New exact solutions of nonlinear wave type PDEs with delay

2023

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The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.

“…The studies [168][169][170][171][172] describe a number of exact solutions and reductions of nonlinear wave equations of the form (11) and related hyperbolic PDEs with constant delay.…”

Section: Delay Wave-type Pdesmentioning

confidence: 99%

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

2023

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“…PDEs with two independent variables, x and t, and constant delay generally admit traveling-wave solutions, u = u(z), where z = kx + λt [55,[106][107][108], and do not have selfsimilar solutions, u = t α U(y), where y = xt β . Additive, multiplicative, and generalized separable solutions and more complex solutions of PDEs with constant or varying delays are obtained in [27,[31][32][33]78,79,[91][92][93][94][95]97,[109][110][111][112][113][114][115][116][117] (for a brief overview of publications on exact solutions, see [97,99]). In contrast, PDEs with one proportional delay do not have traveling-wave solutions but can admit self-similar ones.…”

Section: Type Of Equation Form Of Equation References/commentsmentioning

confidence: 99%

Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay

2023

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This study is devoted to reaction–diffusion equations with spatially anisotropic time delay. Reaction–diffusion PDEs with either constant or variable transfer coefficients are considered. Nonlinear equations of a fairly general form containing one, two, or more arbitrary functions and free parameters are analyzed. For the first time, reductions and exact solutions for such complex delay PDEs are constructed. Additive, multiplicative, generalized, and functional separable solutions and some other exact solutions are presented. In addition to reaction–diffusion equations, wave-type PDEs with spatially anisotropic time delay are considered. Overall, more than twenty new exact solutions to reaction–diffusion and wave-type equations with anisotropic time delay are found. The described nonlinear delay PDEs and their solutions can be used to formulate test problems applicable to the verification of approximate analytical and numerical methods for solving complex PDEs with variable delay.

“…More complex than traveling-wave, exact solutions of nonlinear delay reaction-diffusion equations are obtained in [8-17, 23, 24]. Exact solutions of nonlinear delay equations of the Klein Gordon type and other nonlinear hyperbolic equations are given in [18][19][20][21][22][23][24]. Some exact solutions of differential-difference equations of a viscous incompressible fluid (which generalize the Navier Stokes equations) are described in [25].…”

Section: Differential Equations With Constant Delaymentioning

confidence: 99%

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

2021

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We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).