of engineering and nonlinear science. Mathematics. 2020;8:1692. 10.3390/math8101692Search in Google Scholar
[72] Anco SC. Symmetry properties of conservation laws. Int J Modern Phys B. 2016;30:1640003. 10.1142/S0217979216400038Search in Google Scholar
[73] Anco SC, Kara A. Symmetry invariance of conservation laws. Euro J Appl Math. 2018;29:78–117. 10.1017/S0956792517000055Search in Google Scholar
[74] Physics today: Light’s Orbital Angular Momentum. [Online]. Available: https://physicstoday.scitation.org/doi/10.1063/1.1768672. Search in Google Scholar
[75] University Physics : Types of Collisions. [Online]. Available: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/9-4-types-of-collisions/. Search in Google Scholar
[76] Elastic and Inelastic Collisions. [Online]. Available: https://www.texasgateway.org/resource/83-elastic-and-inelastic-collisions. Search in Google Scholar
[77] Investigate the Relationship Between Force and Acceleration. [Online]. Available. https://classnotes.gidemy.com/topic/investigate-the-relationship-between-force-and-acceleration/. Search in Google Scholar
[78] Understanding the Relationship Between Mass and Weight. [Online]. Available: https://www.arborsci.com/blogs/cool/understanding-the-relationship-between-mass-and-weight. Search in Google Scholar
[79] https://fittrakker.com/scale-of-energy-pools-used-in-energy-physiology-zones/. Search in Google Scholar
[80] Kiika D. “The Basics of Energy Production”: The Phosphagen System. [Online]. Available: https://thesportsedu.com/the-phosphagen-system/. [ Accessed 27 March 2023]. Search in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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- The kinetic relativity theory – hiding in plain sight
- Special Issue on Advanced Energy Materials - Part III
- Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
Articles in the same Issue
- Regular Articles
- Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
- Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
- High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
- Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
- Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
- Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
- The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
- Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
- A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
- Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
- Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
- Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
- Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
- Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
- Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
- Chaotic control problem of BEC system based on Hartree–Fock mean field theory
- Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
- Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
- Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
- Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
- Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
- Model of conversion of flow from confined to unconfined aquifers with stochastic approach
- Study of fractional variable-order lymphatic filariasis infection model
- Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
- Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
- Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
- Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
- Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
- Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
- Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
- A two-stage framework for predicting the remaining useful life of bearings
- Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
- Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
- Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
- In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
- Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
- Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
- A comparative analysis of shielding effectiveness in glass and concrete containers
- Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
- Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
- Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
- Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
- Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
- Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
- Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
- Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
- Fractional view analytical analysis of generalized regularized long wave equation
- Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
- Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
- Estimation of monotone α-quantile of past lifetime function with application
- Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
- Study of fractional telegraph equation via Shehu homotopy perturbation method
- An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
- Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
- An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
- CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
- Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
- A short report on a probability-based interpretation of quantum mechanics
- Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
- Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
- Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
- Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
- A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
- Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
- Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
- Thermal analysis of extended surfaces using deep neural networks
- Steady-state thermodynamic process in multilayered heterogeneous cylinder
- Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
- Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
- Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
- The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
- Research on optimization of combustor liner structure based on arc-shaped slot hole
- Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
- Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
- Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
- Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
- Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
- Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
- A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
- New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
- Physical aspects of quantile residual lifetime sequence
- Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
- Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
- A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
- Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
- A novel tetra hybrid bio-nanofluid model with stenosed artery
- Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
- Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
- A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
- New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
- Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
- Inflation and the principle of equivalence
- The use of a supercontinuum light source for the characterization of passive fiber optic components
- Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
- A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
- Photon balance in the fiber laser model
- Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
- Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
- Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
- Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
- Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
- On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
- Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
- Review Article
- Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
- Special Issue on Predicting pattern alterations in nature - Part II
- A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
- On the existence and numerical simulation of Cholera epidemic model
- Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
- Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
- Prediction of COVID-19 spread with models in different patterns: A case study of Russia
- Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
- Accumulation process in the environment for a generalized mass transport system
- Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
- Special Issue on Nanomaterial utilization and structural optimization - Part II
- Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
- Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
- Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
- Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
- Special Issue on Nanofluids: Synthesis, Characterization, and Applications
- Exact solutions of a class of generalized nanofluidic models
- Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
- Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
- Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
- Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
- Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
- Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
- Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
- Comparative study of couple stress fluid flow using OHAM and NIM
- Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
- Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
- Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
- Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
- Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
- Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
- Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
- Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
- Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
- Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
- Pulsed excitation of a quantum oscillator: A model accounting for damping
- Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
- Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
- Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
- Effect of the gravitational field strength on the rate of chemical reactions
- The kinetic relativity theory – hiding in plain sight
- Special Issue on Advanced Energy Materials - Part III
- Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
