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Journal articles on the topic 'N) equation'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Konopelchenko, B. G., and W. K. Schief. "On an integrable multi-dimensionally consistent 2 n + 2 n -dimensional heavenly-type equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2230 (2019): 20190091. http://dx.doi.org/10.1098/rspa.2019.0091.

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Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2 n + 2 n -dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The ‘universal’ character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev–Petviashvili and Hirota equations and various avatars of the heavenly equation governi

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2

G, Janaki, and Gowri Shankari A. "(Exponential Diophantine Equation n2􀀀1 )u +n2v = w2;n = 2;3;4;5." Indian Journal of Science and Technology 17, no. 2 (2024): 166–70. https://doi.org/10.17485/IJST/v17i2.2544.

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Abstract <strong>Objectives:</strong> Diophantine research focuses on various ways to tackle multivariable and multidegree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to a few exponential Diophantine equations and . Also generalize the Exponential equation , and of the form and explore that it has at least one solution as .<strong> Methods:</strong> Diophantine equations may have finite, infinite or no solutions in integers. There is no universal method for finding so

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Obiero, Beatrice Adhiambo, and Kimtai Boaz Simatwo. "On Certain Results On The Diophantine Equation: \(\sum_{{r}={1}}^{n}w^2_r+\frac{n}{3}d^2=3(\frac{nd^2}{3}+\sum^{\frac{n}{3}}_{r=1}w^2_{3r-1})\)." Journal of Advances in Mathematics and Computer Science 40, no. 2 (2025): 1–7. https://doi.org/10.9734/jamcs/2025/v40i21966.

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Consider a sequence wr in arithmetic progression with a common difference d. he exploration of Diophantine equations, which are polynomial equations seeking integer solutions, has been a fascinating endeavor in number theory. These equations have historically intrigued mathematicians due to their inherent complexities and their importance in understanding the properties of integers. In this study, we investigate a Diophantine equation that relates the sum of squares of integers from specific sequences to a variable d. Specifically, we extend existing results on the Diophantine equation: \(\sum

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Mude, Lao Hussein, Kinyanjui Jeremiah Ndung’u, and Zachary Kaunda Kayiita. "On Sums of Squares Involving Integer Sequence: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)." Journal of Advances in Mathematics and Computer Science 39, no. 7 (2024): 1–6. http://dx.doi.org/10.9734/jamcs/2024/v39i71906.

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Let wr be a given integer sequence in arithmetic progression with a common difference d. The study of diophantine equations, which are polynomial equations seeking integer solutions, has been a very interesting journey in the field of number theory. Historically, these equations have attracted the attention of many mathematicians due to their intrinsic challenges and their significance in understanding the properties of integers. In this current study, we examine a diophantine equation relating the sum of squared integers from specific sequences to a variable d: In particular, the diophantine

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IVANOV, E., S. KRIVONOS, and R. P. MALIK. "N = 2 SUPER W3 ALGEBRA AND N = 2 SUPER BOUSSINESQ EQUATIONS." International Journal of Modern Physics A 10, no. 02 (1995): 253–88. http://dx.doi.org/10.1142/s0217751x95000127.

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We study classical N=2 super W3 algebra and its interplay with N=2 supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs-covariant reduction approach. These techniques have been previously used by us in the bosonic W3 case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general N=2 super Boussinesq equation and two kinds of the modified N=2 super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to ce

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Majumdar, AAK, and GC Ray. "On the Equation Z(n) + SL(n) = n." Chittagong University Journal of Science 41, no. 1 (2021): 106–11. http://dx.doi.org/10.3329/cujs.v41i1.51918.

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This paper considers the Diophantine equation Z(n) + SL(n) = n, where Z(n) is the pseudo Smarandache function and SL(n) is the Smarandache LCM function. The Chittagong Univ. J. Sci. 40(1) : 106-111, 2019

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Zayed, EL Sayed M. E., and Abdul-Ghani Al-Nowehy. "Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method." Open Physics 14, no. 1 (2016): 129–39. http://dx.doi.org/10.1515/phys-2016-0013.

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AbstractIn this article, we apply the generalized Kudryashov method for finding exact solutions of three nonlinear partial differential equations (PDEs), namely: the Biswas-Milovic equation with dual-power law nonlinearity; the Zakharov--Kuznetsov equation (ZK(m,n,k)); and the K(m,n) equation with the generalized evolution term. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, and hyperbolic function solutions. Physical explanations for certain solutions of the three nonlinear PDEs are obtained.

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8

Mohamad- Jawad, Anwar. "The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 17, 2021): 120–39. http://dx.doi.org/10.55562/jrucs.v32i2.327.

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The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota e

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9

Bahyrycz, Anna, and Justyna Sikorska. "On Stability of a General n-Linear Functional Equation." Symmetry 15, no. 1 (2022): 19. http://dx.doi.org/10.3390/sym15010019.

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Let X be a linear space over K∈{R,C}, Y be a real or complex Banach space and f:Xn→Y. With some fixed aji,Ci1…in∈K (j∈{1,…,n}, i,ik∈{1,2}, k∈{1,…,n}), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f(a11x11+a12x12,…,an1xn1+an2xn2)=∑1≤i1,…,in≤2Ci1…inf(x1i1,…,xnin), for all xjij∈X (j∈{1,…,n},ij∈{1,2}). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the

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Domoshnitsky, Alexander, and Roman Koplatadze. "On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/168425.

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The following differential equationu(n)(t)+p(t)|u(σ(t))|μ(t) sign u(σ(t))=0is considered. Herep∈Lloc(R+;R+), μ∈C(R+;(0,+∞)), σ∈C(R+;R+), σ(t)≤t, andlimt→+∞⁡σ(t)=+∞. We say that the equation is almost linear if the conditionlimt→+∞⁡μ(t)=1is fulfilled, while iflim⁡supt→+∞⁡μ(t)≠1orlim⁡inft→+∞⁡μ(t)≠1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new

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Schlage-Puchta, J. C. "The equation ω( n ) = ω( n + 1)". Mathematika 50, № 1-2 (2003): 99–101. http://dx.doi.org/10.1112/s0025579300014820.

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Hua, Ni, and Tian Li-Xin. "The Existence of n Periodic Solutions on One Element n-Degree Polynomial Differential Equation." Advances in Mathematical Physics 2020 (March 16, 2020): 1–7. http://dx.doi.org/10.1155/2020/7034591.

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This paper deals with a class of one element n-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.

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13

Wang, Shuyi, and Fanwei Meng. "Ulam Stability of n-th Order Delay Integro-Differential Equations." Mathematics 9, no. 23 (2021): 3029. http://dx.doi.org/10.3390/math9233029.

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In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our

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14

P.N Seetharaman. "An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation." Indian Journal of Advanced Mathematics 5, no. 1 (2025): 27–31. https://doi.org/10.54105/ijam.a1192.05010425.

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Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n, where n is any integer > 2. Around 1637 Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an Auxiliary equation x 3 + y3 = z3 and create transformation equations.

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P.N, Seetharaman. "An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation." Indian Journal of Advanced Mathematics (IJAM) 5, no. 1 (2025): 27–31. https://doi.org/10.54105/ijam.A1192.05010425.

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<strong>Abstract: </strong>Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Around 1637 Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an Auxiliary equation x 3 + y3 = z3

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Seetharaman, Palamadai Narayanan. "A Proof for Fermat’s Last Theorem." European Journal of Mathematics and Statistics 5, no. 1 (2024): 1–6. http://dx.doi.org/10.24018/ejmath.2024.5.1.347.

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Fermat’s proposition during 1637 that the Diophantine equation xn + yn = zn where x, y, z and n are integers has no solution for n > 2 has come to be known as Fermat’s Last Theorem. Taking the proofs of the theorem by Fermat and Euler for the index n = 4 and n = 3, it would suffice to prove the theorem for the index p which is any prime >3. We consider the equation rp + sp = tp to prove the theorem. We take another auxiliary equation x3 + y3 = z3 to substantiate the proof. Both equations have been combined by means of equivalent equations, into which we have employed the Ramanujan-Nagell

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17

Yilmazer, R., and O. Ozturk. "N-Fractional Calculus Operator Method to the Euler Equation." Issues of Analysis 25, no. 2 (2018): 144–52. http://dx.doi.org/10.15393/j3.art.2018.5730.

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18

Bugajewski, Dariuss, and Daria Wójtowicz. "On the equation $x_{ap}^{(n)}=f(t,x)$." Czechoslovak Mathematical Journal 46, no. 2 (1996): 325–30. http://dx.doi.org/10.21136/cmj.1996.127294.

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19

Aggarwal, Sudhanshu. "On the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\)." Engineering and Applied Science Letters 4, no. 1 (2021): 77–79. https://doi.org/10.30538/psrp-easl2021.0064.

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Nowadays, scholars are very interested to determine the solution of different Diophantine equations because these equations have numerous applications in the field of coordinate geometry, cryptography, trigonometry and applied algebra. These equations help us for finding the integer solution of famous Pythagoras theorem and Pell's equation. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. In the present paper, author studied the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n\) are whole numbers, for

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20

Rubina, L. I., and O. N. Ul'yanov. "On solving non-homogeneous partial differential equations with right-hand side defined on the grid." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 3 (2021): 443–57. http://dx.doi.org/10.35634/vm210307.

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An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential e

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21

Bose, A. K. "An integral equation associated with linear homogeneous differential equations." International Journal of Mathematics and Mathematical Sciences 9, no. 2 (1986): 405–8. http://dx.doi.org/10.1155/s0161171286000509.

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Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

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22

Ratnawati, Ratnawati. "Prediction Of Solubility Of Solid N-Paraffins In Supercritical Fluids Using Modified Redlich-Kwong Equation Of State." REAKTOR 8, no. 1 (2017): 1. http://dx.doi.org/10.14710/reaktor.8.1.1-6.

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Three equation of state are used to predict solubilities of solid n-pafaffins in supercritical fluids. The equations are the Redlich-Kwong, the Soave-Redlich-Kwong, and equation proposed by Hartono et.al. (2003; 2004). Both the last two equations were formed by modificating the Redlich-Kwong equqtion of state. With the binary interactions parameter, kif , equals zero, the equations proposed by Hartono et.al. is better than both the Redlich-Kwong and the Soave-Redlich-Kwong equations of state are. Upon optimization with kif as the adjustable parameter, the equation of state proposed by Hartono

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P.N. Seetharaman. "In Search of an Elementary Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 35–39. https://doi.org/10.54105/ijam.a1190.04010424.

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Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connects these two equations by using the transformation equations. On solving the t

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P.N Seetharaman. "An Alternative Elementary Proof for Fermat's Last Theorem." International Journal of Basic Sciences and Applied Computing 11, no. 8 (2025): 11–16. https://doi.org/10.35940/ijbsac.h0534.11080425.

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Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connect these two equations by using transformation equations. On solving the transf

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MAES, KOEN C., and BERNARD DE BAETS. "DIVISIBLE, RECIPROCAL AUTOMORPHISMS OF THE UNIT INTERVAL." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, no. 02 (2009): 221–35. http://dx.doi.org/10.1142/s0218488509005826.

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The Cauchy equation for automorphisms of the unit interval is fulfilled only by the identity mapping. We consider two weakened forms of this Cauchy equation: the reciprocity equation and the n-divisibility equation. Although the solution sets of both functional equations are quite large, requiring that an automorphism is both reciprocal and n-divisible, drastically increases the set of obligatory fixed points. However, increasing n also enlarges the domain on which the automorphism can be chosen freely. We also describe the solution sets of two functional equations that arise by composing the

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WANG, DENG-SHAN. "A CLASS OF SPECIAL EXACT SOLUTIONS OF SOME HIGH DIMENSIONAL NON-LINEAR WAVE EQUATIONS." International Journal of Modern Physics B 24, no. 23 (2010): 4563–79. http://dx.doi.org/10.1142/s0217979210056621.

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In this paper, the separation transformation approach is extended to some high dimensional non-linear wave equations, such as the (N+1)-dimensional Zhiber–Shabat equation, the generalized (N+1)-dimensional complex non-linear Klein–Gordon equation and the generalized (N+1)-dimensional Toda lattice equation. As a result, a class of special exact solutions of these equations are obtained. The solutions obtained contain one or two arbitrary functions which may lead to abundant structures of the high dimensional non-linear wave equations.

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Korhonen, Risto, and Yueyang Zhang. "Existence of Meromorphic Solutions of First-Order Difference Equations." Constructive Approximation 51, no. 3 (2019): 465–504. http://dx.doi.org/10.1007/s00365-019-09491-0.

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AbstractIt is shown that if $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$f(z+1)n=R(z,f),where R(z, f) is rational in f with meromorphic coefficients and $$\deg _f(R(z,f))=n$$degf(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if $$f(z+1)^n=R(z,f)$$f(z+1)n=R(z,f), where the assumption $$\deg _f(R(z,f))=n$$degf(R(z,f))=n has been discarded, has rationa

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Cinar, Cengiz, Ramazan Karatas, and Ibrahim Yalçınkaya. "On solutions of the difference equation $x_{n+1}=x_{n-3}/(-1+x_{n}x_{n-1}x_{n-2}x_{n-3})$." Mathematica Bohemica 132, no. 3 (2007): 257–61. http://dx.doi.org/10.21136/mb.2007.134123.

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Yan, Guo Xin, Wei Wu, and Shi Jiang Zhu. "Deducing Stream Function N-S Equation from Classic N-S Equation and its Verification." Advanced Materials Research 950 (June 2014): 201–4. http://dx.doi.org/10.4028/www.scientific.net/amr.950.201.

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N-S equation is basic equation of fluid flow. Based on N-S equation, it deduced two different forms of governing equation which refer to turbulent-stream function equation and stream function equation. It analyzed their characteristics including advantages and disadvantages. The sudden expansion flow was calculated using turbulent-stream function N-S equation and stream function N-S equation separately. The results showed that both of them can have good effects.

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P.N Seetharaman. "An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R." Indian Journal of Advanced Mathematics 5, no. 1 (2025): 22–26. https://doi.org/10.54105/ijam.a1191.05010425.

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In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence it would suffice to prove the theorem for the index n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + sp = tp where p is any prime >3 and prove the theorem using the method of contradiction. We have used an Auxiliary equations x

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Zhang, Yingnan, Xingbiao Hu, and Jianqing Sun. "Numerical calculation of N -periodic wave solutions to coupled KdV–Toda-type equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2245 (2021): 20200752. http://dx.doi.org/10.1098/rspa.2020.0752.

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In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

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CHOWDHURY, A. ROY, and A. GHOSE CHOUDHURY. "SU(N) LIE ALGEBRA, EXTENDED TODA LATTICE EQUATION AND THE EXACT SOLUTIONS." Modern Physics Letters A 09, no. 06 (1994): 525–34. http://dx.doi.org/10.1142/s0217732394003750.

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An integro-differential generalization of the Toda lattice equation is proposed via the zero curvature equation belonging to SU(N) Lie algebra. It is shown that the exact solutions for this equation can be constructed by the method of chiral vectors. Explicit results are given for SU(2) and SU(3). We also demonstrate that these equations are connected to the constrained WZW theory and hence Polyakov’s two-dimensional gravity.

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P., N. Seetharaman. "In Search of an Elementary Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 35–39. https://doi.org/10.54105/ijam.A1190.04010424.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connects these two equations by using the transform

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34

P.N, Seetharaman. "An Alternative Elementary Proof for Fermat's Last Theorem." International Journal of Basic Sciences and Applied Computing (IJBSAC) 11, no. 8 (2025): 11–16. https://doi.org/10.35940/ijbsac.H0534.11080425.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connect these two equations by using transformation

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Senthil, Gowri, S. Sudharsan, V. Banu Priya, S. Annadurai, G. Ganapathy, and A. Vijayalakshmi. "Hyers-Ulam Stability of N-Dimensional Additive Functional Equation in Modular Spaces Using Fixed Point Method." International Journal of Analysis and Applications 23 (June 26, 2025): 148. https://doi.org/10.28924/2291-8639-23-2025-148.

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The Hyers–Ulam stability of functional equations is a subject of mathematical research that examines the approximate validity of these equations. This notion investigates if a function that nearly fulfills a specified functional equation must be near a precise solution of that equation. Numerous research have investigated this domain, examining the stability of diverse functional equations under varying situations. In this present work, we investigated Hyers-Ulam stability of a n-dimensional additive functional equation in modular spaces using the fixed point approach with the help of Fatou pr

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Elabbasy, E. M., H. El-Metwally, and E. M. Elsayed. "On the difference equation ${ x_{n+1}=\frac{ax_{n}^{2}+bx_{n-1}x_{n-k}}{cx_{n}^{2}+dx_{n-1}x_{n-k}}}$." Sarajevo Journal of Mathematics 4, no. 2 (2024): 239–48. http://dx.doi.org/10.5644/sjm.04.2.09.

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In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence \begin{equation*}x_{n+1}=\frac{ax_{n}^{2}+bx_{n-1}x_{n-k}}{cx_{n}^{2}+dx_{n-1}x_{n-k}},\;\;\;n=0,1,\dots\end{equation*}where the parameters $a,b,c$ and $d$ are positive real numbers and the initial conditions $ x_{-k},x_{-k+1},\dots,x_{-1}$ and $x_{0}$ are arbitrary positive numbers. 2000 Mathematics Subject Classification. 39A10

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Choy, Jaeyoo, Hahng-Yun Chu, and Ahyoung Kim. "A Remark for the Hyers–Ulam Stabilities on n-Banach Spaces." Axioms 10, no. 1 (2020): 2. http://dx.doi.org/10.3390/axioms10010002.

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In this article, we deal with stabilities of several functional equations in n-Banach spaces. For a surjective mapping f into a n-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for f in n-Banach spaces.

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Dong, Fengfeng, and Lingjun Zhou. "Equations with Peakon Solutions in the Negative Order Camassa-Holm Hierarchy." Advances in Mathematical Physics 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/1679625.

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The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders -3 and -4 are obtained. Indeed, for N≤-2, the peakons of N-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for N-order C

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Aggarwal, S., and S. Kumar. "On the Exponential Diophantine Equation (13^2m) + (6r + 1)ⁿ = z²." Journal of Scientific Research 13, no. 3 (2021): 845–49. http://dx.doi.org/10.3329/jsr.v13i3.52611.

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Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have d

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Pyanylo, Yaroslav, and Galyna Pyanylo. "Analysis of approaches to mass-transfer modeling n non-stationary mode." Physico-mathematical modelling and informational technologies, no. 28, 29 (December 27, 2019): 55–64. http://dx.doi.org/10.15407/fmmit2020.28.055.

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A significant number of natural and physical processes are described by differential equations in partial derivatives or systems of differential equations in partial derivatives. Numerical methods have been found to find their solutions. Partial derivatives systems are solved mainly by reducing the order of the system of equations or reducing it to one differential equation. This procedure leads to an increase in the order of the differential equation. There are various restrictions and errors that can lead to additional solutions, boundary conditions for intermediate derivatives, and so on. T

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P. N Seetharaman. "A Comprehensible Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 29–34. https://doi.org/10.54105/ijam.a1181.04010424.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer

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Seetharaman, Palamadai N. "Direct Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation." European Journal of Mathematics and Statistics 3, no. 6 (2022): 1–7. http://dx.doi.org/10.24018/ejmath.2022.3.6.176.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation: An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3 [1]. We hypothesize that all r, s and t are non-zero integers in the equation: rp + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation: x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero in

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43

Seetharaman, Palamadai N. "A Solution for Fermat’s Last Theorem Using Ramanujan-Nagell Equation." European Journal of Mathematics and Statistics 4, no. 6 (2023): 38–45. http://dx.doi.org/10.24018/ejmath.2023.4.6.340.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B, and C satisfying the equation: An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3 [1]. We hypothesize that all r, s and t are non-zero integers in the equation: rp + sp = tp and establish contradiction. Just to support the proof in the above equation, we have another equation: x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 is a non-zero in

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KOBAYASHI, KEN-ICHIRO. "N = 2 SUPERSYMMETRIC SOLITON EQUATION." Modern Physics Letters A 05, no. 30 (1990): 2515–22. http://dx.doi.org/10.1142/s0217732390002924.

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45

Toppan, Francesco. "The N = 2 Heavenly Equation." Czechoslovak Journal of Physics 54, no. 11 (2004): 1387–92. http://dx.doi.org/10.1007/s10582-004-9806-y.

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Delduc, F., and E. Ivanov. "N = 4 super KdV equation." Physics Letters B 309, no. 3-4 (1993): 312–19. http://dx.doi.org/10.1016/0370-2693(93)90939-f.

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P., N. Seetharaman. "A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation." Indian Journal of Advanced Mathematics (IJAM) 4, no. 2 (2024): 19–24. https://doi.org/10.54105/ijam.A1182.04021024.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x 4 + y 4 = z 4 and x 3 + y 3 = z 3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + s p = t p where p is any prime >3 a

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Kinlaw, Paul, Mitsuo Kobayashi, and Carl Pomerance. "On the equation $\varphi (n)=\varphi (n+1)$." Acta Arithmetica 196, no. 1 (2020): 69–92. http://dx.doi.org/10.4064/aa190627-20-1.

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Hohenboken, W. D., A. Dudley, and D. E. Moody. "A comparison among equations to characterize lactation curves in beef cows." Animal Science 55, no. 1 (1992): 23–28. http://dx.doi.org/10.1017/s0003356100037223.

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AbstractMonthly and fortnightly milk production records were analysed from 59 autumn-calving Angus and Angus × Holstein crossbred cows. Half the cows had been administered 10 mg thyroxine per day from day 60 to 120 of lactation and half were controls. Four published equations to characterize individual lactation curves were compared. These were: (1) log Y(n) = log –a1 + b1log n – c1n (Wood); (2) equation 1 with each log Y(n)2 weighted by Yin)2 (Wood weighted); (3) log [Y(n)/n7 = log l/a3 – k3n(Jenkins); and (4) log Y(n) = a4 – b4n‘(l + 25·5 n’) + c4n2 = d 4/ n (Morant), where Y(n) is milk yiel

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50

Gaillard, Pierre. "Rogue Waves in the Nonlinear Schrödinger, Kadomtsev–Petviashvili, Lakshmanan–Porsezian–Daniel and Hirota Equations." Axioms 14, no. 2 (2025): 94. https://doi.org/10.3390/axioms14020094.

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We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order 2N. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appea

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